H.S.V. de Snoo & H.L. Wietsma (Eds.) Contributions to Mathematics and Statistics Essays in honor of Seppo Hassi  ACTA WASAENSIA 462 H.S.V. de Snoo Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence University of Groningen The Netherlands h.s.v.de.snoo@rug.nl H.L. Wietsma School of Technology and Innovation Department of Mathematics and Statistics University of Vaasa Finland rwietsma@uwasa.fi Publisher University of Vaasa School of Technology and Innovation P.O. Box 700 FI-65101 Vaasa Finland ISBN 978-952-476-959-4 (print) 978-952-476-960-0 (online) http://urn.fi/URN:ISBN:978-952-476-960-0 ISSN 0355-2667 (Acta Wasaensia 462, print) 2323-9123 (Acta Wasaensia 462, online) SEPPO HASSI VFOREWORD With this Festschrift we celebrate the sixtieth birthday of our friend and colleague Professor Seppo Hassi of the University of Vaasa. It consists of papers written by colleagues outside Vaasa, who have been coauthors of Seppo, as well as by colleagues from Vaasa. Although many friends and colleagues have known and worked with Seppo for a long time, quite a few people answered in disbelief "What? Seppo 60?" when first approached about this project. This collection of essays shows our appreciation of Seppo as a friend and as a colleague. From early on, his main activities have been in the branches of mathematics, known as operator theory and spectral theory, although his interests are much broader. Almost all of the included essays reflect these interests. Unfortunately, due to the consequences of the global pandemic some contributions could not be submitted in time to be part of our collection. It is our pleasure to thank all the authors, both for contributing their work to this volume and for their readiness to respond to our questions and suggestions. Furthermore, we are grateful to Heinz Langer, Kenneth Nordström, Seppo Pynnönen, and Franek Szafraniec for answering our queries concerning several points about the past and present of the person to whom this volume is dedicated. Finally, our thanks go to the staff at the University of Vaasa, in particular to Riikka Kalmi, for their efficient production of this collection. Groningen and Vaasa, April 2021 Henk de Snoo and Rudi Wietsma VII SEPPO HASSI, 60 YEARS Seppo Ortamo Hassi was born on July 2, 1961, in Hyvinkää in southern Finland. He re- ceived his secondary school education in Pori, located on the western coast of Finland, and in 1980 he went to the University of Helsinki to be a student in mathematics. There Seppo obtained his master’s degree in 1984. He would stay at the university and eventually in 1985 became an assistant in the Department of Statistics (which was located at Yliopistonkatu, while the Department of Mathematics was located at Alexanterinkatu, a geographical gap). The leading people in the statistics department were Hannu Niemi (a student of Louhivaara, whom we will meet below) and Seppo Mustonen. Mustonen somehow awakened Seppo’s interest in singular values and canonical representations of operators. This eventually led to the dissertation A singular value decomposition of matrices in a space with an indefi- nite scalar product, with Ilppo Simo Louhivaara (1927 - 2008)† as adviser. This thesis in mathematics was approved by the University of Helsinki on January 31, 1991, at the time that Seppo served in the Finnish army (between August 1990 and April 1991). The oppo- nent at the defence was Heinz Langer (originally from Dresden); Langer had first visited Louhivaara in Jyväskylä in 1969 and had been a frequent guest ever since. Prior to finishing his dissertation, Seppo had been invited to participate in the Schur Analysis meeting (October 16 - October 20, 1989) at the Karl Marx Universität in Leipzig, Deutsche Demokratische Republik, organized by Bernd Kirstein and Bernd Fritzsche. This seminar brought together many people from East and West. It took place in the middle of the peace- ful protests against the communist regime that had been going on in Leipzig for some time. Loudspeakers in empty streets would advise the public not to follow the protesting crowds: "They are misguided." On November 9, shortly after the conference, the Berlin wall came down. At the beginning of the conference it turned out that Heinz Langer had left the country and at its closing it was announced that the great mathematician Mark Grigorievich Kreı˘n (1907-1989) had died. One of the people present from the East was Yury L’vovich Smul’yan (1927-1990), whose work played an important role in Seppo’s dissertation and in his later articles. With the dissertation completed, Seppo started some joint work with his colleague Ken- neth Nordström, who was also an assistant in the Department of Statistics. Their interest focussed on antitonicity properties of operators and projections in indefinite inner prod- uct spaces. In the meantime Heinz Langer had obtained a professorship at the Technische Universität Wien in 1991. He invited Seppo to spend some weeks in Vienna in 1992 at the same time that Henk de Snoo from Groningen was also visiting. During that period Langer’s Dutch and Finnish visitors started to work together, which led to many mutual †Louhivaara had been one of the many students of Rolf Herman Nevanlinna (1895-1980). He was also interested in extension theory and indefinite metrics, like his contemporary fellow students Yrjö Kilpi (1924-2010) and Erkki Pesonen (1930-2006). Louhivaara had been a professor of mathematics at the universities in Helsinki and Jyväskylä, before moving to the Freie Universität Berlin. VIII visits to Holland and Finland over the years, up till the present day. It was during a num- ber of subsequent conferences in or visits to Vienna, Pula, Timisoara, Warsaw, Krakow, and Budapest that it was possible to meet old and new acquaintances and lay foundations for future work. It is appropriate to mention in this context Michael Kaltenbäck, Harald Woracek, Henrik Winkler, Andreas Fleige, Franek Szafraniec, Zoltán Sebestyén (thanks to Jan Stochel), Jean-Philippe Labrousse, and last, but not least, Yury Arlinskiı˘, Vladimir Derkach, and Mark Malamud. A sabbatical visit to Groningen and Berlin in the academic year 2000-2001 made it possible to meet the group around Karl-Heinz Förster of the Tech- nische Universität Berlin, which consisted of Peter Jonas and Peter’s students Carsten Trunk and Jussi Behrndt. Peter Jonas was from East Berlin and had come to the Technische Uni- versität via Ilppo Simo Louhivaara at the Freie Universität. Seppo’s visit led to fruitful contacts; also the later December conferences in Berlin were very productive. Seppo would remain at the Department of Statistics in Helsinki until 2001; in the mean- time he had been formally named docent at the Department of Mathematics of the same university. In November-December 2000 there had been a longer visit to Manfred Möller at the University of Witwatersrand in South Africa and it was there that Seppo found out that the University of Vaasa was interested in his person. He obtained a professorship at that university in the spring of 2001. Seppo settled down in Vaasa during the summer and took up the usual teaching and administrative tasks. In the following years the number of coworkers increased with, for instance, Annemarie Luger, Adrian Sandovici, Sergey Belyi, Eduard Tsekanovskiı˘, and Sergii Kuz˙el. As a consequence there has been a regular stream of visitors (all of whom think with a certain melancholy of the old wooden guestrooms of the University of Vaasa). In May 2003 Seppo was the organizer of an Operator Theory Symposium and, a little later, in 2005 he was one of the organizers of the Algorithmic Infor- mation Theory Conference, see Acta Wasaensia 124, 2005. Moreover, Seppo was one of the organizers of the conferences "Boundary relations" and "Operator realizations of analytic functions" at the Lorentz Center in Leiden in 2009 and 2013, respectively. The main mathematical interest of Seppo circles around the topics of spectral theory, bound- ary value problems for differential equations, operator theory and its applications in analy- sis, mathematical physics, and system theory. This keeps him going with great dedication. In particular, right from the beginning Seppo looked into situations involving indefinite in- ner product spaces and this interest also led to several doctoral students, Rudi Wietsma, Dmytro Baidiuk, and Lassi Lilleberg, writing a dissertation on this topic under his direc- tion. Being a rather prolific writer himself, he is furthermore an editor for a number of mathematical journals. When Seppo first arrived in Vaasa he belonged to the Department of Mathematics and Statis- tics. As the century progresses, so does the university. Seppo now belongs to the School of Technology and Innovations, where he is the leader of the Mathematics and Statistics Re- search Group. He is also head of the Doctoral Programme in Technical Sciences. Moreover, there are duties beyond Vaasa. For many years Seppo has been involved with the nationwide IX entrance exam for Technical Sciences and Architecture studies of the member universities in Finland. And then there is the Academy of Finland: for some years now Seppo has been a member of its Research Council for Natural Sciences and Engineering, and a member of its steering group. All these things coming his way are done with his usual attention to detail. Those who deal with Seppo, either as colleagues or as students, know that he provides a listening ear and is ready to help whenever needed. And those who are fortunate enough to work with him recognize his quiet determination. Uninterrupted, he can sit behind his desk for hours, like a sphinx – lost in thought (so we assume). But when he returns back to real life, you know that something is going to happen. On behalf of all his many friends, whether in Vaasa or elsewhere in the world, we congratu- late Seppo on reaching his sixtieth birthday and we wish him, together with his wife Merja and their son Leo, good health and happiness. May there be many more productive years to come! XI CONTENTS FOREWORD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V SEPPO HASSI, 60 YEARS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII CONGRUENCE OF SELFADJOINT OPERATORS AND TRANSFORMATIONS OF OPERATOR-VALUED NEVANLINNA FUNCTIONS Yury Arlinskiı˘ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 A CLASS OF SINGULAR PERTURBATIONS OF THE DIRAC OPERATOR: BOUNDARY TRIPLETS AND WEYL FUNCTIONS Jussi Behrndt, Markus Holzmann, Christian Stelzer, and Georg Stenzel . . . . . . . 15 THE ORIGINAL WEYL-TITCHMARSH FUNCTIONS AND SECTORIAL SCHRÖDINGER L-SYSTEMS Sergey Belyi and Eduard Tsekanovskiı˘ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 PT -SYMMETRIC HAMILTONIANS AS COUPLINGS OF DUAL PAIRS Volodymyr Derkach, Philipp Schmitz, and Carsten Trunk . . . . . . . . . . . . . . . . . . . . . . 55 POSITIVE AND NEGATIVE EXAMPLES FOR THE RIESZ BASIS PROPERTY OF INDEFINITE STURM-LIOUVILLE PROBLEMS Andreas Fleige . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ON PARSEVAL J-FRAMES Alan Kamuda and Sergii Kuz˙el . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 IDEMPOTENT RELATIONS, SEMI-PROJECTIONS, AND GENERALIZED INVERSES Jean-Philippe Labrousse, Adrian Sandovici, Henk de Snoo, and Henrik Winkler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 LIPSCHITZ PROPERTY OF EIGENVALUES AND EIGENVECTORS OF 22 DIRAC-TYPE OPERATORS Anton Lunyov and Mark Malamud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 XII COMPLETENESS AND MINIMALITY OF EIGENFUNCTIONS AND ASSOCIATED FUNCTIONS OF ORDINARY DIFFERENTIAL OPERATORS Manfred Möller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141 PARTIALLY OVERLAPPING EVENT WINDOWS AND TESTING CUMULATIVE ABNORMAL RETURNS IN FINANCIAL EVENT STUDIES Seppo Pynnönen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .153 ON THE KREI˘N-VON NEUMANN AND FRIEDRICHS EXTENSION OF POSITIVE OPERATORS Zoltán Sebestyén and Zsigmond Tarcsay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 THE CHARACTERIZATION OF BROWNIAN MOTION AS AN ISOTROPIC I.I.D.-COMPONENT LÉVY PROCESS Tommi Sottinen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 THE ROLE OF MATHEMATICS AND STATISTICS IN THE UNIVERSITY OF VAASA; THE FIRST FIVE DECADES Ilkka Virtanen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 CONGRUENCE OF SELFADJOINT OPERATORS AND TRANSFORMATIONS OF OPERATOR-VALUED NEVANLINNA FUNCTIONS Yury Arlinskiı˘ Dedicated to my colleague and friend Seppo Hassi on the occasion of his sixtieth birthday 1 Introduction The Banach space of all continuous linear operators acting between Hilbert spaces H and K is denoted by B(H;K) and by B(H) if K = H. Likewise, the group of all invertible operators in B(H) is denoted by GB(H). Let N be a Hilbert space. Recall that a B(N)-valued function M is called a Nevanlinna function (Behrndt, Hassi & de Snoo, 2020) (alternatively, an R-function (Allen & Narcowich, 1976; Derkach & Malamud, 2017; Kac & Kreı˘n, 1968; Shmul’yan, 1971), a Herglotz function (Gesztesy & Tsekanovskiı˘, 2000), or a Herglotz-Nevanlinna function (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011; Arlinskiı˘ & Klotz, 2010)) if it is holomorphic outside the real axis, symmetric M() = M(), and satisfies the inequality Im ImM()  0 for all  2 CnR. The class of Nevanlinna functions is often denoted by R[N]. A function M 2 R[N] admits the integral representation M() = A+B+ Z R  1 t  t t2 + 1  d(t); Z R d(t) t2 + 1 2 B(N);  2 CnR; (1.1) where A = A 2 B(N), 0  B = B 2 B(N), the B(N)-valued function () is nondecreasing and (t) = (t 0), see (Behrndt, Hassi & de Snoo, 2020; Derkach & Malamud, 2017; Kac & Kreı˘n, 1968; Shmul’yan, 1971). The integral is uniformly convergent in the strong topology; cf. (Behrndt, Hassi & de Snoo, 2020; Brodskiı˘, 1969; Kac & Kreı˘n, 1968). It follows from (1.1) that B = s-lim y"1 M(iy) y and ImM(iy) = B y + Z R y t2 + y2 d(t): This implies that limy!1 yImM(iy) exists in the strong resolvent sense as a selfadjoint relation; see, e.g., (Behrndt, et al., 2010). This limit is a bounded selfadjoint operator if and only ifB = 0 andR R d(t) 2 B(N), in which case s-limy!1 yImM(iy) = R R d(t): In this case one can rewrite the integral representation (1.1) in the form M() = E + Z R 1 t  d(t); Z R d(t) 2 B(N); (1.2) where E = limy!1M(iy) in B(N). The class of B(N)-valued Nevanlinna functions M with the integral representation (1.2) for which E = 0 is denoted byR0[N]. In this paper we consider the following subclasses of the classR0[N]. 2 Acta Wasaensia Definition 1.1. Let M belong to the classR0[N]. Then M is said to belong to N [N] if s-lim y!1 iyM(iy) = IN: Moreover, M is said to belong to N0N if M 2 N [N] and M is holomorphic at infinity. If A is a selfadjoint operator in the Hilbert space H and N is a subspace (closed linear manifold) of H, then the compressed resolvent M(), defined as M() = PN(A I)1N;  2 (A); (1.3) belongs to the class N [N]. Moreover, M as in (1.3) belongs to the class N0N  N [N] if and only if the selfadjoint operator A is bounded. Throughout this paper the representation of M 2 N (N) in the form (1.3) is called a realization of the function M . Note that the function M in (1.3) is often called the compressed resolvent, N-resolvent, Weyl function, or m-function; see (Berezansky, 1968; Gesztesy & Simon, 1997). Here, and throughout the paper, the notation T N denotes the restriction of a linear operator T to the set N  dom T and PL denotes the orthogonal projection onto a subspace L in the Hilbert space H. Let H = NK be a decomposition of a Hilbert space H, then a selfadjoint operator A 2 H is called minimal with respect to N, or N-minimal, if H = span  N+ (A I)1N :  2 C n R : The next theorem follows from (Brodskiı˘, 1969: Theorem 4.8) and Naı˘mark’s dilation theorem (Brodskiı˘, 1969: Theorem 1, Appendix I); see (Arlinskiı˘, Hassi & de Snoo, 2006) and (Arlinskiı˘ & Klotz, 2010) for the case M 2 N0N. Theorem 1.2. The following assertions are valid: (1) IfM 2 N [N], then there exist a Hilbert spaceH containingN as a subspace and a selfadjoint operator A in H, such that A isN-minimal and M() is of the form (1.3) for  in the domain of M . If M 2 N0N, then the selfadjoint operator A is bounded. (2) If A1 and A2 are selfadjoint operators in the Hilbert spaces H1 and H2, respectively, N is a common subspace of H1 and H2, A1 and A2 are N-minimal, and PN(A1 IH1)1N = M() = PN(A2 IH2)1N;  2 CnR; then there exists a unitary operator U mapping H1 onto H2 such that UN = IN and UA1 = A2U: The following linear transformations ! of the complex plane C will play an important role in this paper. Let !() = a+ b;  2 C; (1.4) where a 2 R+, b 2 R, then ! has the property !(R) = R; !(C) = C; !(C+) = C+: Acta Wasaensia 3 Note that if !1() = a1+ b1 and !2() = a2+ b2, then !1  !2() := !1(!2()) = a1a2+ a1b2 + b1: (1.5) Hence, these transformations form a group with respect to composition. The inverse transformation ![1] corresponding to !() = a+ b is ![1]() = a1 ba1: (1.6) This group is denoted by G. For !() = a+ b 2 G we define a! := a. It follows from (1.5) that a!1!2 = a!2!1 = a!1a!2 : Hence, the function G 3 ! 7! a! 2 R+ is a character on the group G. For a function !() = a+ b 2 G define the following transformations G! on N (N): M() 7! G!(M)() := a!M() (I + ( !())M())1 : (1.7) The properties of this transformation are discussed in the theorem below. For this theorem also recall that two linear operators X and Y in H are said to be congruent, if there exists U 2 GB(H) such that Y = UXU ; see, e.g., (Patel, 1983). In the case of unbounded X and Y , the above equality means that dom Y = U1 dom X and Y U1f = UXf; for all f 2 dom X: The main goal of this paper is to prove the following theorem. Theorem 1.3. For the transformations G! defined in (1.7), where ! is given by (1.4), the following assertions are valid: (1) For each ! 2 G the transformation G! is well-defined and maps N (N) into N (N), and N0N into N0N. (2) The set fG! : ! 2 Gg is a group with respect to composition: G!2(G!1(M)) = G!1!2(M); !1; !2 2 G; G1! (M) = G![1](M); ! 2 G; M 2 N (N): In particular, for each ! 2 G the transformation G! mapsN (N) bijectively ontoN (N), and N0N bijectively onto N 0 N. (3) If A is a N-minimal realization of M 2 N (N) and if !() = a+ b 2 G; then any minimal realization of the function G!(M) is congruent toAbPN. Moreover, if !() = +b, b 6= 0; then any minimal realization of the function G!(M) is unitarily equivalent to A bPN. Note that the transformations N0N 3M() 7!MB() := M() (IN +BM())1 2 N0N; B = B 2 B(N); 4 Acta Wasaensia have been considered in Arlinskiı˘, Hassi & de Snoo (2006) and Arlinskiı˘ & Klotz (2010). The transformations R[N] 3 m() 7! m() + t 1 tm() 2 R[N]; t 2 R [ f1g; of scalar Nevanlinna functions and their connections with selfadjoint extensions of symmetric op- erators with deficiency indices (1; 1) have been studied in Behrndt, Hassi, de Snoo, Wietsma & Winkler (2013). Other transformations of Nevanlinna functions, or Nevanlinna families, and their fixed points have been examined in Arlinskiı˘ (2017; 2020) and Arlinskiı˘ & Hassi (2019). This paper is organized as follows. In Section 2 we study properties of congruent operators; in particular, it is shown that congruence preserves the deficiency indices of densely defined closed symmetric operators. In Section 3 we define and examine special transformations of linear operators, which are used in Section 4 in the proof of Theorem 1.3. 2 Properties of congruent operators Proposition 2.1. The following assertions are valid: (1) If the closed densely defined operators X and Y are congruent, then the adjoint operators X and Y  are congruent. (2) Congruence preserves the notions densely defined, closed, maximal dissipative, maximal ac- cumulative, and selfadjoint. (3) If the closed densely defined symmetric operatorsX and Y are congruent, then the deficiency indices of X and Y coincide. Proof. (1) If X and Y are densely defined and Y = UXU , then Y  = UXU; (2.1) as easily follows. (2) Let Y = UXU , U 2 GB(H). Then (dom Y )? = U(dom X)? and it follows that X and Y are both densely defined or non-densely defined. Next let X be a closed operator and suppose that ffng and fUXUfng are Cauchy sequences. Then, due to assumption U 2 GB(H), it follows that fUfng and fXUfng are Cauchy sequences. Since X is closed, we get that g = lim n!1Ufn 2 dom X and Xg = limn!1Ufn. Hence U1g = lim n!1 fn 2 dom Y and Y U 1g = UXU(U1g) = lim n!1U XUfn: Thus, Y is closed. The equality (Y f; f) = (XUf;Uf); f 2 dom Y , yields that congruence preserves the notions Hermitian, dissipative, and accumulative. Thanks to (2.1) congruence preserves selfadjointness. Acta Wasaensia 5 Finally, it is well-known that X is maximal dissipative if and only if X is dissipative and X is accumulative. Hence, we conclude that congruence also preserves the notions maximal dissipative and maximal accumulative. (3) Suppose that X is a closed densely defined symmetric operator whose deficiency indices are (n+(X); n(X)). Consider a maximal dissipative extension eX of X . Then for any , Im < 0, the following direct decomposition of dom X holds: dom X = dom eX _+N(X): (2.2) Here N(X) = ker(X I) is the defect subspace of X corresponding to . In particular, if h 2 dom X, then there exists ~h 2 dom eX and ' 2 N(X) such that h = eh+ ' and Xh = eXeh+ ': (2.3) Next we will describe the defect subspace N(Y ) for the symmetric operator Y congruent to X: Y = UXU: For this purpose, set eY = U eXU: Then eY is a maximal dissipative extension of Y , see part (2) of this proposition. Hence, from the equality eY I = U eXU I = (U eX U1)U; it follows that the operator (U eX U1)1 exists, is bounded, is defined on the whole H for Im < 0, and maps H onto dom eX . Let f 2 N(Y ). Then 0 = (Y  I)f = (UX U1)Uf: (2.4) As h := Uf belongs to dom X, (2.2)-(2.3) imply that the following decomposition holds h = h eX + ' and Xh = eXh eX + '; h eX 2 dom eX; ' 2 N(X): Consequently, (Y  I)f = (UXU I)f = (UX U1)Uf = (UX U1)h = UXh U1h = U( eXh eX + ') U1(h eX + ') = (U eX U1)h eX + (U U1)': Combining the preceding result with (2.4) yields h eX = (U eX U1)1(U1 U)': 6 Acta Wasaensia Hence, h =  I + (U eX U1)1(U1 U)'; f = U 1  I + (U eX U1)1(U1 U)': Thus,N(Y )  U1  I + (U eX U1)1(U1 U)N(X): One can verify that the con- verse inclusion is also true. Therefore N(Y ) = U 1  I + (U eX U1)1(U1 U)N(X): This equality yields that dimN(Y ) = dimN(X), Im < 0. Similarly, using the decomposition dom X = dom eX _+N; Im < 0; we obtain the equality dimN(Y ) = dimN(X). Consequently, the deficiency indices of Y are given by (n+(X); n(X)). 3 Special transformations of operators Let H be an infinite-dimensional separable complex Hilbert space, let N be a subspace of H, and set N? := H N. For z 2 C n (R [ iR) define the operator Uz;N 2 B(H) as follows Uz;N := PN? + i Im z Re z PN = I z Re z PN: (3.1) It is clear from the first equality in (3.1) that Uz;N Uiz;N = PN? PN. Moreover, it follows from the second equality in (3.1) that ran Uz;N = H and that Uz;N 2 GB(H); in fact, one has U1z;N = I iz Im z PN = Uiz;N: (3.2) Hence, one also sees immediately that Uz;N = Uz;N and U1z;N = Uiz;N: (3.3) Observe that Uz;N Uz;N =  I z Re z PN  I z Re z PN  = I + jzj2 2(Re z)2 (Re z)2 PN; (3.4) so that Uz;N 2 GB(H) is unitary if and only if (Re z)2 = (Im z)2. For z 2 C n (R [ iR) we define the transformation Fz;N on the set of all linear operators A in H as follows 8><>: dom Fz;N(A) = Uz;N dom A; Fz;N(A)f = A+ iz Im z PN(A zI) ! U1z;Nf; f 2 dom Fz;N(A): (3.5) Acta Wasaensia 7 Lemma 3.1. Let A be an operator and let z 2 C n (R [ iR). Then the operator Fz;N(A) satisfies Fz;N(A) = U1z;N A jzj 2 Re z PN ! U1z;N; (3.6) i.e., Fz;N(A) is congruent to the operator A (jzj2=Re z)PN. Moreover, if jIm zj = jRe zj, then Fz;N(A) is unitarily equivalent to the operator A (jzj2=Re z)PN. Proof. It follows from (3.1) that A+ iz Im z PN(A zI) =  I + i z Im z PN  A+ i jzj2 Im z PN = Uiz;NA+ Uiz;NUz;Ni jzj2 Im z PN = Uiz;N " A+ Uz;Ni jzj2 Im z PN # = Uiz;N A jzj 2 Re z PN ! : Consequently, the first statement about the congruence now follows from the definition of Fz;N(A) and (3.3). The last statement follows from the identity (3.4). It is clear from the definition in (3.5), that (dom Fz;N(A))? = U1z;N (dom A)?: Thus, the operator Fz;N(A) is densely defined if and only if the operator A is densely defined. Furthermore, the domain of dom Fz;N(A) is closed if and only if dom A is closed. The next corollary collects the basic properties of the transformation Fz;N. Corollary 3.2. The transformation Fz;N in (3.5) possesses the following properties: (1) dom Fz;N(A) \ dom Fz;N(B) = f0g if and only if dom A \ dom B = f0g. (2) The operatorFz;N(A) is bounded or closed if and only ifA is bounded or closed, respectively. (3) The operator Fz;N(A) is symmetric, dissipative, or accumulative if and only if A is symmet- ric, dissipative, or accumulative, respectively. Moreover, maximality with respect to these properties is preserved and selfadjointness is also preserved. (4) The following relation holds (Fz;N(A)) = Fz;N(A): (5) The following identities hold dom Fz;N(A) \N? = dom A \N? and PN?Fz;N(A)N? = PN?AN?: (6) If A is a closed densely defined symmetric operator, then the deficiency indices of Fz;N(A) coincide with the deficiency indices of A. 8 Acta Wasaensia Proof. (1) Due to the identity dom Fz;N(A) \ dom Fz;N(B) = Uz;N (dom A \ dom B) ; we obtain the equivalence dom Fz;N(A) \ dom Fz;N(B) = f0g () dom A \ dom B = f0g: (2) – (5) These statements follow from Lemma 3.1, because Fz;N(A) is congruent to the operator A(z;N) given by (3.6), and A(z;N) is the additive perturbation of A by the bounded selfadjoint operator (jzj2=Re z)PN. (6) It is well known that the additive perturbation of a symmetric operator by a bounded selfadjoint operator preserves deficiency indices, see, e.g., (Akhiezer & Glazman, 1981). Let N be a subspace of the Hilbert space H. For a linear operator A in H and  2 (A) we define the transform Tz;N(A; ) of A by Tz;N(A; ) := PN? + i Re z Im z PN(A z()I)(A I)1; (3.7) where z() is defined by z() :=  Im z Re z !2 + jzj2 Re z : (3.8) From the definition in (3.7) it is clear that Tz;N(A; ) 2 B(H), since  2 (A). Note that with respect to the orthogonal decomposition H = N? N one has Tz;N(A; ) =  IN? 0 Az;N(A; ) Bz;N(A; )  :  N? N  !  N? N  ; (3.9) where Az;N(A; ) and Bz;N(A; ) are defined by Az;N(A; ) := PNTz;N(A; )N?; Bz;N(A; ) := PNTz;N(A; )N; so that Az;N(A; ) 2 B(N?;N) and Bz;N(A; ) 2 B(N). In particular, it is useful to observe that the compression PNTz;N(A; )N has the form PNTz;N(A; )N = i Re z Im z I + ( z())PN(A )1  ; (3.10) cf. (3.7). The properties of the transform Tz;N(A; ) and its compression Bz;N(A; ) toN are stated in the following theorem. Theorem 3.3. Let A be a linear operator in the Hilbert space H, let z 2 C n (R [ iR), and let Fz;N(A) be defined as in (3.5). Let  2 (A) and let the transformation Tz;N(A; ) of A be defined as in (3.7). Then the following identity holds Fz;N(A) I = Tz;N(A; )(A I)U1z;N;  2 (A): (3.11) Consequently, for  2 (A) the following statements are equivalent: Acta Wasaensia 9 (i)  2 (Fz;N(A)); (ii) the operator Tz;N(A; ) belongs to GB(H); (iii) the operator PNTz;N(A; )N belongs to GB(N). Moreover, for  2 (Fz;N(A)) \ (A), one has (Fz;N(A) I)1 = Uz;N(A )1Tz;N(A; )1; (3.12) while the compression of (Fz;N(A) I)1 to N is given by PN(Fz;N(A) I)1N = Im z Re z !2 PN(A I)1 IN + ( z())PN(A I)1N 1 : (3.13) In particular, if the operator A is a maximal dissipative, maximal accumulative, or selfadjoint in the Hilbert space H, then (3.12) and (3.13) hold for each proper subspaceN, for each z 2 Cn (R[ iR), and for each  in C, C+, or C [ C+, respectively. Proof. Let bA := Fz;N(A). It follows from dom bA = Uz;N dom A that any bf 2 dom bA is of the form bf = Uz;NfA with a unique fA 2 dom A and conversely. From (3.5) one therefore sees that for all bf 2 dom bA ( bA I) bf = A+ iz Im z PN(A zI) ! fA  I z Re z PN ! fA = (A )fA + iz Im z PN(A zI)fA +  z Re z PNfA = (A I)fA + iz Im z PN(A I)fA + iz Im z ( z) +  z Re z ! PNfA = I + iz Im z PN + iz Im z ( z) +  z Re z ! PN(A I)1 ! (A I)fA: By writing I = PN? +PN, we see that the first factor in the right-hand side of the last term is given by PN? +  I + iz Im z  PN + i Re z2 jzj2 Re z Re z Im z PN(A )1 = PN? + i Re z Im z PN  A + Re z 2 jzj2 Re z (Re z)2  (A )1 = PN? + i Re z Im z PN(A z())(A )1 = Tz;N(A; ); where the following identities were used I + iz Im z = i Re z Im z and z() =  Re z2 jzj2 Re z (Re z)2 : Therefore, (3.11) has been shown. 10 Acta Wasaensia (i), (ii) This equivalence follows from (3.11). (ii), (iii) This equivalence follows from (3.9) and (3.11). The resolvent formula (3.12) follows from (3.11). In order to see (3.13), first observe from (3.1) and (3.10) that PN Uz;N = i Im z Re z PN and B1z;N() = 1 i Im z Re z I + ( z())PN(A )1 1 : Therefore, it is seen as a consequence of (3.9) and (3.12) that PN (Fz;N(A) I)1N = PN Uz;N(A )1T 1z;N N = PN Uz;N(A )1B1z;N() = Im z Re z !2 PN(A )1 I + ( z())PN(A )1 1 ; which gives (3.13). Next let A be a maximal dissipative operator. Then by Proposition 3.2 the operator bA := Fz;N(A) is maximal dissipative too. Therefore the open lower half-plane C belongs to the resolvent set of A and bA. As has been proven above, the operators Tz;N(A; ) and Bz;N(A; ) belong to GB(H) and GB(N), respectively for all  2 C. Hence the identities (3.12) and (3.13) are valid for all  2 C. The proofs of the statements for a maximal accumulative or selfadjoint operator A can be established in a similar way. Corollary 3.4. Let A be a selfadjoint operator in the Hilbert space H and letN be a subspace of H. Then A is N-minimal if and only if Fz;N(A) is N-minimal. Proof. It follows from (3.9) and (3.11), and from the invertibility of Bz;N(A; ) in H, that (Fz;N(A) I)1 = Uz;N (A )1  IN? 0 Bz;N(A; )1Az;N(A; ) Bz;N(A; )1  : Thanks to the invertibility of Uz;N in H, the statement is clear from the above identity. Lemma 3.5. Let A be an operator in the Hilbert space H and let z1; z2 2 C n (R [ iR). Then Fz2;N(Fz1;N(A)) = PN? Re z1 Re z2 Im z1 Im z2 PN !  0@A 0@ jz1j2 Re z1 + jz2j2 Re z2 Im z1 Re z1 !21APN 1A PN? Re z1 Re z2Im z1 Im z2PN ! : Thus, the operator Fz2;N(Fz1;N(A)) is congruent to the operator A 0@ jz1j2 Re z1 + jz2j2 Re z2 Im z1 Re z1 !21APN: Acta Wasaensia 11 Proof. Let A be a linear operator, then it follows from (3.6) that Fz2;N(Fz1;N(A)) = Fz2;N( bA1) = U1z2;N bA1 jz2j2 Re z2 PN ! U1z2;N = U1z2;N U1z1;N A jz1j 2 Re z1 PN ! U1z1;N jz2j2 Re z2 PN ! U1z2;N = U1z2;N U1z1;N A jz1j 2 Re z1 PN jz2j2 Re z2 Uz1;NPNUz1;N ! U1z1;N U1z2;N; which, thanks to (3.2) and (3.3), gives the required result. One can easily verify the identity F(z);N  Fz;N = Fz;N  F(z);N = id; where id is the identity transformation on the set of all linear operators in H and the function (z) is defined as (z) := iz Re z Im z = Re z + i (Re z) 2 Im z = z + i z2 Im z ; z 2 C n (R [ iR): Remark 3.6. Let S be a closed densely defined symmetric operator in the Hilbert space H, let z 2 C n (R[ iR), and letNz = ker(S zI) 6= f0g be the deficiency subspace of S corresponding to z. Define the associated operators US(z) by US(z) := Uz;Nz = PN?z + i Im z Re z PNz = I z Re z PNz : Then the symmetric operator S(z) = US(z)1 S jzj 2 Re z PNz ! US(z)1 has been studied in Arlinskiı˘ (2021) and it was established that S(z) preserves various properties of S. When the deficiency indices of S are equal, then a bijection of the set of all selfadjoint extensions of S onto the set of all selfadjoint extensions of S(z) was established. 4 Proof of Theorem 1.3 This section provides a proof of Theorem 1.3. It is based on the general constructions in Section 3, which are applied under the assumption that the underlying operator is selfadjoint. (1) Let M 2 N (N) be arbitrary. Then by Theorem 1.2 there exists a selfadjoint operator A in the Hilbert space H, containing N as a subspace, realizing M as follows M() = PN(A I)1N;  2 C n R: (4.1) 12 Acta Wasaensia Let the transformation !() = a+ b 2 G, where a 2 R+ and b 2 R n f0g, be arbitrary, cf. (1.4). Then define z! 2 C n (R [ iR) as z! := b 1 + a + i p ab 1 + a (4.2) so that z! () = !(), see (3.8). Note that, conversely, a and b in (4.2) can be expressed in terms of z! as a = Im z! Re z! !2 and b = jz!j2 Re z! : (4.3) For the selfadjoint operator A in (4.1) let bA := Fz!;N(A), where z! is given by (4.2) and Fz;N is the transformation defined in (3.5). Then bA is selfadjoint by Corollary 3.2, and Theorem 3.3 implies that for  2 CnR PN( bA I)1N = Im z Re z !2 PN(A I)1 IN + ( z! ())PN(A I)1N 1 = a!M() (I + ( !())M())1 (4.4) = G!(M)(); see (4.1), (4.3), and (1.7). Now the representation (4.4) for G!(M) shows that it belongs to the class N (N), since it is the compression of the resolvent of the selfadjoint operator bA. If the representing operator A is additionally assumed to be bounded, then Corollary 3.2 implies that also bA is bounded and, hence, G!(M) 2 N0N. (2) Let !k() = ak+ bk 2 G be arbitrary, for k = 1; 2. Then one observes !1  !2() = !1() a!1( !2()); (4.5) cf. (1.5). For M 2 N (N) we have by (1) that M!1 := G!1(M) 2 N (N). Therefore, observe that G!2(G!1(M))() = G!2(M!1)() = a!2M!1() (I + ( !2())M!1())1 = a!1a!2M() (I + ( !1())M())1  h I + ( !2())a!1M() (I + ( !1())M())1 i1 = a!1a!2M() (I + ( !1())M())1 (I + ( !1())M())  [(I + ( !1())M()) + a!1( !2())M()]1 = a!1a!2M()  I +  !1() + a!1( !2()  M() 1 = a!1!2M() I +  !1  !2()  M() 1 = G!1!2(M)(); where the penultimate identity follows thanks to (4.5). This shows that the first identity in Theo- rem 1.3 (2) holds. That identity implies the second identity in view of (1.5) and (1.6). Let cM 2 N (N) be arbitrary. Then by the above composition result M := G![1](cM) 2 N (N) and G!(M) = cM , showing that G is surjective. Likewise, if M1;M2 2 N (N) satisfy the equality G!(M1) = G!(M2), then composing the preceding equality with G![1] yields thatM1 = M2. Thus G! is bijective on the set N (N). The bijectivity of G! restricted to the set N0N can be established in exactly the same manner. Acta Wasaensia 13 (3) Let A be a N-minimal realization of M 2 N (N) and let !() = a + b 2 G be arbitrary. Moreover, define zw as in (4.2) and let bA := Fz!;N(A). Then the identity (4.4) holds. Lemma 3.1 now yields that the operator bA = Fz!;N(A) is congruent to the operator A(z;N) = A jzj 2 Re z PN = A bPN; see also (4.2) and (4.3), via U1z;N = PN? i Re z Im z PN = PN? i 1p a PN: Moreover, the operator A is N-minimal if and only if the operator bA := Fz!;N(A) is N-minimal, see Corollary 3.4. Finally, recall that by Theorem 1.2 (2) any N-minimal realization of G!(M) is unitary equivalent to bA = Fz!;N(A) and, hence, is congruent to A bPN. This establishes the first part of this assertion. Next assume that !() =  + b. If z! is such that z! () = !(), then equation (4.3) implies that (Re z)2 = (Im z)2. Therefore (3.3) yields that the operator U1z;N is equal to PN?+ iPN and, hence, is unitary. Consequently, Lemma 3.1 shows that bA := Fz!;N(A) is unitary equivalent to A bPN. This establishes the second part of the assertion by Theorem 1.2 (2). Remark 4.1. The composition formula G!2  G!1 = G!1!2 in Theorem 1.3 has a counterpart for the operator representations. With the transformations !1() = a1+ b1 and !2() = a2+ b2; define the corresponding parameters z1 = b1 1 + a1 + i p a1b1 1 + a1 and z2 = b2 1 + a2 + i p a2b2 1 + a2 ; cf. (4.2). Then the composition of Fz2;N(Fz1;N(A)) in Lemma 3.5 is given in terms of !1 and !2 by  PN? 1p a1a2 PN  A (a1b2 + b1)PN  PN? 1p a1a2 PN  : Note that a1b2 + b1 is the constant term of the composition !1  !2, see (1.5). References Akhiezer, N.I. & Glazman, I.M. (1981). Theory of Linear Operators in Hilbert Space. Monographs and Studies in Mathematics, vol. 9, 10. Boston, Mass.: Pitman. Allen, G.D. & Narcowich, F.J. (1976). R-operators. I. Representation theory and applications. Indi- ana Univ. Math. J. 25, 945–963. Arlinskiı˘, Yu.M. (2017). Transformations of Nevanlinna operator-functions and their fixed points. Methods Funct. Anal. Topology 23, 212–230. Arlinskiı˘, Yu.M. (2020). Compressed resolvents, Schur functions, Nevanlinna families and their transformations. Complex Anal. Oper. Theory 14, paper 63, 59 pp. 14 Acta Wasaensia Arlinskiı˘, Yu.M. (2021). Cloning of symmetric operators. Complex Anal. Oper. Theory 15, paper 8, 42 pp. Arlinskiı˘, Yu., Belyi, S. & Tsekanovskiı˘, E. (2011). Conservative Realizations of Herglotz- Nevanlinna Functions. Operator Theory: Advances and Applications, vol. 217. Basel: Birkhäuser. Arlinskiı˘, Yu. & Hassi, S. (2019). Holomorphic operator valued functions generated by passive selfadjoint systems. Oper. Theory Adv. Appl. 272, 1–42. Arlinskiı˘, Yu.M., Hassi, S. & de Snoo, H.S.V. (2006). Q-functions of quasi-selfadjoint contractions. Oper. Theory Adv. Appl. 163, 23–54. Arlinskiı˘, Yu. & Klotz, L. (2010). Weyl functions of bounded quasi-selfadjoint operators and block operator Jacobi matrices. Acta Sci. Math. (Szeged) 76, 585–626. Behrndt, J., Hassi, S., de Snoo, H. & Wietsma, R. (2010). Monotone convergence theorems for semi- bounded operators and forms with applications. Proc. Roy. Soc. Edinburgh Sect. A 140, 927–951. Behrndt, J., Hassi, S. & de Snoo, H. (2020). Boundary Value Problems, Weyl Functions, and Differ- ential Operators. Monographs in Mathematics, vol. 108. Cham: Birkhäuser. Behrndt, J., Hassi, S., de Snoo, H., Wietsma, R. & Winkler, H. (2013). Linear fractional transforma- tions of Nevanlinna functions associated with a nonnegative operator. Complex Anal. Oper. Theory 7, 331–362. Berezansky, Yu.M. (1968). Expansions in Eigenfunctions of Selfadjoint Operators. Providence, R.I.: Amer. Math. Soc. Brodskiı˘, M.S. (1969). Triangular and Jordan Representations of Linear Operators. Moscow: Nauka. (In Russian. English translation: Translations of Mathematical Monographs, vol. 32 (1971). Providence, R.I.: Amer. Math. Soc.) Derkach, V.A. & Malamud, M.M. (2017). Extension Theory of Symmetric Operators and Boundary Value Problems. Proceedings of Institute of Mathematics of NAS of Ukraine, vol. 104. (In Russian.) Gesztesy, F. & Simon, B. (1997). m-functions and inverse spectral analysis for finite and semi-finite Jacobi matrices. J. Anal. Math. 73, 267–297. Gesztesy, F. & Tsekanovskiı˘, E.R. (2000). On matrix-valued Herglotz functions. Math. Nachr. 218, 61–138. Kac, I.S. & Kreı˘n, M.G. (1968). R-functions – analytic functions mapping the upper halfplane into itself. Supplement to the Russian edition of F.V. Atkinson, Discrete and Continuous Boundary Prob- lems, Mir, Moscow 1968. (In Russian. English translation: Amer. Math. Soc. Transl. Ser. 2, vol. 103 (1974), 1–18.) Patel, S.M. (1983). On congruency of operators. Publ. Inst. Math., Nouv. Ser. 33, 169–171. Shmul’yan, Yu.L. (1971). The operator R-functions. Sibirsk. Mat. Z. 12, 442–451. (In Russian. English translation: Sib. Math. J. 12 (1971), 315–322.) Stuttgart, Germany E-mail address: yury.arlinskii@gmail.com Acta Wasaensia 15 A CLASS OF SINGULAR PERTURBATIONS OF THE DIRAC OPERATOR: BOUNDARY TRIPLETS AND WEYL FUNCTIONS Jussi Behrndt, Markus Holzmann, Christian Stelzer, and Georg Stenzel Dedicated to our friend and colleague Seppo Hassi on the occasion of his 60th birthday! 1 Introduction Singular perturbations of self-adjoint operators play an important role in the description of ideal- ized quantum systems, where a localized short-range potential is often replaced by a more singular model potential. More precisely, assume that A0 is a self-adjoint differential operator in an L2- Hilbert space which is viewed as the Hamiltonian of an unperturbed quantum system and suppose that V is some potential such that the formal sum AV = A0 + V describes the quantum system under investigation. Standard operator theory techniques ensure that for potentials V belonging to certain function spaces the perturbed operator AV is again self-adjoint; we refer the reader to the monographs of Reed & Simon (1972; 1975; 1979; 1978) or Kato (1995). However, a detailed spec- tral analysis of AV is typically very difficult, and for this reason the potential V is often replaced by an idealized perturbation term of -type, which is then regarded as an approximation of the real model, see (Behrndt et al., 2017; Exner, 2008). On the one hand, this procedure may simplify the spectral analysis considerably, see (Albeverio et al., 2005; Behrndt, Langer & Lotoreichik, 2013; Brasche et al., 1994; Holzmann & Unger, 2020), but, on the other hand, it may lead to new technical difficulties in the mathematically rigorous definition of the Hamiltonian itself. In the case that A0 is the Laplacian in an L2-space and the -potential is supported on hypersurfaces in Rd (e.g., curves in R2, or surfaces in R3) the standard quadratic form approach is useful. Roughly speaking, the perturbed operator A = A0 +  is in this situation viewed as the self-adjoint operator corresponding to the form a[f; g] = (rf;rg)L2 + Z   f j gj dx; (1.1) where (rf;rg)L2 is the quadratic form defined on the Sobolev spaceH1 associated with the Lapla- cian, and the singular perturbation is encoded in the additive form perturbation with  denoting the support of the -distribution,  is some real (position dependent) coefficient, and f j and gj denote the traces of the Sobolev space functions f and g, respectively, defined in an appropriate way. Of course, one has to impose certain assumptions on the support  of the -potential and the coeffi- cient  to ensure that a in (1.1) is a densely defined closed semibounded form (which then gives rise to a self-adjoint operator A ); we refer to (Brasche et al., 1994; Exner, 2008; Exner & Kovarik, 2015; Herczyn´ski, 1989; Stollmann & Voigt, 1996) for a detailed treatment and further references. A different approach to the operator A is via extension theory techniques in general, and bound- ary triplet methods in particular, see the recent monograph (Behrndt, Hassi & de Snoo, 2020) and (Derkach, Hassi & Malamud, 2020; Derkach et al., 2000; 2006; 2009; 2012; Derkach, Hassi & de Snoo, 2001; 2003) by Seppo Hassi and his coauthors for an extensive treatment of boundary triplets and further developments. For the case of point interactions it is well known what type of transmis- sion or jump conditions the functions in the domain of A satisfy; cf. (Albeverio et al., 2005) for a 16 Acta Wasaensia comprehensive treatment of point interactions. In the case that the -distribution is supported on a hypersurface we refer to (Behrndt, Langer & Lotoreichik, 2013), where quasi boundary triplets were used for the first time to define A as a self-adjoint restriction of a Laplacian that is decoupled along the support . As in the case of point interactions, also in the multi-dimensional setting one ends up with transmission and jump conditions for the functions in the domain of A along the support  of the -distribution, see also (Behrndt et al., 2020; 2018; Mantile, Posilicano & Sini, 2016). In conclusion, for the case that A0 is the Laplacian (or some more general semibounded Schrödinger operator) nowadays one may efficiently apply form techniques or boundary triplet methods to define and study the perturbed operator A ; depending on the particular problem under consideration one method may prove more useful than the other. Now assume that the unperturbed operator A0 is the Dirac operator instead of the Laplacian or the Schrödinger operator. While the Dirac operator describes a similar physical system as the Laplace operator including relativistic effects (see Section 3 for more details), the mathematical situation is entirely different: The free Dirac operator A0 is not semibounded from below and, hence, standard quadratic form methods are not applicable. Therefore, it is most natural to try to apply boundary triplet techniques, since these methods do not require any type of semiboundedness of the operators under consideration. In fact, Dirac operators with singular interactions supported on points and spheres were already treated with direct methods in (Albeverio et al., 2005; Dittrich, Exner & Šeba, 1989; Gesztesy & Šeba, 1987), but for more general supports of the singular potential only recently a series of papers was published (Arrizabalaga, Mas & Vega, 2014; 2015; 2016), which in turn led to our publications (Behrndt et al., 2018; Behrndt & Holzmann, 2020; Behrndt, Holzmann & Mas, 2020; Behrndt et al., 2020) employing the quasi boundary triplet technique. We also emphasize the recent papers (Behrndt et al., 2019; 2020; Holzmann, Ourmières-Bonafos & Pankrashkin, 2018; Mas & Pizzichillo, 2018; Ourmières-Bonafos & Vega, 2018; Pankrashkin & Richard, 2014) where closely related techniques were used to study Dirac operators with -shell interactions. The main objective of this note is to provide boundary triplets for Dirac operators with Lorentz scalar interactions supported on a point in the one-dimensional case, and supported on curves and surfaces in the two- and three-dimensional situation. This operator is formally given by A = A0 +  0; where 0 is a Dirac matrix defined in Section 3, and  0 describes the Lorentz scalar -shell in- teraction supported on . The one-dimensional setting with a single point interaction is particularly easy to treat and we discuss in Section 4 a possible choice of an ordinary boundary triplet, which was also used in Pankrashkin & Richard (2014). We compute the corresponding -field and Weyl function, and give an expression for the resolvent of the singularly perturbed one-dimensional Dirac operator. In the multi-dimensional setting one observes typical analytic difficulties with trace maps and integration by parts formulas on maximal operator domains, similar to the case of the Laplacian or more general elliptic operators; cf. (Behrndt & Langer, 2007; 2012). It is convenient to extend the notion of ordinary boundary triplet in such a way that these analytic difficulties can be circumvented. As in the case of symmetric second order elliptic operators, the concepts of quasi boundary triplets and generalized boundary triplets are useful and fit in this setting very well. In the present manuscript we allow some flexibility in the domain of the boundary maps and obtain a family of quasi boundary triplets that reduce to a generalized boundary triplet in the limit case, where the parameter describing regularity of the operator domain is minimal; cf. Theorem 5.3. As in the one-dimensional situation, we provide the corresponding -fields and Weyl functions, we discuss the self-adjointness of the operator A , and list some of its spectral properties. An interesting issue in the multi-dimensional Acta Wasaensia 17 setting is the regularity of the support  of the Lorentz scalar -perturbation: From C2-curves and hypersurfaces treated earlier in Arrizabalaga, Mas & Vega (2014; 2015); Behrndt et al. (2018; 2019); Behrndt & Holzmann (2020); Ourmières-Bonafos & Vega (2018) and piecewise C2-curves studied in Pizzichillo & Van Den Bosch (2019), we make a substantial step towards more rough supports and discuss in Theorem 5.4 the case that  is the boundary of a bounded Lipschitz domain. The paper is organized as follows. In Section 2 we briefly recall some basic definitions and abstract facts about ordinary, generalized, and quasi boundary triplets. Section 3 is devoted to regular Dirac operators: We collect some required notations, state the well-known properties of the unperturbed Dirac operator A0, and shortly describe the physical interpretation of the objects of interest. In Section 4 we study the one-dimensional case, provide an ordinary boundary triplet suitable to treat singular perturbations of the free Dirac operator in R, and investigate Dirac operators with Lorentz scalar -point interactions. Finally, Section 5 is devoted to the multi-dimensional case. We construct a family of quasi boundary triplets that are suitable to prove the self-adjointness of Dirac operators with Lorentz scalar -shell interactions supported on arbitrary closed compact Lipschitz smooth hypersurfaces in R2 and R3. 2 Ordinary, generalized, and quasi boundary triplets In this section we briefly recall basic definitions of ordinary and generalized boundary triplets, quasi boundary triplets, and some related techniques in extension and spectral theory of symmetric and self-adjoint operators in Hilbert spaces. The concepts will be presented such that they can be ap- plied directly to Dirac operators with singular interactions in the next sections. We refer the reader to (Behrndt, Hassi & de Snoo, 2020; Behrndt & Langer, 2007; 2012; Brüning, Geyler & Pankrashkin, 2008; Derkach & Malamud, 1991; 1995; Gorbachuk & Gorbachuk, 1991) for more details on bound- ary triplet techniques. Throughout this section H denotes a complex Hilbert space with inner prod- uct (; )H and S is a densely defined closed symmetric operator with adjoint S. Definition 2.1. Let T be a linear operator in H such that T = S. A triplet fG;0;1g consisting of a Hilbert space G and linear mappings 0;1 : dom T ! G is called a quasi boundary triplet for S if it has the following properties: (i) For all f; g 2 dom T the abstract Green’s identity (Tf; g)H (f; Tg)H = (1f;0g)G (0f;1g)G is true. (ii) The range of = (0;1)> is dense in G  G. (iii) The restriction A0 := T  ker 0 is a self-adjoint operator inH. If (i) and (iii) hold, and the mapping 0 : dom T ! G is surjective, then fG;0;1g is called a generalized boundary triplet; if (i) and (iii) hold, and the mapping = (0;1) > : dom T ! G  G is surjective, then fG;0;1g is called an ordinary boundary triplet. 18 Acta Wasaensia Note that the above (non-standard) definition of generalized and ordinary boundary triplets is equiva- lent to the usual one given in, e.g., (Behrndt, Hassi & de Snoo, 2020; Brüning, Geyler & Pankrashkin, 2008; Derkach & Malamud, 1991; 1995; Gorbachuk & Gorbachuk, 1991), see (Behrndt & Langer, 2007: Corollary 3.2 & Corollary 3.7). In particular, if fG;0;1g is an ordinary boundary triplet, then T = S. Note that a quasi boundary triplet, generalized boundary triplet, or ordinary boundary triplet for S exists if and only if the defect numbers dim ker(S  i) coincide, i.e., if and only if S admits self-adjoint extensions in H. Moreover, the operator T in Definition 2.1 is in general not unique. Next, we recall the definition of the -field and the Weyl function associated with the quasi boundary triplet fG;0;1g. These mappings will allow us to describe spectral properties of self-adjoint extensions of S. With A0 = T  ker 0 the direct sum decomposition dom T = dom A0 _+ ker(T ) = ker 0 _+ ker(T );  2 (A0); (2.1) holds. The definition of the -field and Weyl function for quasi boundary triplets is in accordance with the definition for ordinary and generalized boundary triplets in Derkach & Malamud (1991; 1995). Definition 2.2. Assume that T is a linear operator in H satisfying T = S and let fG;0;1g be a quasi boundary triplet for S. Then the corresponding -field and Weyl function M are defined by (A0) 3  7! () := 0  ker(T ) 1 and (A0) 3  7!M() := 1 0  ker(T ) 1 ; respectively. From (2.1) we see that the -field is well defined and that ran () = ker(T ) holds for all  2 (A0). Moreover, dom () = ran 0 is dense in G by Definition 2.1. With the help of the abstract Green’s identity in Definition 2.1 (i) one verifies that () = 1(A0 )1;  2 (A0): (2.2) Thus () is a bounded and everywhere defined operator from H to G. Therefore, () is a, in general not everywhere defined, bounded operator; cf. (Behrndt & Langer, 2007: Proposition 2.6) or (Behrndt & Langer, 2012: Proposition 6.13). If fG;0;1g is a generalized or ordinary boundary triplet, then () is automatically bounded and everywhere defined. Next, we state some useful properties of the Weyl function M corresponding to the quasi boundary triplet fG;0;1g; see, e.g., (Behrndt & Langer, 2007: Proposition 2.6) for proofs of these state- ments. For any  2 (A0) the operator M() is densely defined in G with dom M() = ran 0 and ran M()  ran 1. Moreover, for all ;  2 (A0) and ' 2 ran 0 one has M()'M()' = ( ) () ()': (2.3) Therefore, we see that M()  M() for any  2 (A0) and hence M() is a closable, but, in general, unbounded linear operator in G. If fG;0;1g is a generalized or ordinary boundary triplet, then M() is bounded and everywhere defined. Acta Wasaensia 19 In the main part of this paper we are going to use ordinary boundary triplets, generalized boundary triplets, quasi boundary triplets, and their Weyl functions to define and study self-adjoint exten- sions of the underlying symmetry S. Let again T be a linear operator in H such that T = S, let fG;0;1g be a quasi boundary triplet for S, and let # be a linear operator in G. Then we define the extension A# of S by A# := T  ker(1 #0); (2.4) i.e., f 2 dom T belongs to dom A# if and only if f satisfies 1f = #0f . If # is a symmetric operator in G, then Green’s identity implies (A#f; g)H (f;A#g)H = (#0f;0g)G (0f; #0g)G = 0 (2.5) for all f; g 2 dom A# and, hence, the extension A# is symmetric inH. Of course, one is mostly interested in the self-adjointness of A#. If fG;0;1g is an ordinary boundary triplet, then the situation is simple: Here one has a one-to-one correspondence between self-adjoint realizations A# as in (2.4) and self-adjoint operators and relations # in G. In particular, if # is a self-adjoint operator in G, then A# is self-adjoint inH, see, e.g., (Behrndt, Hassi & de Snoo, 2020: Theorem 2.1.3) for more details. If fG;0;1g is a generalized or a quasi boundary triplet, then the self-adjointness of # does, in general, not imply the self-adjointness of A#, or vice versa. However, the following theorem, where we also state an abstract version of the Birman-Schwinger principle and a Kreı˘n type resolvent formula for canonical extensions A#, will allow us to give conditions for the self-adjointness of A#; for the proof we refer to (Behrndt & Langer, 2007: Theorem 2.8) or (Behrndt & Langer, 2012: Theorem 6.16). Theorem 2.3. Let T be a linear operator in H satisfying T = S, let fG;0;1g be a quasi boundary triplet for S with A0 = T  ker 0, and denote the associated -field and Weyl function by and M , respectively. Let A# be the extension of S associated with an operator # in G as in (2.4). Then the following statements hold for all  2 (A0): (i)  2 p(A#) if and only if 0 2 p(#M()). Moreover, ker(A# ) =  ()' : ' 2 ker(#M()) : (ii) If  =2 p(A#), then g 2 ran (A# ) if and only if ()g 2 ran (#M()). (iii) If  =2 p(A#), then (A# )1g = (A0 )1g + () #M()1 ()g holds for all g 2 ran (A# ). Assertion (ii) of the previous theorem shows how the self-adjointness of an extension A# can be proven if fG;0;1g is a generalized or a quasi boundary triplet. If # is symmetric in G, then A# is symmetric inH by (2.5), and hence A# is self-adjoint if, in addition, ran (A#  i) = H. According to Theorem 2.3 (ii) the latter is the case if ran (i)  ran (#M(i)). 20 Acta Wasaensia 3 Some facts about Dirac operators In this section, a brief introduction to Dirac operators will be presented. These operators correspond to the right-hand side of the Dirac equation. The free Dirac equation was derived by P. Dirac when linearising the relativistic energy-momentum relationship E2 = dX j=1 p2j +m 2; (3.1) where E denotes the energy and p = (p1; : : : ; pd) denotes the momentum. Here, and in the subse- quent sections, d is the space dimension and m > 0 is the mass of the particle. Furthermore, the speed of light c and Planck’s constant ~ are set to 1 for simplicity. This can always be realized by a suitable choice of units. The usual linearization approach, as it is carried out for instance in Thaller (1992), corresponds to0@E dX j=1 jpj m 0 1A0@E + dX j=1 jpj +m 0 1A = 0 (3.2) with matrices j 2 CNN , whereN = 2[(d+1)=2] and [] is the Gauss bracket. For the cases relevant to us we have N = 2 for d 2 f1; 2g and N = 4 for d = 3. A comparison of (3.2) with the energy- momentum relationship (3.1) shows that the matrices j must be chosen such that they satisfy the anti-commutation relations k j + j k = 2kjIN for all k; j 2 f0; 1; : : : ; dg; (3.3) where In denotes the n  n-identity matrix. For d 2 f1; 2g the matrices j can be chosen as the Pauli spin matrices 1 = 1 =  0 1 1 0  ; 2 = 2 =  0 i i 0  ; and 0 = 3 =  1 0 0 1  ; and for d = 3 as the so-called Dirac matrices j =  0 j j 0  and 0 =  I2 0 0 I2  : If one now applies the usual substitution rules i @@t and i @@xj for E and pj in one of the factors in (3.2), one obtains the free Dirac equation i @ @t = 0@i dX j=1 j @ @xj +m 0 1A ; which describes a particle with spin 1=2, such as an electron, that moves in Rd. Here, and in the following, we use for x = (x1; : : : ; xd) 2 Rd the formal notations  x := dX j=1 jxj and  r := dX j=1 j @ @xj : Acta Wasaensia 21 As in the case of the Schrödinger equation, one now defines the free Dirac operator as the right-hand side of the free Dirac equation by A0f := (i(  r) +m 0) f; dom A0 = H1(Rd;CN ): (3.4) With the help of the Fourier transform it is not difficult to verify thatA0 is self-adjoint inL2(Rd;CN ) with purely essential spectrum (A0) = (1;m] [ [m;1); (3.5) cf. (Thaller, 1992) or (Weidmann, 2003). From a physical point of view there are possible energy states of the system that are negative and these energies are not bounded from below. This led to the discovery of anti-particles, as, e.g., in the case of the electron, the positron. To derive an explicit representation of the resolvent (A0 )1 for  2 (A0), one uses that (3.3) implies the relation (A0 )(A0 + ) =  +m2 2 IN ; where  is the free Laplace operator defined on dom () = H2(Rd). This implies (A0 )1 = i(  r) +m 0 + IN( +m2 2)1IN : (3.6) Using the well-known form of the resolvent of, one finds that (A0)1 is an integral operator in L2(Rd;CN ). In order to describe its integral kernel G;d(x y), we write Kj for the modified Bessel functions of the second kind and k() = p 2 m2 and () = +m k() = +mp 2 m2 ; (3.7) here p z is chosen for z 2 C n [0;1) such that Impz > 0. For d 2 f1; 2; 3g the integral kernel G;d is explicitly given by G;1(x) = i 2 eik()jxj  () sgn (x) sgn (x) ()1  ; G;2(x) = k() 2 K1 ik()jxj  xjxj + 12K0 ik()jxjI2 +m3; (3.8) G;3(x) =  I4 +m 0 + (1 ik()jxj) i(  x)jxj2  1 4jxje ik()jxj; cf. (Albeverio et al., 2005; Behrndt et al., 2020; Thaller, 1992; Weidmann, 2003). Next, we consider external potential fields in which the particle moves. Since we are studying relativistic effects, these potentials must be invariant under Lorentz transformations. For a given scalar potential s the quantity V = s 0 is Lorentz invariant as shown in Thaller (1992). This motivates the following formal ansatz for the Dirac operator corresponding to a relativistic quantum particle with spin 1=2 moving in an external field consisting of a scalar potential s: A = A0 + s 0: Of particular interest are strongly localized fields, i.e., fields that only have an effect in a small neighborhood of a set   Rd with measure 0. An example of a field of this kind is the quark 22 Acta Wasaensia confinement inside a nucleon in the form of the MIT bag model. To describe these strongly localized fields it is often a useful simplification to replace them by -potentials which are supported on . In the following we consider a Lorentz scalar potential which is strongly localized in a neighborhood of the hypersurface   Rd and approximate it by a -potential supported on . Applying the formal ansatz above for the Dirac operator yields the formal expression A = A0 +  0 (3.9) with interaction strength  2 R. In the following sections, this operator will be defined in a math- ematically rigorous way and its properties will be studied. Recall from (3.5) that the free Dirac operator A0 is not bounded from below and hence the usual form approach to construct self-adjoint realizations with singular perturbations is not applicable. 4 One-dimensional Dirac operators with Lorentz scalar -point interactions In this section, one-dimensional Dirac operators with Lorentz scalar -interactions supported on  = f0g will be investigated. The following results are well known, see for instance (Pankrashkin & Richard, 2014), but are presented here for the sake of completeness. In particular, the methods used and the results obtained in the discussion will serve as a motivation for the analysis of two- and three-dimensional Dirac operators in the following section. As already mentioned in the previous section, it is well known that the free Dirac operator A0f = i1 d dx f +m3f; dom A0 = H 1(R;C2); is self-adjoint in the Hilbert space L2(R;C2). In accordance with (3.9), Lorentz scalar -interactions will now be considered, which are represented by the formal expression A = A0 + 3: (4.1) Here  2 R corresponds to the constant interaction strength. Following the usual construction of self-adjoint realizations of the expression above as in Albeverio et al. (2005), one first defines the symmetric operator Sf := i1 d dx f +m3f; dom S := H10 (0;1);C2H10 ((1; 0);C2): It can be shown that the adjoint operator S acts in the same way as S, but has the larger domain dom S = H1((0;1);C2)H1((1; 0);C2): In the next step, self-adjoint extensions of S are defined by restricting S to a suitable domain of definition. This domain is characterized by imposing certain coupling conditions on  = f0g, which are found by a formal integration of the expression (4.1). In the present case the coupling conditions Acta Wasaensia 23 for a spinor f = (f1; f2) have the form i (f2(0+) f2(0)) =  2 (f1(0+) + f1(0)) ; i (f1(0+) f1(0)) =  2 (f2(0+) + f2(0)) : (4.2) Next, we define the two linear mappings 0;1 : dom S ! C2 by the assignments 0f := i  f2(0+) f2(0) f1(0+) f1(0)  and 1f := 1 2  f1(0+) + f1(0) f2(0+) + f2(0)  : (4.3) Using these boundary maps one obtains the equivalent representation 0f + 31f = 0; f 2 dom S; of the coupling conditions in (4.2). Proposition 4.1. The triplet fC2;0;1g is an ordinary boundary triplet for S. Proof. Integration by parts and a straightforward computation shows that the abstract Green’s iden- tity in Definition 2.1 is valid. If one defines the function f(x) = i 2  c2 c1  sgn (x)ejxj +  c3 c4  ejxj; x 2 R; for a given vector (c1; c2; c3; c4) 2 C4, then f 2 dom S and the surjectivity of the mapping (0;1) > : dom S ! C4 follows. This shows (ii) in Definition 2.1. Finally, to show that Def- inition 2.1 (iii) holds, notice that the restriction A0 = S  ker 0 corresponds to the free Dirac operator. Hence, it follows that the triplet is an ordinary boundary triplet. Using the ordinary boundary triplet from Proposition 4.1, one can now define the operator A = S   ker(0 + 31); which is interpreted as the realization of the formal expression (4.1) on the basis of the coupling conditions (4.2). Due to  2 R it follows immediately thatA is a self-adjoint operator inL2(R;C2); see the discussion before Theorem 2.3 with # = 13, which is self-adjoint. Next we derive an explicit resolvent formula for A and characterize its spectrum. For this purpose, the first step is to determine the -field and the Weyl function of the ordinary boundary triplet from Proposition 4.1. To simplify the presentation, we first define the two functions f1(x) := i 2  () sgn (x)  eik()jxj and f2(x) := i 2  sgn (x) ()1  eik()jxj with k() and () defined as in (3.7). Note that these functions form a basis of ker(S ) for all  2 (A0) and are mapped to the basis vectors (1; 0) and (0; 1) of C2 by 0. A simple computation now shows that the -field is given by ()  1 2  (x) = 1f1(x) + 2f2(x) = i 2 eik()jxj  () sgn (x) sgn (x) ()1  1 2  24 Acta Wasaensia for (1; 2) 2 C2 and x 2 R, while the Weyl function corresponds to the matrix M() = i 2  () 0 0 ()1  : Note that the x-dependent part in the representation of the -field corresponds to the Green’s function of the free Dirac operator. This will remain valid also in the multi-dimensional considerations in the next section. Using the above representations of the -field and the Weyl function the next result follows from Theorem 2.3. Proposition 4.2. For all  2 (A ) \ (A0) and f 2 L2(R;C2) the resolvent formula (A )1f(x) = (A0 )1f(x) +  2 (2 + i())  () sgn ()  eik()jj; f  L2(R;C2)  () sgn (x)  eik()jxj () 2 (2() i)  sgn () ()1  eik()jj; f  L2(R;C2)  sgn (x) ()1  eik()jxj is valid for all x 2 R. Furthermore, the spectrum of A is given by ess(A ) = (1;m] [ [m;1); disc(A ) = ( ;; if   0;n m 424+2 o ; if  < 0: Proof. From Theorem 2.3 (iii) the representation (A )1f = (A0 )1f ()3(I + M()3)1 ()f follows for all  2 (A )\(A0). After a simple calculation, using the above expressions for the - field and the Weyl function, one obtains the claimed resolvent representation for all f 2 L2(R;C2). The statement about the essential spectrum follows from the fact that bothA andA0 are self-adjoint extensions of the operator S, which has the finite defect indices (2; 2). It remains to show the claim about the discrete spectrum. Notice first that disc(A )  (m;m)  (A0): Thus, it follows from Theorem 2.3 (i) that  2 disc(A ) if and only if 0 2 (I + M()3). The eigenvalues of this matrix can be determined in an elementary way and one obtains the defining equations 2 + i() = 0 or 2() i = 0: If the first equality holds, then there exist an eigenvalue if and only if  < 0 (due to the choice of the complex square root in (3.7)). This eigenvalue is then given by 1 = m 4 2 4 + 2 : If the second equation holds, then a similar reasoning yields the eigenvalue 2 = 1. Acta Wasaensia 25 5 Boundary triplets for two- and three-dimensional Dirac operators with singular interactions In this section we use boundary mappings similar to those in Section 4 to construct boundary triplets for Dirac operators with -shell interactions in R2 and R3. However, by translating the boundary mappings in (4.3) directly to the higher-dimensional setting one obtains a generalized or quasi boundary triplet instead of an ordinary boundary triplet. Before we can introduce the boundary triplets, some preliminaries related to function spaces and trace theorems are needed. For smooth surfaces similar boundary triplets and Sobolev spaces were used in (Behrndt et al., 2018; 2019; Behrndt & Holzmann, 2020; Holzmann, Ourmières-Bonafos & Pankrashkin, 2018) and (Behrndt et al., 2020; Benguria et al., 2017; Ourmières-Bonafos & Vega, 2018), respectively; it is one of the main goals of this note to extend these constructions to closed Lipschitz smooth hypersurfaces. As an application we prove that Dirac operators with Lorentz scalar -shell interactions supported on general compact Lipschitz hypersurfaces are self-adjoint. 5.1 Sobolev spaces for Dirac operators and related trace theorems As in Section 3 the space dimension is denoted by d 2 f2; 3g while N := 2[(d+1)=2], where [] is the Gauss bracket. Consequently, we have N = 2 for d = 2 and N = 4 for d = 3. Let 0; : : : ; d be the d+ 1 anti-commuting CNN -valued Dirac matrices defined in Section 3. Throughout this subsection let  Rd be a bounded or unbounded Lipschitz domain with compact boundary and denote by  the unit normal vector field at @ . For s 2 [0; 1] we define the space Hs ( ;CN ) :=  f 2 Hs( ;CN ) : (  r)f 2 L2( ;CN ) ; where the derivatives are understood in the distributional sense and Hs( ;CN ) is the standard L2- based Sobolev space of order s of CN -valued functions, and we endow it with the norm kfk2Hs ( ;CN ) := kfk 2 Hs( ;CN ) + k(  r)fk2L2( ;CN ): One can show with standard techniques that Hs ( ;CN ) is a Hilbert space and that C10 ( ;CN ) is dense in Hs ( ;CN ); cf. (Benguria et al., 2017: Lemma 2.1), (Behrndt & Holzmann, 2020: Lemma 3.2), or (Ourmières-Bonafos & Vega, 2018: Proposition 2.12) for similar arguments. More- over, with the help of the Fourier transform one sees that Hs (Rd;CN ) = H1(Rd;CN ) for any s 2 [0; 1]. In the following lemma we state a trace theorem for Hs ( ;CN ) when s  12 . Lemma 5.1. For s 2 [ 12 ; 1] the map C10 ( ;CN ) 3 f 7! f j@ extends to a unique continuous operator D : Hs ( ;CN )! Hs1=2(@ ;CN ). Proof. For s 2 ( 12 ; 1] the claim follows from the classical trace theorem (McLean, 2000: Theo- rem 3.38), as Hs ( ;CN ) is continuously embedded in Hs( ;CN ). For s = 12 we consider for s1; s2 2 R the Hilbert space Hs1;s2 ( ;C N ) :=  f 2 Hs1( ;CN ) : f 2 Hs2( ;CN ) (5.1) 26 Acta Wasaensia endowed with the norm kfk2 H s1;s2  ( ;CN ) := kfk2Hs1 ( ;CN ) + kfk2Hs2 ( ;CN ): It follows from Gesztesy & Mitrea (2011: Lemma 3.1) that there exists a continuous trace map from H 1=2;1  ( ) toL 2(@ ). Since (3.3) implies ( r)2 =  in the distributional sense,H1=2 ( ;CN ) is continuously embedded in H1=2;1 ( ;CN ). This yields the claim also for s = 1 2 . Using Lemma 5.1 as well as the fact that C10 ( ;CN ) is dense in Hs ( ;CN ), one can show for all f; g 2 Hs ( ;CN ), s 2 [ 12 ; 1], the following integration by parts formulaZ i(  r)f  g dx = Z @ i(  )f  g d + Z f  i(  r)g dx: (5.2) In the construction of boundary triplets for Dirac operators with singular interactions some families of integral operators related to the fundamental solution G;d given in (3.8) are required. Assume that   Rd is a closed bounded Lipschitz hypersurface and that + is the bounded Lipschitz domain with @ + = , let  be the unit normal vector field at  pointing outwards of +, and let := Rd n +. We introduce the potential operator  : L2(;CN ) ! L2(Rd;CN ) for  =2 (1;m] [ [m;1) by '(x) := Z  G;d(x y)'(y) d(y); ' 2 L2(;CN ); x 2 Rd n ; (5.3) and the strongly singular boundary integral operator C : L2(;CN ) ! L2(;CN ) by the follow- ing limit C'(x) := lim "&0 Z nB(x;") G;d(x y)'(y) d(y); ' 2 L2(;CN ); x 2 ; (5.4) where B(x; ") is the ball of radius " centered at x. Both operators  and C are well defined and bounded, see (Arrizabalaga, Mas & Vega, 2014: Lemma 3.3) and the references there. Moreover, for  2 (m;m) the operator C is self-adjoint in L2(;CN ). In the next lemma we improve the mapping properties for . Lemma 5.2. For any  2 (A0) the operator  gives rise to a bounded map  : L 2(;CN )! H1=2 (Rd n ;CN ): Proof. Let SL() = ()1 0D be the single layer potential for , where 0D is the dual of the Dirichlet trace operator. Using that (3.3) implies (  r)2 =  in the distributional sense one gets  = i  r+m 0 + IN )SL(2 m2)IN ; see also (3.6). Since SL(2 m2) : L2()! H3=2;0 (Rd n ) is bounded, where H3=2;0 (Rd n ) is defined by (5.1) (this follows, e.g., from (Gesztesy & Mitrea, 2009: Equation (2.127))), the claimed result follows. Acta Wasaensia 27 Finally, we note that for ' 2 L2(;CN ) the trace of ', which is well defined by Lemmas 5.1 and Lemma 5.2, is given by D' =  i 2 (  )'+ C'; (5.5) where D denotes the trace operator for ; this can be shown in the same way as in (Arrizabalaga, Mas & Vega, 2014: Lemma 3.3) or (Behrndt et al., 2020: Proposition 3.4). 5.2 Quasi boundary triplets and generalized boundary triplets for Dirac operators with singular interactions In this subsection we follow ideas from Section 4 and introduce a family of quasi boundary triplets for Dirac operators; similar constructions can also be found in Behrndt et al. (2018) and Behrndt & Holzmann (2020). Let +  Rd be a bounded Lipschitz domain and set := Rd n +,  := @ + = @ . We denote by  the unit normal vector field at  that is pointing outwards of +. In the following we will often denote the restriction of a function f defined on Rd onto  by f. We introduce for s 2 [0; 1] the operators T (s) in L2(Rd;CN ) by T (s)f := (i(  r) +m 0)f+  (i(  r) +m 0)f; dom T (s) := Hs ( +;CN )Hs ( ;CN ); and S := T (s)  H10 (Rd n ;CN ), which is given more explicitly by Sf = (i(  r) +m 0)f; dom S = H10 (Rd n ;CN ): The operator S is densely defined, closed, and symmetric. Using standard arguments and distribu- tional derivatives one verifies that S = T (0) and (T (0)) = S: Next, we introduce for s 2 [ 12 ; 1] the mappings (s)0 ;(s)1 : dom T (s) ! L2(;CN ) by (s) 0 f := i(  )(f+j fj) and (s)1 f := 1 2 (f+j + fj); (5.6) and note that (s)0 and (s) 1 are well defined due to Lemma 5.1. In order to characterize the range of (s) 0 , we introduce the space Hs (;CN ) :=  ' 2 L2(;CN ) : (  )' 2 Hs(;CN ) ; whereHs(;CN ) denotes the standard Sobolev space on  ofCN -valued functions. If  isC1;s+"- smooth for some " > 0, then Hs (;CN ) = Hs(;CN ), cf. (Behrndt, Holzmann & Mas, 2020: Lemma A.2). In the following theorem we show that the mappings (s)0 and (s) 1 in (5.6) give rise to a quasi boundary triplet for S and we compute the associated -field and Weyl function. Recall that A0 is the free Dirac operator defined in (3.4), and that  and C are the mappings introduced in (5.3) and (5.4), respectively. 28 Acta Wasaensia Theorem 5.3. Let s 2 [ 12 ; 1]. Then the following statements hold: (i) The triplet  L2(;CN );(s)0 ; (s) 1 is a quasi boundary triplet for S = T (s) such that T (s)  ker (s)0 = A0, and one has ran (s) 0 = H s1=2 (;CN ): (5.7) In particular,  L2(;CN );(1=2)0 ; (1=2) 1 is a generalized boundary triplet. (ii) For  2 (A0) = C n ((1;m] [ [m;1)) the values (s)() of the -field are given by (s)() =   Hs1=2 (;CN ): Each (s)() is a densely defined bounded operator from L2(;CN ) to L2(Rd;CN ) and an everywhere defined bounded operator from Hs1=2 (;CN ) to Hs (Rd n ;CN ). Moreover, (s)() : L2(Rd;CN )! L2(;CN ) is compact. (iii) For  2 (A0) = C n ((1;m] [ [m;1)) the values M (s)() of the Weyl function are given by M (s)() = C  Hs1=2 (;CN ): Each M (s)() is a densely defined bounded operator in L2(;CN ) and a bounded every- where defined operator from Hs1=2 (;CN ) to Hs1=2(;CN ). Proof. Let s 2 [ 12 ; 1] be fixed. First, we show that  L2(;CN );(s)0 ; (s) 1 is a quasi boundary triplet. For this we note that T (s) = T (0) = S, as C10 ( ;CN ) is dense in H0 ( ;CN ), while the norm in H0 (R3 n ;CN ) and the graph norm induced by T (0) are equivalent. Next, we verify that Green’s identity in Definition 2.1 (i) is fulfilled. For this let f = f+  f; g = g+  g 2 dom T (s) = Hs ( +;CN )Hs ( ;CN ): Then integration by parts (5.2) applied in  yields (i(  r) +m 0)f; g  L2( ;CN ) f; (i(  r) +m 0)g  L2( ;CN ) =  i(  )fj; gjL2(;CN ); where it is used that  is the normal vector field pointing outwards of . By adding these two formulas for + and one arrives at Green’s identity. Next, we show that T (s)  ker (s)0 = A0. As the free Dirac operator A0 is self-adjoint, this shows that T (s)  ker (s)0 is self-adjoint. The inclusion A0  T (s)  ker (s)0 is clear. To verify the converse inclusion, let f 2 ker (s)0 . Then Green’s identity yields for any ' 2 C10 (Rd;CN ) f;i(  r)' L2(Rd;CN ) = (T (s) m 0)f; '  L2(Rd;CN ): (5.8) Hence, (  r)f 2 L2(Rd;CN ), which shows f 2 Hs (Rd;CN ) = H1(Rd;CN ) = dom A0. Therefore, we conclude that T (s)  ker (s)0 = A0 holds. Acta Wasaensia 29 It remains to prove that ran ((s)0 ; (s) 1 ) is dense in L 2(;CN ) L2(;CN ). For this, we prove ran ( (s) 0  ker (s) 1 ) = H 1=2 (;CN ) (5.9) and ran ( (s) 1  ker (s) 0 ) = H 1=2(;CN ): (5.10) To establish the inclusion "" in (5.9) we note first that any function f 2 ker (s)1 satisfies the equality f+j = fj. One can show as in (5.8) that f+(f) 2 Hs (Rd;CN ) = H1(Rd;CN ) and thus, f 2 H1( +;CN )  H1( ;CN ). Therefore, the definition of (s)0 yields the claimed inclusion. For the converse inclusion "" let ' 2 H1=2 (;CN ). Choose f 2 H1( ;CN ) such that fj =  i2 (  )'. Then f 2 ker (s)1 and (s)0 f = '. Since this can be done for all functions ' 2 H1=2 (;CN ), we have shown (5.9). To verify (5.10) note that the inclusion "" follows from ker (s)0 = H1(Rd;CN ) and the definition of (s)1 . For the converse inclusion "" let ' 2 H1=2(;CN ). Choose f 2 H1(Rd;CN ) such that f j = '. Then f 2 ker (s)0 and (s)1 f = '. Since this can be done for all ' 2 H1=2(;CN ), we have verified (5.10). Hence, we have shown that  L2(;CN );(s)0 ; (s) 1 is indeed a quasi boundary triplet for all s 2 [ 12 ; 1]. Thus, except for formula (5.7), assertion (i) has been shown. Equation (5.7) will be proved together with items (ii) and (iii). Next, we show that (s)() is compact for all s. For this purpose, recall that formula (2.2) implies that (s)() = (s)1 (A0 )1. Since (A0 )1 : L2(Rd;CN )! H1(Rd;CN ) is bounded, we see that (s)() is actually independent of s and, furthermore, that (s)() : L2(Rd;CN )! H1=2(;CN ) is also bounded. Since H1=2(;CN ) is compactly embedded in L2(;CN ), the claimed compact- ness of (s)() follows. The remaining assertions will be proved in three steps for various values of s 2 [ 12 ; 1]. Step 1 for s = 12 . Consider for ' 2 L2(;CN ) the function f := ', see (5.3). Then one has f 2 H1=2 (Rd n ;CN ) = dom T (1=2) by Lemma 5.2, and by (5.5) we get (1=2)0 f = '. Therefore, ran (1=2)0 = L 2(;CN ), which is (5.7) for s = 12 . Moreover, as G;d in (3.8) is a fundamental solution for the Dirac equation, the definition of  shows that T (1=2) f = 0 in Rd n : Consequently, (1=2)() = . Finally, using the definition of (1=2) 1 and (5.5), it follows that M (1=2)() = C and thus, M (1=2)() is bounded in L2(;CN ). This shows all claims for s = 12 . Step 2 for s = 1. First note that dom T (1) = H1(Rdn;CN ), the definition of (1)0 , and (5.9) imply ran (1) 0 = H 1=2 (;CN ). As fL2(;CN );(1)0 ;(1)1 g is a restriction of the triplet for s = 12 , we deduce from the already shown results that (1)() = (1=2)()  ran (1)0 =   H1=2 (;CN ) and M (1)() = M (1=2)()  ran (1)0 = C  H1=2 (;CN ): 30 Acta Wasaensia Using the closed graph theorem and the fact that H1=2 (;CN ) and H1(Rd n;CN ) are boundedly embedded in L2(;CN ) and L2(Rd;CN ), respectively, one gets that (1)() : ran (1) 0 = H 1=2 (;CN )! dom T (1) = H1(Rd n ;CN ) is bounded as well. The mapping properties of the trace map yield that also M (1)() : ran (1) 0 = H 1=2 (;CN )! ran (1)1 = H1=2(;CN ) is bounded. Hence, all claimed statements for s = 1 have been shown. Step 3 for s 2 ( 12 ; 1). First we note that an interpolation argument shows that  : H s1=2(;CN )! Hs (Rd n ;CN ) = dom T (s) is bounded. Together with (5.5) this implies that ran (s)0 = H s1=2 (;CN ), i.e., (5.7), holds for s 2 ( 12 ; 1). Hence, we have (s)() =   Hs1=2 (;CN ) and the trace theorem shows that M (s)() = (s) 1 (s)() : Hs1=2 (;CN )! Hs1=2(;CN ) is bounded. Thus, all claims are proved. In the next theorem we study the self-adjointness of a Dirac operatorA with a Lorentz scalar -shell interaction of strength  2 R n f0g, which is formally given by i(  r) + m 0 +  0. In a similar way as in (4.2) we define the operator A by A := T (1=2)  ker (1=2) 0 +  0 (1=2) 1  : The operator A is given more explicitly by Af = (i(  r) +m 0)f+  (i(  r) +m 0)f; dom A = n f 2 H1=2 (Rd n ;CN ) : i(  )(f+j fj) +  2 0(f+j + fj) = 0 o : This operator was investigated under various assumptions in Behrndt et al. (2019; 2020); Holzmann, Ourmières-Bonafos & Pankrashkin (2018); Pizzichillo & Van Den Bosch (2019). In the following theorem it is shown, for the first time, that A is self-adjoint when the interaction support   Rd is an arbitrary closed bounded Lipschitz smooth hypersurface. Theorem 5.4. For any  2 Rnf0g the operatorA is self-adjoint in L2(Rd;CN ) and the following statements hold: (i) For  2 (A ) the resolvent of A is given by (A )1 = (A0 )1   1  0 + C 1   : (ii) ess(A ) = ess(A0) = (1;m] [ [m;1). (iii) disc(A ) is finite and  2 disc(A ) if and only if 0 2 p( 1 0 + C). Acta Wasaensia 31 Remark 5.5. By Theorem 5.4 the operator A is self-adjoint defined on a subset of H1=2 (Rd n ;CN ): If  is a smooth hypersurface, then it is known that A is self-adjoint and dom A  H1(Rd n ;CN ); see (Behrndt et al., 2020; Holzmann, Ourmières-Bonafos & Pankrashkin, 2018). However, for more general Lipschitz smooth hypersurfaces this smoothness in the operator domain can not be expected, as was shown explicitly in (Le Treust & Ourmières-Bonafos, 2018: Remark 1.9) in the two-dimensional setting for polygonal domains. Proof of Theorem 5.4. In order to show the self-adjointness of A , it suffices, according to Theo- rem 2.3 and the discussion following it, to verify that ran (1=2) 1 (A0  i)1  = H1=2(;CN )  ran  1  0 +M (1=2) i  = ran  1  0 + Ci  : (5.11) In order to see this, we prove that 1 0 +Ci is bijective in L2(;CN ). First, we note that 1 0 +Ci is injective, as otherwise the symmetric operator A would have the non-real eigenvalue i by Theorem 2.3. Next, by (2.3) we have that Ci = M (1=2)(i) = M (1=2)(0) i (1=2)(0) (1=2)(i) = C0 +Ki and note that Ki = i (1=2)(0) (1=2)(i) is compact in L2(;CN ) due to Theorem 5.3 (ii). Next, we compute 1  0 + Ci 2 = 1 2 IN + C20 + 1  0Ci + Ci 0  +K2i + C0Ki +KiC0: Since C0 is self-adjoint, the operator 12 IN +C20 is a strictly positive self-adjoint operator and, hence, it is a Fredholm operator with index zero. Next, due to the anti-commutation relation (3.3) it is not difficult to see that 1  0Ci + Ci 0  = 1  (2m 2i 0)S(m2 1); where S() is the single layer boundary integral operator for  . According to (Holzmann & Unger, 2020: Lemma 3.4) the latter operator is compact. Since also Ki is compact, we conclude that 1  0 + Ci 2 must be a Fredholm operator with index zero. Since 1 0 + Ci is injective, we conclude that 1  0 + Ci 2 is also injective and hence, as it has Fredholm index zero, it must be surjective. Therefore 1 0 + Ci is also bijective. This shows that (5.11) holds and thus, A is self-adjoint. Next, by Theorem 2.3 the claimed resolvent formula in (i) holds for  = i. The map 1 0 + Ci is bijective and, hence, boundedly invertible. This, the mapping properties of i and i from Theorem 5.3, and Kreı˘n’s resolvent formula imply assertion (ii). The resolvent formula in item (i) for  2 (A ) is now a direct consequence of Theorem 2.3. 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Institut für Angewandte Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria E-mail address: behrndt@tugraz.at, holzmann@math.tugraz.at, christian.stelzer09@gmail.at, and gstenzel@math.tugraz.at Acta Wasaensia 37 THE ORIGINAL WEYL-TITCHMARSH FUNCTIONS AND SECTORIAL SCHRÖDINGER L-SYSTEMS Sergey Belyi and Eduard Tsekanovskiı˘ Dedicated with great pleasure to Seppo Hassi on the occasion of his 60th birthday 1 Introduction This paper is part of an ongoing project studying the realizations of the original Weyl-Titchmarsh function m1(z) and its linear-fractional transformation m (z) associated with a Schrödinger op- erator in L2[`;+1). In this project the Herglotz-Nevanlinna functions m1(z) and 1=m1(z) as well as m (z) and 1=m (z) are being realized as impedance functions of L-systems with a dis- sipative Schrödinger main operator Th, Imh > 0. For the sake of brevity we will refer to these L-systems as Schrödinger L-systems. The formal definition, exposition, and discussion of general and Schrödinger L-systems are presented in Sections 2 and 4. We capitalize on the fact that all Schrödinger L-systems ;h form a two-parametric family whose members are uniquely defined by a real-valued parameter  and a complex boundary value h of the main dissipative operator. The focus of this paper is on the case when the realizing Schrödinger L-systems are based on a non-negative symmetric Schrödinger operator with deficiency indices (1; 1) that has an accretive state-space operator. It is known that in this case the impedance functions of such L-systems are Stieltjes functions, see (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011). Here we study the situation when the realizing Schrödinger L-systems are also sectorial and the Weyl-Titchmarsh functions m (z) fall into sectorial classes S and S 1; 2 of Stieltjes functions that are discussed in detail in Section 3. Section 5 provides us with the general realization results (obtained in Belyi & Tsekanovskiı˘ (2021)) for the functions m1(z), 1=m1(z), and m (z). It is shown that m1(z), 1=m1(z), and m (z) can be realized as impedance functions of Schrödinger L-systems 0;i, 1;i, and tan ;i, respectively. The main results of the paper are contained in Section 6. There we apply the realization theorems from Section 5 to Schrödinger L-systems that are based on a non-negative symmetric Schrödinger operator to obtain additional properties. Utilizing the results presented in Section 4, we derive some new features of Schrödinger L-systems tan ;i whose impedance functions fall into particular sectorial classes S 1; 2 with 1 and 2 explicitly described. The results are given in terms of the parameter that appears in the definition of the function m (z). Moreover, the knowledge of the limit value m1(0) and the value of allow us to find the angle of sectoriality of the main and state-space operators of the realizing L-system. This, in turn, leads to connections to Kato’s problem about sectorial extensions of sectorial forms. The paper is concluded with an example that illustrates the main results and concepts. The present work is a further development of the theory of open physical systems conceived by M. Livs˘ic (1966). 38 Acta Wasaensia 2 Preliminaries For a pair of Hilbert spacesH1 andH2 we denote by [H1;H2] the set of all bounded linear operators from H1 to H2. Let _A be a closed, densely defined, symmetric operator in a Hilbert space H with inner product (f; g), where f; g 2 H. Any non-symmetric operator T in H such that _A  T  _A will be called a quasi-self-adjoint extension of _A. Consider the rigged Hilbert spaceH+  H  H, whereH+ = dom _A and (f; g)+ = (f; g) + ( _A f; _Ag); f; g 2 dom A; see (Berezansky, 1968; Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011). Let R be the Riesz-Berezansky operator R from H onto H+ such that (f; g) = (f;Rg)+ and kRgk+ = kgk, where f 2 H+, g 2 H, see (Berezansky, 1968; Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011). Identifying the space conjugate to H with H, we get that if A 2 [H+;H], then A 2 [H+;H]. An operator A 2 [H+;H] is called a self-adjoint bi-extension of a symmetric operator _A if A = A and A  _A. Let A be a self-adjoint bi-extension of _A and let the operator A^ inH be defined as follows dom A^ = ff 2 H+ : Af 2 Hg; A^ = A dom A^: Then the operator A^ is called a quasi-kernel of a self-adjoint bi-extension A, see (Tsekanovskiı˘ & S˘muljan, 1977) and (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011: Section 2.1). A self-adjoint bi-extension A of a symmetric operator _A is called t-self-adjoint if its quasi-kernel A^ is a self-adjoint operator in H, see (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011: Definition 4.3.1). An operator A 2 [H+;H] is called a quasi-self-adjoint bi-extension of an operator T if A  T  _A and A  T   _A. We will be mostly interested in the following type of quasi-self-adjoint bi-extensions. Let T be a quasi-self-adjoint extension of _A with nonempty resolvent set (T ). A quasi-self-adjoint bi- extension A of an operator T is called a ()-extension of T if ReA is a t-self-adjoint bi-extension of _A, see (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011: Definition 3.3.5). In what follows we assume that _A has deficiency indices (1; 1). In this case it is known that every quasi-self-adjoint extension T of _A admits ()-extensions, see (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011). The description of all ()- extensions via the Riesz-Berezansky operator R can be found in Arlinskiı˘, Belyi & Tsekanovskiı˘ (2011: Section 4.3). Recall that a linear operator T in a Hilbert spaceH is called nonnegative, accretive, or dissipative if (Tf; f)  0, Re (Tf; f)  0, or Im (Tf; f)  0 for all f 2 dom T respectively, see (Kato, 1966). An accretive operator T is called -sectorial if there exists a value 2 (0; =2) such that (cot )jIm (Tf; f)j  Re (Tf; f); f 2 dom T; (2.1) see (Kato, 1966). We say that the angle of sectoriality is exact for a -sectorial operator T if tan = supf2domT jIm (Tf; f)j Re (Tf; f) : An accretive operator is called extremal accretive if it is not -sectorial for any 2 (0; =2). A ()-extension A of T is called accretive if Re (Af; f)  0 for all f 2 H+. This is equivalent to the real part ReA = (A+ A)=2 being a nonnegative t-self-adjoint bi-extension of _A. Acta Wasaensia 39 The following definition is a "lite" version of the definition of an L-system given for a scattering L-system with a one-dimensional input-output space. It is tailored to the case when the symmetric operator of an L-system has deficiency indices (1; 1). The general definition of an L-system can be found in Arlinskiı˘, Belyi & Tsekanovskiı˘ (2011: Definition 6.3.4), see also (Belyi et al., 2006) for a non-canonical version. Definition 2.1. An array  =  A K 1 H+  H  H C  (2.2) is called an L-system if: (1) T is a dissipative quasi-self-adjoint extension of a symmetric operator _A with deficiency indices (1; 1); (2) A is a ()-extension of T ; (3) ImA = KK, where K 2 [C;H] and K 2 [H+;C]. The operators T and A in the above definition are called the main and state-space operator, respec- tively, of the L-system , and K is called a channel operator. It is easy to see that the operator A of the system (2.2) is such that ImA = (; ),  2 H and pick Kc = c  , c 2 C, see (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011). The L-system  in (2.2) is called minimal if the operator _A is a prime operator in H, i.e., if there does not exist a non-trivial reducing invariant subspace of H on which it induces a self-adjoint operator. Minimal L-systems of the form (2.2) with a one-dimensional input-output space were also considered in Belyi, Makarov & Tsekanovskiı˘ (2015). We associate with an L-system  the function W(z) = I 2iK(A zI)1K; z 2 (T ); which is called the transfer function of the L-system . We also consider the function V(z) = K (ReA zI)1K; (2.3) which is called the impedance function of an L-system  of the form (2.2). The transfer function W(z) of the L-system  and the impedance function V(z) of the form (2.3) are connected by the following relations valid for Im z 6= 0 and z 2 (T ): V(z) = i[W(z) + I] 1[W(z) I]; W(z) = (I + iV(z)) 1(I iV(z)): An L-system  of the form (2.2) is called an accretive L-system if its state-space operator A is accretive, that is Re (Af; f)  0 for all f 2 H+, see (Belyi & Tsekanovskiı˘, 2008; Dovzhenko & Tsekanovskiı˘, 1990). An accretive L-system is called sectorial if the operator A is sectorial, i.e., if it satisfies (2.1) for some 2 (0; =2) and all f 2 H+. 40 Acta Wasaensia 3 Sectorial classes and their realizations A scalar function V (z) is called a Herglotz-Nevanlinna function if it is holomorphic on C n R, sym- metric with respect to the real axis, i.e., V (z) = V (z), z 2 C n R, and satisfies the positivity condition ImV (z)  0, z 2 C+. The class of all Herglotz-Nevanlinna functions that can be real- ized as impedance functions of L-systems and connections with Weyl-Titchmarsh functions can be found in Arlinskiı˘, Belyi & Tsekanovskiı˘ (2011), Belyi, Makarov & Tsekanovskiı˘ (2015), Derkach, Malamud & Tsekanovskiı˘ (1989), Gesztesy & Tsekanovskiı˘ (2000), and references therein. The following definition can be found in Kac & Krein (1974): a scalar Herglotz-Nevanlinna function V (z) is a Stieltjes function if it is holomorphic in Ext[0;+1) and Im [zV (z)] Im z  0: It is known that a Stieltjes function V (z) admits the following integral representation V (z) = + 1Z 0 dG(t) t z ; (3.1) where  0 and G(t) is a non-decreasing function on [0;+1) with R1 0 dG(t) 1+t < 1, see (Kac & Krein, 1974). We are going to focus on the class S0(R) of scalar Stieltjes functions such that the measure G(t) in the representation (3.1) is of unbounded variation, see (Belyi & Tsekanovskiı˘, 2008; Dovzhenko & Tsekanovskiı˘, 1990; Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011). It was shown in Arlinskiı˘, Belyi & Tsekanovskiı˘ (2011), see also (Belyi & Tsekanovskiı˘, 2008), that such a function V (z) can be realized as the impedance function of an accretive L-system  of the form (2.2) with a densely defined symmetric operator if and only if it belongs to the class S0(R). Now we are going to consider sectorial subclasses of scalar Stieltjes functions introduced in Alpay & Tsekanovskiı˘ (2000). Let 2 (0; 2 ). The sectorial subclass S of Stieltjes functions consists of all scalar functions V (z) for which nX k;l=1  zkV (zk) zlV (zl) zk zl (cot ) V (zl)V (zk)  hkhl  0; for arbitrary sequences of complex numbers fzkg, Im zk > 0, and fhkg. For 0 < 1 < 2 < 2 , we have S 1  S 2  S; where S denotes the class of all Stieltjes functions (which corresponds to the case = 2 ). Let  be a minimal L-system of the form (2.2) with a densely defined non-negative symmetric operator _A. Then the impedance function V(z) defined by (2.3) belongs to the class S if and only if the operator A of the L-system  is -sectorial, see (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011). Let 0  1 < 2 , 0 < 2  2 , and 1  2. We say that a scalar Stieltjes function V (z) belongs to the class S 1; 2 if tan 1 = lim x!1V (x) and tan 2 = limx!0 V (x): The following connection between the classes S and S 1; 2 can be found in Arlinskiı˘, Belyi & Acta Wasaensia 41 Tsekanovskiı˘ (2011). Let  be an L-system of the form (2.2) with a densely defined non-negative symmetric operator _A with deficiency numbers (1; 1) and let A be a -sectorial ()-extension of T , then the impedance function V(z), defined by (2.3), belongs to the class S 1; 2 , tan 2  tan . Moreover, the main operator T is ( 2 1)-sectorial with the exact angle of sectoriality 2 1. In the case when is the exact angle of sectoriality of the operator T we have that V(z) 2 S0; , see (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011). It also follows that, under this set of assumptions, the impedance function V(z) is such that = 0 in the representation (3.1). Now let  be an L-system of the form (2.2), where A is a ()-extension of T and _A is a closed densely defined non-negative symmetric operator with deficiency numbers (1; 1). It was proved in Arlinskiı˘, Belyi & Tsekanovskiı˘ (2011) that if the impedance function V(z) belongs to the class S 1; 2 and 2 6= =2, then A is -sectorial, where tan = tan 2 + 2 p tan 1(tan 2 tan 1): Let  be an L-system satisfying the above conditions. Then the operator A is a -sectorial ()- extension of a -sectorial operator T with the exact angle 2 (0; =2) if and only if V(z) 2 S0; , see Arlinskiı˘, Belyi & Tsekanovskiı˘ (2011). Moreover, the angle is determined by the formula tan = Z 1 0 dG(t) t ; where G(t) is the measure from integral representation (3.1) of V(z). 4 L-systems with Schrödinger operator and their impedance functions Let H = L2[`;+1), `  0, and let l(y) = y00 + q(x)y, where q is a real locally summable function on [`;+1). Suppose that the symmetric operator _Ay = y00 + q(x)y; y(`) = y0(`) = 0; (4.1) has deficiency indices (1,1). LetD be the set of functions y for which y and y0 are locally absolutely continuous, and l(y) 2 L2[`;+1). Provide the spaceH+ = dom _A = D with the scalar product (y; z)+ = Z 1 `  y(x)z(x) + l(y)l(z)  dx; y; z 2 D: LetH+  L2[`;+1)  H be the corresponding triplet of Hilbert spaces. Consider the operators Thy = l(y) = y00 + q(x)y; hy(`) y0(`) = 0; and  T hy = l(y) = y00 + q(x)y; hy(`) y0(`) = 0; (4.2) where Imh > 0. Let _A be a symmetric operator of the form (4.1) with deficiency indices (1,1) generated by the differential expression l(y) = y00 + q(x)y. Moreover, let 'k(x; ), k = 1; 2, be 42 Acta Wasaensia the solutions of the following Cauchy problems:8<: l('1) = '1; '1(`; ) = 0; '01(`; ) = 1; and 8<: l('2) = '2; '2(`; ) = 1; '02(`; ) = 0: It is well known that there exists a function m1() introduced by H. Weyl (1909; 1910) for which '(x; ) = '2(x; ) +m1()'1(x; ) belongs to L2[`;+1), see (Naimark, 1968; Levitan, 1987). The function m1() is not a Herglotz- Nevanlinna function, but m1() and 1=m1() are. Now we shall construct an L-system based on a non-self-adjoint Schrödinger operator Th with Imh > 0. It was shown in Arlinskiı˘ & Tsekanovskiı˘ (2004) and Arlinskiı˘, Belyi & Tsekanovskiı˘ (2011) that the set of all ()-extensions of a non-self-adjoint Schrödinger operator Th of the form (4.2) in L2[`;+1) can be represented in the form A;h y = y00 + q(x)y 1  h [y 0(`) hy(`)] [(x `) + 0(x `)]; A;h y = y00 + q(x)y 1  h [y 0(`) hy(`)] [(x `) + 0(x `)]: (4.3) Moreover, the formulas (4.3) establish a one-to-one correspondence between the set of all ()- extensions of a Schrödinger operator Th of the form (4.2) and all  2 [1;+1]. One can easily check that the ()-extension A in (4.3) of the non-self-adjoint dissipative Schrödinger operator Th, Imh > 0, of the form (4.2) satisfies the condition ImA;h = A;h A;h 2i = (; g;h)g;h; (4.4) where g;h = (Imh) 1 2 j hj [(x `) +  0(x `)]: (4.5) Here (x `) and 0(x `) are the delta-function and its derivative at the point `, respectively. Furthermore, (y; g;h) = (Imh) 1 2 j hj [y(`) y 0(`)]; where y 2 H+, g 2 H, and H+  L2[`;+1)  H is the triplet of Hilbert spaces discussed above. It was also shown in Arlinskiı˘, Belyi & Tsekanovskiı˘ (2011) that the quasi-kernel A^ of ReA;h is given by  A^y = y00 + q(x)y; y0(`) = y(`); where  = Reh jhj2  Reh : (4.6) Let E = C and K;hc = cg;h, where c 2 C. Then it is clear that K;hy = (y; g;h); y 2 H+; (4.7) Acta Wasaensia 43 and ImA;h = K;hK;h, see (4.4). Therefore the array ;h =  A;h K;h 1 H+  L2[`;+1)  H C  (4.8) is an L-system with the main operator Th, Imh > 0, of the form (4.2), the state-space operator A;h of the form (4.3), and with the channel operator K;h of the form (4.7). It was established in Arlinskiı˘ & Tsekanovskiı˘ (2004) and Arlinskiı˘, Belyi & Tsekanovskiı˘ (2011) that the transfer and impedance functions of ;h are W;h(z) =  h  h m1(z) + h m1(z) + h and V;h(z) = (m1(z) + ) Imh ( Reh)m1(z) + Reh jhj2 ; respectively. It was shown in Arlinskiı˘, Belyi & Tsekanovskiı˘ (2011: Section 10.2) that if the pa- rameters  and  are related as in (4.6), then the two L-systems ;h and ;h of the form (4.8) have the following properties W;h(z) = W;h(z); V;h(z) = 1 V;h(z) ; with  = Reh jhj2  Reh : 5 Realizations of m1(z), 1=m1(z), and m (z) It is known that the original Weyl-Titchmarsh function m1(z) has the property that m1(z) is a Herglotz-Nevanlinna function, see (Levitan, 1987; Naimark, 1968). The question whetherm1(z) can be realized as the impedance function of a Schrödinger L-system is answered in the following theorem. Theorem 5.1 (Belyi & Tsekanovskiı˘ (2021)). Let _A be a symmetric Schrödinger operator of the form (4.1) with deficiency indices (1; 1) inH = L2[`;1). Ifm1(z) is the Weyl-Titchmarsh function of _A, then the Herglotz-Nevanlinna function m1(z) can be realized as the impedance function of a Schrödinger L-system ;h of the form (4.8) with  = 0 and h = i. Conversely, let ;h be a Schrödinger L-system of the form (4.8) with the symmetric operator _A such that V;h(z) = m1(z) for all z 2 C and  2 R [ f1g. Then the parameters  and h defining ;h are given by  = 0 and h = i. A similar result was proved for the function 1=m1(z). Theorem 5.2 (Belyi & Tsekanovskiı˘ (2021)). Let _A be a symmetric Schrödinger operator of the form (4.1) with deficiency indices (1; 1) inH = L2[`;1). Ifm1(z) is the Weyl-Titchmarsh function of _A, then the Herglotz-Nevanlinna function 1=m1(z) can be realized as the impedance function of a Schrödinger L-system ;h of the form (4.8) with  =1 and h = i. Conversely, let ;h be a Schrödinger L-system of the form (4.8) with the symmetric operator _A such that V;h(z) = 1 m1(z) for all z 2 C and  2 R [ f1g. Then the parameters  and h defining ;h are given by  =1 and h = i. 44 Acta Wasaensia We note that both L-systems 0;i and 1;i, obtained in Theorems 5.1 and 5.2, share the same main operator  Ti y = y00 + q(x)y; y0(`) = i y(`): (5.1) Now we recall the definition of the Weyl-Titchmarsh functions m (z). Let _A be a symmetric operator of the form (4.1) with deficiency indices (1,1) generated by the differential expression l(y) = y00 + q(x)y. Moreover, let ' (x; z) and  (x; z) be the solutions of the following Cauchy problems: 8<: l(' ) = z' ; ' (`; z) = sin ; '0 (`; z) = cos ; and 8<: l( ) = z ;  (`; z) = cos ; 0 (`; z) = sin : Then it is known that there exists a function m (z), analytic in C, for which (x; z) =  (x; z) +m (z)' (x; z) (5.2) belongs to L2[`;+1), see (Danielyan, 1990; Naimark, 1968; Titchmarsh, 1962). It is easy to see that if = , then m(z) = m1(z). The functions m (z) and m1(z) are connected by m (z) = sin +m1(z) cos cos m1(z) sin ; (5.3) see (Danielyan, 1990; Titchmarsh, 1962). We know that for any real the function m (z) is a Herglotz-Nevanlinna function, see (Naimark, 1968; Titchmarsh, 1962). Also, modifying (5.3) slightly, we obtain m (z) = sin +m1(z) cos cos +m1(z) sin = cos + 1m1(z) sin sin 1m1(z) cos : (5.4) The following realization theorem for the Herglotz-Nevanlinna functionsm (z) is similar to The- orem 5.1. Theorem 5.3 (Belyi & Tsekanovskiı˘ (2021)). Let _A be a symmetric Schrödinger operator of the form (4.1) with deficiency indices (1; 1) in H = L2[`;1). If m (z) is the function of _A described in (5.2), then the Herglotz-Nevanlinna function m (z) can be realized as the impedance function of a Schrödinger L-system ;h of the form (4.8) with  = tan and h = i: (5.5) Conversely, let ;h be a Schrödinger L-system of the form (4.8) with the symmetric operator _A such that V;h(z) = m (z); for all z 2 C and  2 R [ f1g. Then the parameters  and h defining ;h are given by (5.5). In case =  we obtain  = 0, m(z) = m1(z), and the realizing Schrödinger L-system 0;i is described in Belyi & Tsekanovskiı˘ (2021: Section 5). In case = =2, then we obtain  =1, m (z) = 1=m1(z), and the realizing Schrödinger L-system is 1;i, see (Belyi & Tsekanovskiı˘, 2021: Section 5). Assuming that 2 (0; ] and neither =  nor = =2, we give the description Acta Wasaensia 45 of a Schrödinger L-system  ;i realizing m (z) as follows: tan ;i =  Atan ;i Ktan ;i 1 H+  L2[`;+1)  H C  ; (5.6) where Atan ;i y = l(y) 1 tan i [y 0(`) iy(`)][(tan )(x `) + 0(x `)]; Atan ;i y = l(y) 1 tan + i [y0(`) + iy(`)][(tan )(x `) + 0(x `)]; (5.7) and Ktan ;i c = c gtan ;i, c 2 C, with gtan ;i = (tan )(x `) + 0(x `): Also, Vtan ;i(z) = m (z); Wtan ;i(z) = tan i tan + i  m1(z) i m1(z) + i = (e2 i) m1(z) i m1(z) + i : The realization theorem for the Herglotz-Nevanlinna function 1=m (z) is similar to Theorem 5.2 and can be found in Belyi & Tsekanovskiı˘ (2021). 6 Non-negative Schrödinger operators and sectorial L-systems Now let us assume that _A is a densely defined non-negative symmetric operator of the form (4.1) with deficiency indices (1,1) generated by the differential expression l(y) = y00 + q(x)y. Theorem 6.1 (Tsekanovskiı˘ (1980; 1981; 1987)). Let _A be a nonnegative symmetric Schrödinger operator of the form (4.1) with deficiency indices (1; 1) inH = L2[`;1) and let the operator Th be given by (4.2). Then the following statements hold: (1) the operator _A has more than one non-negative self-adjoint extension, i.e., the Friedrichs extension AF and the Kreı˘n-von Neumann extension AK do not coincide, if and only if m1(0) <1; (2) the operator Th, h = h, coincides with the Kreı˘n-von Neumann extension AK if and only if h = m1(0); (3) the operator Th is accretive if and only if Reh  m1(0); (4) the operator Th, h 6= h, is -sectorial if and only if Reh > m1(0); (5) the operator Th, h 6= h, is accretive, but not -sectorial for any 2 (0; 2 ) if and only if Reh = m1(0); (6) if the operator Th, Imh > 0, is -sectorial, then the exact angle can be calculated via tan = Imh Reh+m1(0) : (6.1) 46 Acta Wasaensia For the remainder of this paper we assume that m1(0) < 1. Then, according to Theorem 6.1 above, the operator Th, Imh > 0, is accretive and/or sectorial, see also (Arlinskiı˘ & Tsekanovskiı˘, 2009; Tsekanovskiı˘, 1980; 1992). It was shown in Arlinskiı˘, Belyi & Tsekanovskiı˘ (2011) that if Th, Imh > 0, is an accretive Schrödinger operator of the form (4.2), then for all real  satisfying the following inequality   (Imh) 2 m1(0) + Reh + Reh; (6.2) the formulas (4.3) define the set of all accretive ()-extensionsA;h of the operator Th. Moreover, an accretive ()-extension A;h of a sectorial operator Th with exact angle of sectoriality 2 (0; =2) also preserves the same exact angle of sectoriality if and only if  = +1 in (4.3), see (Belyi & Tsekanovskiı˘, 2019: Theorem 3). Also, A;h is an accretive ()-extension of Th that is not - sectorial for any 2 (0; =2) if and only if in (4.3)  = (Imh)2 m1(0) + Reh + Reh; (6.3) see (Belyi & Tsekanovskiı˘, 2019: Theorem 4). An accretive operator Th has a unique accretive ()-extension A1;h if and only if Reh = m1(0). Then this unique ()-extension has the form A1;hy = y00 + q(x)y + [hy(`) y0(`)] (x `); A1;hy = y00 + q(x)y + [hy(`) y0(`)] (x `): (6.4) Now consider the functionsm (z) described by (5.2)-(5.3) and associated with the non-negative op- erator _A above. The parameter in the definition of m (z) affects the L-system realizing m (z) as follows: if the non-negative symmetric Schrödinger operator satisfies m1(0)  0, then the L-system tan ;i of the form (5.6) realizing the function m (z) is accretive if and only if tan  (m1(0))1; (6.5) see (Belyi & Tsekanovskiı˘, 2021: Theorem 6.3). Note that if m1(0) = 0 in (6.5), then = =2 and m 2 (z) = 1=m1(z). Also, from Belyi & Tsekanovskiı˘ (2021: Theorem 6.2) we know that if m1(0)  0, then 1=m1(z) is realized by an accretive system 1;i. Having established criteria for an L-system realizingm (z) to be accretive, we can look into more of its properties. There are two choices for an accretive L-system tan ;i: it is either (1) accretive sectorial or (2) accretive extremal. In the case (1) the operator Atan ;i of the form (5.7) is 1- sectorial with some angle of sectoriality 1 that can only exceed the exact angle of sectoriality of Ti. In the case (2) the state-space operator Atan ;i is extremal (not sectorial for any 2 (0; =2)) and is a ()-extension of Ti that itself can be either -sectorial or extremal. These possibilities were described in detail in Belyi & Tsekanovskiı˘ (2021: Theorem 6.4). In particular, it was shown that for the accretive L-system tan ;i realizing the function m (z) the following is true: (1) If m1(0) = 0, then there exists only one accretive L-system 1;i realizingm (z). This L-system is extremal and its main operator Ti is extremal as well. (2) If m1(0) > 0, then Ti is -sectorial for 2 (0; =2) and (a) if tan = 1=m1(0), then tan ;i is extremal; (b) if 1m1(0) < tan < +1, then tan ;i is 1-sectorial with 1 > ; (c) if tan = +1, then 1;i is -sectorial. Acta Wasaensia 47 0 m 1 m∞ (−0) m 1:, i - extremal - preserves angle1:, i =+∞[ = tan" Figure 1. Accretive L-systems ;i. Figure 1 above describes the dependence of the properties of L-systems realizing m (z) on the value of  and, hence, on . The bold part of the real line depicts values of  = tan that produce accretive L-systems ;i. Additional analytic properties of the functions m1(z), 1=m1(z), and m (z) were described in Belyi & Tsekanovskiı˘ (2021: Theorem 6.5). It was proved there that under the current set of assumptions we have: (1) The function 1=m1(z) is a Stieltjes function if and only if m1(0)  0. (2) The function m1(z) is never a Stieltjes function.† (3) The function m (z) given by (5.3) is a Stieltjes function if and only if 0 < 1 m1(0)  tan : As the case that the realizing L-system tan ;i is accretive maximal does not require any further elaboration, we will now restrict ourselves to the case when it is accretive sectorial. To begin with, let  be an L-system of the form (4.8), whereA is a ()-extension (4.3) of the accretive Schrödinger op- erator Th. Here we summarize and list some known facts about possible accretivity and sectoriality of : • The operator A;h of ;h is accretive if and only if (6.2) holds, see (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011). • If an accretive operator Th, Imh > 0, is -sectorial, then (6.1) holds, see Theorem 5.1. Conversely, if Imh > 0 and Reh > m1(0); then the operator Th of the form (4.2) is -sectorial and is given by (6.1). • The operator Th is accretive but not -sectorial for any 2 (0; =2) if and only if the equality Reh = m1(0) holds. • If ;h is such that  = +1, then V1;h(z) belongs to the class S0; . In the case when  6= +1 we have V;h(z) 2 S 1; 2 , see (Belyi, 2011). • The operator A;h is a -sectorial ()-extension of the operator Th (with the same angle of sectoriality) if and only if  = +1 in (4.3), see (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011; Belyi & Tsekanovskiı˘, 2019). †It will be shown in an forthcoming paper that if m1(0)  0, then the function m1(z) is actually an inverse Stieltjes function. 48 Acta Wasaensia • If the operator Th is -sectorial with the exact angle of sectoriality , then it admits only one -sectorial ()-extension A;h with the same angle of sectoriality . Consequently,  = +1 and A;h = A1;h has the form (6.4). • A ()-extension A;h of the operator Th is accretive but not -sectorial for any 2 (0; =2) if and only if the value of  in (4.3) is defined by (6.3). Note that it follows from the above that any -sectorial operator Th with the exact angle of sec- toriality 2 (0; =2) admits only one accretive ()-extension A;h that is not -sectorial for any 2 (0; =2). This extension takes the form (4.3) with  given by (6.3). Now let us consider a function m (z) and a Schrödinger L-system tan ;i of the form (5.6) that realizes it. According to Belyi & Tsekanovskiı˘ (2021: Theorem 6.4 & Theorem 6.5) this L-system tan ;i is sectorial if and only if tan > 1 m1(0) : (6.6) Furthermore, if the L-system tan ;i is assumed to be -sectorial, then its impedance function Vtan ;i(z) = m (z) belongs to S . The following theorem provides more refined properties of m (z) for this case. Theorem 6.2. Let tan ;i be the accretive L-system of the form (5.6) realizing the functionm (z) associated with the non-negative operator _A, and let Atan ;i be a -sectorial ()-extension of Ti defined by (5.1). Then the function m (z) belongs to the class S 1; 2 , where tan 2  tan , and tan 1 = cot and tan 2 = tan +m1(0) (tan )m1(0) 1 : (6.7) Moreover, the operator Ti is ( 2 1)-sectorial with the exact angle of sectoriality 2 1. Proof. First recall from (5.3) that m (z) = sin +m1(z) cos cos +m1(z) sin = tan +m1(z) (tan )m1(z) 1 : (6.8) Since tan ;i is -sectorial, (6.6) holds and its impedance function Vtan ;i(z) = m (z) belongs to S and to S 1; 2 . In order to prove (6.7), pass to the limits in (6.8) tan 1 = lim x!1(m (x)) = limx!1 tan m1(x) + 1 tan 1m1(x) = cot ; where we used limx!1m1(x) = +1, see (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011: Section 10.3), and tan 2 = lim x!0 (m (x)) = tan +m1(0) (tan )m1(0) 1 : In order to show the rest, we apply (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011: Theorem 9.8.4): if A is a -sectorial ()-extension of a main operator T of an L-system , then the impedance function V(z) belongs to the class S 1; 2 , tan 2  tan , and T is ( 2 1)-sectorial with the exact angle of sectoriality 2 1. It can also be checked directly that (6.7) (under condition (6.6)) implies 0 < 2 1 < =2 and, hence, the definition of ( 2 1)-sectoriality applies correctly. Acta Wasaensia 49 Now we state and prove the following. Theorem 6.3. Let tan ;i be an accretive L-system of the form (5.6) that realizes m (z), where Atan ;i is a ()-extension of a -sectorial operator Ti with exact angle of sectoriality . Moreover, let  2 (arctan 1 m1(0)  ; 2 ) be a fixed value that defines Atan ;i via (4.3), and assume that m (z) 2 S 1; 2 . Then a ()-extension Atan ;i of Ti is -sectorial for any 2 [ ; =2) with tan = tan 1 + 2 p tan 1 tan 2; tan > tan : (6.9) Moreover, if = =2, then = 2 1 =  = arctan  1 m1(0)  : Proof. First note that the conditions imply that tan  2 ( 1m1(0) ;+1). Thus, according to Be- lyi & Tsekanovskiı˘ (2021: Theorem 6.4; part 2(c)) a ()-extension Atan ;i is -sectorial for some 2 (0; =2). Then we can apply Theorem 6.2 to confirm that m (z) 2 S 1; 2 , where 1 and 2 are described by (6.7). The equality in (6.9) follows from Belyi & Tsekanovskiı˘ (2019: Theo- rem 8) applied to the L-system tan ;i with  = tan , see also (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011: Theorem 9.8.7). Since Atan ;i is a -sectorial extension of a -sectorial operator Ti, we have tan  tan  with equality possible only when  = tan = 1, see (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011; Belyi & Tsekanovskiı˘, 2019). Since we chose 2 [ ; =2), it follows that tan 6=1 and, hence, tan > tan , which confirms the second part of (6.9). If we assume that = =2, then m (z) = 1=m1(z) is realized by the L-system 1;i (see The- orem 5.2) that preserves the angle of sectoriality of its main operator Ti, see (Belyi & Tsekanovskiı˘, 2021: Theorem 6.4) and Figure 1. Therefore, = . If we combine this fact withm (z) 2 S 1; 2 and apply Theorem 6.2 we also obtain that = 2 1. Finally, since Ti is -sectorial, (6.1) yields tan  = 1m1(0) . 0 α tanβ tan θ α∗ pi2α0 Figure 2. Angle of sectoriality . Here 0 = arctan 1 m1(0)  . Note that Theorem 6.3 provides us with a value which serves as a universal angle of sectoriality for the entire indexed family of ()-extensions Atan ;i of the form (5.6) as depicted in Figure 2. That figure clearly shows that if = =2, then tan = tan . 50 Acta Wasaensia 7 Example We conclude this paper with a simple illustration. Consider the differential expression with the Bessel potential l = d 2 dx2 + 2 1=4 x2 ; x 2 [1;1); in the Hilbert spaceH = L2[1;1) and assume that  > 0. The minimal symmetric operator _A in( _Ay = y00 + 21=4x2 y; y(1) = y0(1) = 0; has defect numbers (1; 1). Let  = 3=2. It is known that in this case m1(z) = iz 32 p z 32 ip z + i 1 2 = p z iz + ip z + i = 1 izp z + i and m1(0) = 1, see (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011). The minimal symmetric operator then becomes  _Ay = y00 + 2x2 y; y(1) = y0(1) = 0: (7.1) The main operator Th of the form (4.2) is written for h = i as Ti y = y00 + 2x2 y; y0(1) = i y(1): (7.2) It will be shared by the whole family of L-systems realizing the functions m (z) described by (5.2)-(5.3). This operator is accretive and -sectorial due to Reh = 0 > m1(0) = 1. Its exact angle of sectoriality is given by tan = Imh Reh+m1(0) = 1 0 + 1 = 1 or =  4 ; see (6.1). A family of L-systems tan ;i of the form (5.6) that realizes m (z) described by (5.2)–(5.4) as m (z) = ( p z iz + i) cos + (pz + i) sin ( p z iz + i) sin (pz + i) cos ; was constructed in Belyi & Tsekanovskiı˘ (2021). According to Belyi & Tsekanovskiı˘ (2021: Theo- rem 6.3) the L-systems tan ;i in (5.6) are accretive if 1 = 1 m1(0)  tan < +1: Using Belyi & Tsekanovskiı˘ (2021: Theorem 6.4; part (2c)), we get that the realizing L-system tan ;i in (5.6) preserves the angle of sectoriality and becomes 4 -sectorial if  = tan = +1 or = =2. Therefore the L-system 1;i =  A1;i K1;i 1 H+  L2[1;+1)  H C  ; Acta Wasaensia 51 where A1;i y = y00 + 2 x2 y [y0(1) iy(1)] (x 1); A1;i y = y00 + 2 x2 y [y0(1) + iy(1)] (x 1); and K1;ic = cg1;i, c 2 C, with g1;i = (x 1), realizes m2 (z) = 1=m1(z). Furthermore, V1;i(z) = m2 (z) = 1 m1(z) = p z + ip z iz + i ; W1;i(z) = (ei) m1(z) i m1(z) + i = (1 i)pz iz + 1 + i (1 + i) p z iz 1 + i : (7.3) This L-system 1;i is clearly accretive according to Belyi & Tsekanovskiı˘ (2021: Theorem 6.2), which can also be independently confirmed by direct evaluation (ReA1;i y; y) = ky0(x)k2L2 + 2ky(x)=xk2L2  0: Moreover, according to Belyi & Tsekanovskiı˘ (2021: Theorem 6.4), see also (Arlinskiı˘, Belyi & Tsekanovskiı˘, 2011: Theorem 9.8.7), the L-system 1;i is 4 -sectorial. Taking into account that (ImA1;i y; y) = jy(1)j2, see formula (4.5), we obtain inequality (2.1) with = 4 , that is (ReA1;i y; y)  j(ImA1;i y; y)j, or ky0(x)k2L2 + 2ky(x)=xk2L2  jy(1)j2: In addition, we have shown that the -sectorial form (Tiy; y) defined on dom Ti can be extended to the -sectorial form (A1;i y; y) defined on H+ = dom _A, see (7.1)–(7.2), having the exact angle of sectoriality = =4 (i.e., exact for both forms). A general problem of extending sectorial sesquilinear forms to sectorial ones was mentioned by T. Kato (1966). It can be easily seen that the function m 2 (z) in (7.3) belongs to the sectorial class S0;  4 of Stieltjes functions. References Alpay, D. & Tsekanovskiı˘, E. (2000). Interpolation theory in sectorial Stieltjes classes and explicit system solutions. Linear Algebra Appl. 314, 91–136. Arlinskiı˘, Yu., Belyi, S. & Tsekanovskiı˘ E. (2011). Conservative Realizations of Herglotz- Nevanlinna functions. Operator Theory: Advances and Applications, vol. 217. Basel: Birkhäuser Verlag. Arlinskiı˘, Yu. & Tsekanovskiı˘, E. (2004). Linear systems with Schrödinger operators and their trans- fer functions. Oper. Theory Adv. Appl. 149, 47–77. Arlinskiı˘, Yu. & Tsekanovskiı˘, E. (2009). M. Krein’s research on semi-bounded operators, its con- temporary developments, and applications. Oper. Theory Adv. Appl. 190, 65–112. Belyi, S. (2012). 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Department of Mathematics, Troy University, Troy, AL 36082, USA E-mail address: sbelyi@troy.edu Department of Mathematics, Niagara University, Lewiston, NY 14109, USA E-mail address: tsekanov@niagara.edu Acta Wasaensia 55 PT -SYMMETRIC HAMILTONIANS AS COUPLINGS OF DUAL PAIRS Volodymyr Derkach, Philipp Schmitz, and Carsten Trunk Dedicated to our friend and colleague Seppo Hassi on the occasion of his 60th birthday 1 Introduction In the seminal paper (Bender & Boettcher, 1998) a new view of quantum mechanics was proposed. This new view differs from the old one in that the restriction on the Hamiltonian to be Hermitian is relaxed: now the Hamiltonian isPT -symmetric. HereP is parity and T is time reversal. Since 1998, PT -symmetric Hamiltonians have been analyzed intensively by many authors. In Mostafazadeh (2002) PT -symmetry was embedded into the more general mathematical framework of pseudo- Hermiticity or, what is the same, self-adjoint operators in Kreı˘n spaces, see (Langer & Tretter, 2004; Azizov & Trunk, 2012; Hassi & Kuzhel, 2013; Leben & Trunk, 2019). For a general introduction to PT -symmetric quantum mechanics we refer to the overview paper of Mostafazadeh (2010) and to the books of Moiseyev (2011) and Bender (2019). A prominent class consists of the PT -symmetric Hamiltonians H := 1 2 p2 (iz)N+2; where N is a positive integer, see (Bender, Brody & Jones, 2002). The associated eigenvalue prob- lem is defined on a contour in the complex plane which is contained in a specific area in the complex plane, the so-called Stokes wedges, see (Bender & Boettcher, 1998), y00(z) (iz)N+2y(z) = y(z); z 2 ; (1.1) where  2 C is the eigenvalue parameter. Recall that a Stokes wedge Sk, k = 0; : : : ; N + 3, is an open sector in the plane with vertex zero, Sk :=  z 2 C : N + 2 2N + 8  + 2k 2 4 +N  < arg(z) < N + 2 2N + 8  + 2k 4 +N   ; see (Bender et al., 2006). The boundary of Sk consists of two rays from the origin, the so-called Stokes lines. PT -symmetry forces to lie in two Stokes wedges, which are symmetric with respect to the imaginary axis. In Mostafazadeh (2005) the contour in equation (1.1) was parameterized by a real parameter. In Bender et al. (2006) and in Jones & Mateo (2006) this approach was extended to different parame- terizations and contours. Here we choose, for simplicity, to be a wedge-shaped contour, := fxei sgn x : x 2 Rg; (1.2) for some angle  2 (=2; =2), see Figure 1. 56 Acta Wasaensia Re Im 4 3 2 1 1 2 3 4 1 2 3 4  Figure 1. The complex contour . Let z : R ! C parameterize via z(x) := xei sgn x. Then y solves (1.1) for z 6= 0 if and only if the pair of functions u+ and u, given by u(x) := y(z(x)), x 2 R, solves a[u] = u; x 2 R; a+[u+] = u+; x 2 R+; (1.3) where the differential expressions a are given by a[u+] = e2iu00 (ix)N+2ei(N+2)u: (1.4) In what follows we assume that lies in Stokes wedges and then, by Leben & Trunk (2019), the differential expressions a are in the limit-point case at1 according to the classification in Brown et al. (1999), which is a refinement of the classification in Sims (1957). We mention, that the limit- circle case can be treated in a similar way as in Azizov & Trunk (2010; 2012). The theory of PT -symmetry claims that the main object, the Hamiltonian, commutes under the joint action of the parity P and the time reversal T , (Pf)(x) := f(x); (T f)(x) := f(x): (1.5) The time reversal T applied to the differential expressions a gives rise to new differential expres- sions b = T aT defined on R b[v] = e2iv00 (ix)N+2ei(N+2)v: (1.6) In Section 3 we introduce the minimal operators A and B associated with a and b in L2(R) and show that hAf; gi = hf;Bgi; for all f 2 dom A; g 2 dom B: (1.7) Here h.; .i stands for the usual inner products in the Hilbert spaces L2(R). Condition (1.7) shows that the pairs (A+; B+) and (A; B) form dual pairs, see Section 2.1 for details. An extension the- ory for dual pairs based on the boundary triple technique was developed by Malamud & Mogilevskiı˘ (2002). This is a generalization of the boundary triple approach to the extension theory of symmet- ric operators which was developed by Calkin (1939); Kocˇhubeı˘ (1975); Gorbachuk & Gorbachuk (1991); Derkach & Malamud (1991), and others. For recent developments of the method of bound- Acta Wasaensia 57 ary triples and its application to the extension theory of differential operators, see the monographs by Derkach & Malamud (2017) and by Behrndt, Hassi & de Snoo (2020). Following this approach, we construct in Theorem 3.1 boundary triples for dual pairs (A+; B+) and (A; B). As our interest is focused on the Hamiltonian in L2(R) and not on the differential expressions a and b, which are defined on the semi-axes, we extend the coupling method for symmetric operators from Derkach et al. (2000) to the case of dual pairs and create a new dual pair (A;B) of operators defined on R. This dual pair (A;B) is called the coupling of the dual pairs (A+; B+) and (A; B), see Theorem 2.5 and Definition 2.6 below. We show that the operator PT intertwines the dual pairs (A+; B+) and (A; B), i.e., PT A+ = APT and PT B+ = BPT : Due to our construction of the coupling, these relations imply that the operator A is PT -symmetric PT A = APT : Moreover, the operator A turns out to be P-symmetric in the Kreı˘n space (H; [; ]) with the funda- mental symmetry P in H = L2(R). In Leben & Trunk (2019) it was shown that the extension H0 of A, defined as a restriction of the adjoint A+ to the domain dom H0 =  u+  u 2 dom A+ : u+(0) u(0) = e2iu0+(0) e2iu0(0) = 0 ; is a PT -symmetric and P-selfadjoint operator in the Kreı˘n space (H; [; ]). Here A+ stands for the adjoint with respect to the Kreı˘n space inner product [.; .]. In Theorem 3.2 below, which is the main result of this note, we find a one-parameter family fH g 2R of PT -symmetric and P-selfadjoint extensions of A in the Kreı˘n space (H; [; ]) with domain dom H =  u+  u 2 dom A+ : u+(0) u(0) = 0; e2iu0+(0) e2iu0(0) = u+(0) : Theorem 3.2 is based on the abstract construction of the coupling (A;B) of two dual pairs (A+; B+) and (A; B) in Theorem 2.5 and the description of allPT -symmetric andP-selfadjoint extensions of A given in Theorem 2.14. Summing up, the results presented here promote the use of boundary triple techniques for dual pairs and techniques from Sturm–Liouville theory for complex potentials in the study of PT -symmetric quantum mechanics. This is in line with Leben & Trunk (2019) and it is, to some extent, a surprise that in the physical literature the techniques presented here were never exploited. It is the aim of this paper to recall those techniques and, hence, provide a mathematically sound setting of the (nowadays) classical Bender–Boettcher-theory. 2 Coupling of dual pairs and parity In this section we recall known facts about dual pairs of linear operators, their boundary triples and corresponding Weyl functions, and coupling from Malamud & Mogilevskiı˘ (2002). However, our notations differ slightly from that paper; we mainly follow the notations of Baidiuk, Derkach & Hassi (2021). 58 Acta Wasaensia Moreover, throughout this paper we use the following notations. By R+ and R we denote the set of all positive and negative reals, respectively. For z 2 C, z denotes the complex conjugate of z. All operators in this paper are densely defined linear operators in some Hilbert spaces. For such operators T , we use the common notation dom T , ran T , and kerT for the domain, the range, and the null-space, respectively, of T . Moreover, as usual, (T ), (T ), and p(T ) stand for the resolvent set, the spectrum, and the point spectrum, respectively, of T . The inner product in a Hilbert space is usually denoted by h.; .i and the adjoint of the operator T by T . The set of all bounded and everywhere defined operators in a Hilbert space H is denoted by L(H). 2.1 Dual pairs of linear operators and Weyl functions Definition 2.1. A pair (A;B) of densely defined closed linear operators A and B in a Hilbert space (H; h.; .i) is called a dual pair, if hAf; gi hf;Bgi = 0 for all f 2 dom A; g 2 dom B: (2.1) The equality (2.1) means that A  B and B  A: Clearly, if (A;B) is a dual pair, then (B;A) is also a dual pair. Definition 2.2. Let (A;B) be a dual pair in a Hilbert space H, let H1, H2 be auxiliary Hilbert spaces, and let B =  B1 B2  : dom B ! H1 H2 and A =  A1 A2  : dom A ! H1 H2 (2.2) be linear operators. Then the triple (H1 H2;A;B) is called a boundary triple for the dual pair (A;B) if: (1) the mappings B and A in (2.2) are surjective; (2) the following identity holds for every f 2 dom B; g 2 dom A, hBf; gi hf;Agi = hB1 f;A1 giH1 hB2 f;A2 giH2 : It is easily seen that if a triple (H1 H2;A;B) is a boundary triple for a dual pair (A;B), then the following identity also holds hAg; fi hg;Bfi = hA2 g;B2 fiH2 hA1 g;B1 fiH1 ; f 2 dom B; g 2 dom A; (2.3) and, hence, the triple (H2 H1; (B)T ; (A)T ) :=  H2 H1;  B2 B1  ;  A2 A1  (2.4) is a boundary triple for the dual pair (B;A). The boundary triple (2.4) is called transposed with respect to the boundary triple (H1 H2;A;B). Acta Wasaensia 59 A linear operator eA is called a proper extension of a dual pair (A;B) if A  eA  B: The proper extension A2 of A is defined as the restriction of B to the set dom A2 = ff 2 dom B : B2 f = 0g: (2.5) Similarly, the proper extension B1 of B is defined as the restriction of A to the set dom B1 = ff 2 dom A : A1 f = 0g: (2.6) For every z 2 (A2) the following decomposition holds dom B = dom A2 uNz(B); where Nz(B) := ker (B zI); and, consequently, the mapping B2 jNz(B) : Nz(B)! H2 is boundedly invertible, see (Malamud & Mogilevskiı˘, 2002) for details. In a similar way, for every z 2 (B1) the following decomposition holds dom A = dom B1 uNz(A); where Nz(A) := ker (A zI); and, hence, the mapping A1 jNz(A) : Nz(A)! H1 is boundedly invertible for z 2 (B1). Moreover, in light of (2.3), (2.5), and (2.6), one has that B1 = A2 and, hence, in particular the following identity holds (B1) = (A2): Definition 2.3. The operator functions (z) := (B2 jNz(B))1 and M(z) := B1 (B2 jNz(B))1; z 2 (A2); are called the -field and the Weyl function, respectively, of the dual pair (A;B), corresponding to the boundary triple  = (H1 H2;A;B). Clearly, the operator functions T (z) := (A1 jNz(A))1 and MT (z) := A2 (A1 jNz(A))1; z 2 (B1); are the -field and the Weyl function, respectively, of the dual pair (B;A), corresponding to the transposed boundary triple (H2 H1; (B)T ; (A)T ), cf. (2.4). Notice that MT (z) = M(z); z 2 (B1) = (A2): Let  be a linear relation from H1 to H2, i.e., a subspace of H1  H2, see, e.g., Arens (1961). Consider the restriction A of B to the subspace dom A = ff 2 dom B : Bf 2 g: The following statement describes some spectral properties of the extension A. Lemma 2.4. Let (A;B) be a dual pair in a Hilbert space H, let (H1 H2;A;B) be a boundary triple for the dual pair (A;B), let M be the corresponding Weyl function, let  be a linear relation fromH1 toH2, and let z 2 (A2). Then the following statements hold: 60 Acta Wasaensia (i) A is the restriction of A  to dom A = ff 2 dom A : Af 2 g: (ii) z 2 p(A) if and only if 0 2 p(IH2 M(z)). In this case ker (A zI) = (z)ker (IH2 M(z)): (iii) z 2 (A) if and only if 0 2 (IH2 M(z)). 2.2 Coupling of dual pairs Theorem 2.5. Let (A+; B+) and (A; B) be dual pairs in Hilbert spaces H+ and H, respec- tively, let (H1  H2;A ;B) be a boundary triple for the dual pair (A; B), and let M be the corresponding Weyl function. Denote by A and B the restrictions of the operators A+  A and B+ B to the domains dom A = fg+  g : g 2 dom A; A+1 g+ = A1 gg (2.7) and dom B = ff+  f : f 2 dom B; B+2 f+ = B2 fg; (2.8) respectively. Then the following statements hold: (i) The operators A := (A) and B := (B) are restrictions of the operators B and A, respectively, to the domains dom A = ff+  f : f 2 dom B;B+2 f+ = B2 f = B+1 f+ + B1 f = 0g; (2.9) dom B = fg+g : g 2 dom A;A+1 g+ = A1 g = A+2 g+ +A2 g = 0g; (2.10) and (A;B) is a dual pair in H+  H. (ii) The triple  = (H1 H2;A;B) with Ag = A+ 1 g+ A+ 2 g+ + A 2 g ! and Bf = B+ 1 f+ + B 1 f B+ 2 f+ ! ; f 2 dom B; g 2 dom A; is a boundary triple for the dual pair (A;B). (iii) The Weyl function M(z) corresponding to the boundary triple  = (H1  H2;A;B) is given by M(z) = M+(z) +M(z); z 2 (A2); (2.11) where A2 is defined by (2.5). Proof. The proof of this theorem consists of three parts: (i) and (ii) are established in (a) and (b), and (iii) is proven in (c). Acta Wasaensia 61 (a) Let f = f+  f 2 dom (B+  B), g = g+  g 2 dom (A+  A). Then it follows from the equalities hB+f+; g+i hf+; A+g+i = hB+1 f+;A+1 g+iH1 hB+2 f+;A+2 g+iH2 ; hBf; gi hf; Agi = hB1 f;A1 giH1 hB2 f;A2 giH2 ; that h(B+ B)f; gi hf; (A+ A)gi = hB+1 f+;A+1 g+iH1 hB+2 f+;A+2 g+iH2 + hB1 f;A1 giH1 hB2 f;A2 giH2 : (2.12) The equality (2.9) follows from (2.12) since the mappings A : dom A ! H1 H2 are surjec- tive. Similarly, (2.10) follows from (2.12) since the mappings B : dom B ! H1  H2 are surjective. (b) Next, for f 2 dom B and g 2 dom A the equation (2.12) takes the form hBf; gi hf; (A)gi = hB+1 f+ + B1 f;A+1 g+iH1 hB+2 f+;A+2 g+ + A2 giH2 : This proves that (A;B) is a dual pair in H+  H and that (ii) holds. (c) It follows from (2.8) that the -field of (A;B) corresponding to the boundary triple  takes the form (z) = +(z) (z); where (z) are -fields of (A; B) corresponding to the boundary triples (H1H2;A ;B). Now formula (2.11) follows from the definition of the Weyl function, see Definition 2.3. Definition 2.6. The dual pair (A;B) constructed in (2.9) and (2.10) is called the coupling of the dual pairs (A+; B+) and (A; B) relative to the triples (H1 H2;A+ ;B+) and (H1 H2;A ;B): 2.3 Real dual pairs and real boundary triples Let T be a conjugation (time reversal) operator in a Hilbert space (H; h.; .i), i.e., T is antilinear, T 2 = IH, and hT f; T gi = hg; fi for all f; g 2 H: In what follows, we suppose thatH1 andH2 coincide: H1 = H2 = H. Definition 2.7. Let T and jH be conjugations in H and H, respectively. A dual pair (A;B) in H is called T -real if T dom A = dom B and T A = BT : (2.13) A boundary triple (H2;A;B) for (A;B) is called (jH; T )-real if jHB1 = A 2 T and jHB2 = A1 T : 62 Acta Wasaensia Observe that the conditions (2.13) are clearly equivalent to T dom A = dom B and T A = BT : Lemma 2.8. Let (A;B) be a T -real dual pair and let (H2;A;B) be a (jH; T )-real boundary triple for (A;B). Then the corresponding Weyl function M(z) satisfies the condition M(z) = jHM(z)jH; z 2 (A2): In what follows we consider a Hilbert space H decomposed into an orthogonal sum H = H+  H (2.14) of two subspaces H with conjugations T 2 L(H). Then the orthogonal sum T = T+  T (2.15) is a conjugation in H. Theorem 2.9. Let a Hilbert space H and a conjugation T in H be such that (2.14) and (2.15) hold. Moreover, let (A; B) be T-real dual pairs in the Hilbert spaces H. Finally, with jH a conjugation inH, let (H2;A ;B) be (jH; T )-real boundary triples for (A; B), and let A0 := A+ A and B0 := B+ B: Then the following statements hold: (i) The dual pair (A0; B0) is T -real and the boundary triple ((HH)2;A0 ;B0) with A0 = A+  A and B0 = B+  B is (jHH; T )-real, where jHH := jH  jH. (ii) The coupling (A;B) of the dual pairs (A+; B+) and (A; B), constructed in (2.9) and (2.10) is T -real. (iii) The boundary triple (H2;A;B) from Theorem 2.5 is (jH; T )-real. 2.4 Parity and P-selfadjoint operators Definition 2.10. Let H be Hilbert spaces and H = H+  H. An operator P 2 L(H) will be called an (abstract) parity operator if P = P; P2 = IH; and PH = H: Now consider a Hilbert space H = H+H with a parity operator P and a conjugation T 2 L(H), such that T P = PT and T H = H: (2.16) Acta Wasaensia 63 The conditions (2.16) mean that the operator T admits the representation as an orthogonal sum T = T+  T of two conjugations T+ and T in Hilbert spaces H+ and H, respectively. Lemma 2.11. Let P be a parity operator in H = H+  H and let T be a conjugation in H such that (2.16) holds. Let (A; B) be T-real dual pairs in the Hilbert spaces H, such that PA+ = BP and PB+ = AP: (2.17) Then the following statements hold: (i) PT dom A+ = dom A, PT dom B+ = dom B, and PT A+ = APT ; PT B+ = BPT ; (2.18) (ii) P dom A+ = dom B, P dom B+ = dom A, and PA+ = BP; PB+ = AP: Proof. (i) Since the dual pairs (A; B) are real with respect to T, one has T+A+ = B+T+; TA = BT: (2.19) Let f+ 2 dom A+. Then by (2.19) T f+ 2 dom B+ and B+T f+ = T A+f+. Next by (2.17) PT f+ 2 dom A and APT f+ = PB+T f+ = PT A+f+: The proofs of the inclusionPT dom A  dom A+ and of the second equality in (2.18) are similar. (ii) Applying P to the left and right of the equalities in (2.17) and using the identity P2 = IH yields A+P = PB and B+P = PA. From these identities the assertions in (ii) are immediate. Definition 2.12. A closed linear operator A in H is said to be PT -symmetric if for all f 2 dom A we have PT f 2 dom A and PT Af = APT f: Consider the Kreı˘n space (H; [; ]) with an indefinite inner product given by [f; g] := hPf; giH: (2.20) For the definition of a Kreı˘n space we refer to the books of Azizov & Iokhvidov (1989) and Bognar (1974). Recall that a densely defined linear operator A in H is called P-symmetric if [Af; g] = [f;Ag] for all f; g 2 dom A: Denote by A+ the adjoint operator in (H; [; ]), i.e., A+ = PAP . For a P-symmetric operator A one has A  A+. The operator A is called P-selfadjoint if A = A+. The following definition of a boundary triple for the P-symmetric operator A was presented in Derkach (1995). Definition 2.13. LetH be an auxiliary Hilbert space and let 1;2 be linear operators from dom A+ to H. The triple (H;1;2) is called a boundary triple for the P-symmetric operator A if the following conditions are satisfied: 64 Acta Wasaensia (i) the mapping :=  1 2  from dom A+ toH2 is surjective; (ii) the following identity holds for every f; g 2 dom A+ [A+f; g] [f;A+g] = h1f;2giH h2f;1giH: In the next theorem we show that the coupling operator A is P-symmetric and PT -symmetric, and describe the set of all P-selfadjoint and PT -symmetric extensions of the operator A. Theorem 2.14. Let P be a parity operator in H = H+  H, let T be a conjugation in H such that (2.16) holds, and let (A; B) be T-real dual pairs in the Hilbert spaces H such that (2.17) holds. With jH a conjugation in H, let (H2;A ;B) be (jH; T )-real boundary triples for (A; B), such that B+ 1 B+ 2 ! f+ = A 2 A 1 ! Pf+ and B 1 B 2 ! f = A+ 2 A+ 1 ! Pf; f 2 dom B: (2.21) Moreover, let (A;B) be the coupling of the dual pairs (A+; B+) and (A; B) given by (2.9), (2.10), and let  be a linear relation inH. Then the following statements hold: (i) The operator A is PT -symmetric, P-symmetric, and A+ = B. (ii) The triple (H;B1 ;B2 ) is a boundary triple for the P-symmetric operator A. (iii) The extension A of the operator A, given by dom A =  f 2 dom B :  1f 2f  2   ; A = B jdomA ; is P-selfadjoint if and only if  = . (iv) A is PT -symmetric if and only if  = jHjH. 3 PT -symmetric Hamiltonians Here we return to the investigation of the non-Hermitian PT -invariant Hamiltonians presented in the introduction, that is, we study equation (1.1) on the wedge shaped contour , cf. (1.2). By substituting z(x) := xei sgn x into (1.1) one obtains the two differential expressions given by (1.3) and (1.4). Assume that the differential expressions a in (1.4) are in the limit point case at 1. As presented in Section 1, this is the case if and only if the angle  of the wedge satisfies  6= N + 2 2N + 8  + 2k 4 +N  for k = 0; : : : ; N + 3: (3.1) Then by Leben & Trunk (2019: Lemma 1) the differential expressions b in (1.6) are also in the limit point case at 1. Define the operators A and B associated with a and b in L2(R) as Af := a[f] and Bg := b[g] for f 2 dom A; g 2 dom B; Acta Wasaensia 65 respectively, with the domains dom A := fu 2 L2(R) : a[u] 2 L2(R); u0 2 ACloc(R); u(0) = u0(0) = 0g; dom B := fv 2 L2(R) : b[v] 2 L2(R); v0 2 ACloc(R); v(0) = v0(0) = 0g: These operators are in some sense the minimal operators. It follows from Leben & Trunk (2019: Proposition 1 & Theorem 3) that the (maximal) operators A and B   are generated by differential expressions in L2(R) where the roles of a and b are switched in the sense that the differential expressions a are now related to B and the differential expressions b are related to A  : Bf := a[f] and A  g := b[g] for f 2 dom B; g 2 dom A; with dom B := fu 2 L2(R) : a[u] 2 L2(R); u0 2 ACloc(R)g; dom A := fv 2 L2(R) : b[v] 2 L2(R); v0 2 ACloc(R)g: Theorem 3.1. The pairs (A; B) and (A+; B+) are dual pairs. The triple (C2;A+ ;B+), B+u+ =  e2iu0+(0) u+(0)  and A+v+ =  v+(0) e2iv0+(0)  ; u+ 2 dom B+; v+ 2 dom A+; is a boundary triple for the dual pair (A+; B+). The triple (C2;A ;B), Bu = e2iu0(0) u(0)  and Av =  v(0) e2iv0(0)  ; u 2 dom B; v 2 dom A; is a boundary triple for the dual pair (A; B). Proof. Integration by parts and (Leben & Trunk, 2019: Proposition 1) show hAu; vi = hu; Bvi; u 2 dom A; v 2 dom B: This proves the first statement. It follows from (Leben & Trunk, 2019: Proposition 1) that for u+ 2 dom B+ and v+ 2 dom A+ hB+u+; v+i hu+; A+v+i = e2i Z 1 0 u00+(x)v+(x) dx+ e 2i Z 1 0 u+(x)v00+(x) dx = e2i(u0+(0)v+(0) u+(0)v0+(0)): Hence, (C2;A+ ;B+) is a boundary triple for the dual pair (A+; B+). The statement for the dual pair (A; B) is shown in the same way. Recall that the coupling (A;B) of the dual pairs (A+; B+) and (A; B) consists of a pair of operators A = (B+ B)jdom A and B = (A+ A)jdom B with the domains dom A = fu+  u : u 2 dom B; u+(0) = u(0) = e2iu0+(0) e2iu0(0) = 0 g; (3.2) dom B = fu+  u : u 2 dom A; u+(0) = u(0) = e2iu0+(0) e2iu0(0) = 0 g; (3.3) see Theorem 2.5. 66 Acta Wasaensia We define the parity P and time reversal T as in (1.5). The parity P gives rise to a new inner product [; ] = hP; i (see also (2.20)), which was considered in many papers, we mention only (Mostafazadeh, 2010). It is easy to see that the parity P and the time reversal T satisfy (2.16), where H := L2(R). Due to Theorem 2.14, the operator A is PT -symmetric and P-symmetric in the Kreı˘n space (L2(R); [; ]) = (L2(R)  L2(R+); [; ]). The (Kreı˘n space) adjoint A+ of A coincides with B = (B+ B)jdomB , where dom B = fu+  u : u 2 dom B; u+(0) = u(0)g: An application of Theorem 2.14 gives a one-parameter family fH g 2R of PT -symmetric and P-selfadjoint extensions of A in the Kreı˘n space (L2(R); [; ]). This is the main result of this note. Theorem 3.2. Let the angle  satisfies (3.1) and letA be the coupling operator constructed in (3.2). Then the following statements are true: (i) A boundary triple (C;1;2) for the P-symmetric operator A is given by 1u = e 2iu0+(0) e2iu0(0) and 2u = u+(0); u = u+  u 2 dom B: (ii) The extension H of the operator A, defined as a restriction of A+ to the domain dom H =  u+  u 2 dom B : e2iu0+(0) e2iu0(0) = u+(0) ; is P-selfadjoint if and only if 2 R. (iii) H is PT -symmetric if and only if 2 R. Proof. By construction the dual pairs (A+; B+) and (A; B) are T-real and the parity operator P intertwines the operators A+, B and A, B+, that is, (2.17) holds. Moreover, the boundary triples (C2;A+ ;B+) and (C2;A ;B) are also (jC; T )-real and satisfy the condition (2.21). Here jC stands for the usual complex conjugation in C. Hence, all assumptions in Theorem 2.14 are satisfied and the statements in Theorem 3.2 follow directly from Theorem 2.14. In Leben & Trunk (2019) only the extension for the parameter value = 0 was considered. More precisely, there it was shown that H0 is an extension of A with domain dom H0 =  u+  u : u 2 dom B; u+(0) u(0) = e2iu0+(0) e2iu0(0) = 0 which is PT -symmetric and P-selfadjoint. The family H , 2 R, of extensions obtained in Theo- rem 3.2 is in some sense an analogue of the -interaction for the differential operation a. Acknowledgements: The research of V.D. was supported by the German Research Foundation (DFG), grant TR 903/22-1. The research of Ph.S. has been supported by DFG, grant TR 903/19-2. References Arens, R. (1961). Operational calculus of linear relations. Pacific J. Math. 11, 9–23. 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Edinburgh Sect. A 143, 141–167. 68 Acta Wasaensia Jones, H.F. & Mateo, J. (2006). Equivalent Hermitian Hamiltonian for the non-Hermitian x4 po- tential. Phys. Rev. D 73, 085002, 4 pp. Kocˇhubeı˘, A.N. (1975). Extentions of symmetric operators and of symmetric binary relations. Mat. Zametki 17, 41–48. (In Russian.) Langer, H. & Tretter, C. (2004). A Krein space approach to PT -symmetry. Czechoslovak J. Phys. 54, 1113–1120. Leben, F. & Trunk, C. (2019). Operator-based approach to PT -symmetric problems on a wedge- shaped contour. Quantum Stud. Math. Found. 6, 315–333. Malamud, M.M. & Mogilevskiı˘, V.I. (2002). Kreı˘n type formula for canonical resolvents of dual pairs of operators. Methods Funct. Anal. Topol. 8, 72–100. Moiseyev, N. (2011). Non-Hermitian Quantum Mechanics. Cambridge: Cambridge University Press. Mostafazadeh, A. (2002). Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian. J. Math. Phys. 43, 205–214. Mostafazadeh, A. (2005). Pseudo-Hermitian description of PT-symmetric systems defined on a com- plex contour. J. Phys. A 38, 3213-–3234. Mostafazadeh, A. (2010). Pseudo-Hermitian representation of quantum mechanics. Int. J. Geom. Methods Mod. Phys. 7, 1191–1306. Sims, A.R. (1957). Secondary conditions for linear differential operators of the second order. J. Math. Mech. 6, 247–285. Institut für Mathematik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany E-mail address: volodymyr.derkach@tu-ilmenau.de, philipp.schmitz@tu-ilmenau.de, and carsten.trunk@tu-ilmenau.de Acta Wasaensia 69 POSITIVE AND NEGATIVE EXAMPLES FOR THE RIESZ BASIS PROPERTY OF INDEFINITE STURM-LIOUVILLE PROBLEMS Andreas Fleige Dedicated to Seppo Hassi on the occasion of his 60th birthday 1 Introduction We consider the indefinite Sturm-Liouville eigenvalue problem f 00 = rf on [1; 1]; with a weight function r changing its sign at 0. It is known that, depending on the behavior of r and on the type of the imposed self-adjoint boundary conditions, the so-called Riesz basis property of the eigenfunctions in the Hilbert space L2jrj[1; 1] can be valid or not. More precisely, let r 2 L1[1; 1] be a real function with a single so-called turning point, i.e., with a sign change, at 0, say r(x) < 0 a.e. on [1; 0); r(x) > 0 a.e. on (0; 1]: Due to the sign change of r we cannot expect that the eigenfunctions of the corresponding Sturm- Liouville eigenvalue problem with self-adjoint boundary conditions form an orthonormal basis in the Hilbert space L2jrj[1; 1] with the inner product (f; g) = Z 1 1 fgjrj dx; f; g 2 L2jrj[1; 1]: However, the eigenfunctions may form a Riesz basis in this space, i.e., an orthonormal basis with respect to some inner product equivalent to (; ). In this case, we say that the eigenvalue problem has the Riesz basis property. This Riesz basis property has been intensively studied during the last decades, see the overview paper (Fleige, 2015). It was first observed by Binding and C´urgus (2004) that the type of the boundary conditions plays an important role. In fact, for the same weight function r, the eigenvalue problem with Dirichlet boundary conditions f 00 = rf on [1; 1]; f(1) = f(1) = 0; (1.1) can have the Riesz basis property, while the eigenvalue problem with antiperiodic boundary condi- tions f 00 = rf on [1; 1]; f(1) + f(1) = 0; f 0(1) + f 0(1) = 0: (1.2) does not have the Riesz basis property, This result was sharpened in C´urgus, Fleige & Kostenko (2013: Theorem 4.10). Under a certain oddness condition a necessary and sufficient criterion on the weight function r was presented for the Riesz basis property of an eigenvalue problem with quite general boundary conditions. In case of the eigenvalue problem (1.1) this criterion only involves the 70 Acta Wasaensia behavior of r at the turning point, whereas in case of the eigenvalue problem (1.2) also the boundary comes into play, see Theorem 2.4 below. In some sense the paper (C´urgus, Fleige & Kostenko, 2013) was the endpoint of a number of papers improving the conditions for the Riesz basis property, at least for the case of a regular differential expression, e.g., (C´urgus & Langer, 1989; Parfenov, 2003; 2005; Pyatkov, 2005). Note that further developments were mainly concerned with singular differential expressions and criteria involving the Titchmarsh-Weyl function, see e.g., (Kostenko, 2013; C´urgus, Derkach & Trunk, 2020). The present paper can be regarded as a certain addition to C´urgus, Fleige & Kostenko (2013). The main intention is a presentation of examples for the validity and non-validity of the Riesz basis prop- erty of eigenvalue problems (1.1) and (1.2), illustrating the different arguments for various settings. Finally, an outline of the paper is given. In Section 2 we recall some conditions concerning the weight function r appearing in the eigenvalue problems (1.1) and (1.2) from the paper (C´urgus, Fleige & Kostenko, 2013; Parfenov, 2003; 2005; Pyatkov, 2005). In Section 3 all possible, positive and negative, cases are obtained by certain modifications of the same "bad" weight function. Never- theless, on the left of the turning point this "bad" weight always remains unchanged. Note that for the proof of these examples also a new result had to be added to the general theory in Section 2. 2 Some known conditions and a slight extension of these results First of all, it should be mentioned that for the eigenvalue problems (1.1) and (1.2) there cannot appear any root functions except the eigenfunctions. This follows, for example, by means of Kreı˘n space methods. To see this let AD and Aa denote the operators in L2jrj[1; 1] associated with (1.1) and (1.2). Thus, in other words, ADf = 1rf 00, when f belongs to dom AD, defined by dom AD =  f 2 L2jrj[1; 1] : 1 r f 00 2 L2jrj[1; 1]; f(1) = f(1) = 0 ; and, likewise, Aaf = 1rf 00, when f belongs to dom Aa, defined by dom Aa = ff 2 L2jrj[1; 1] : 1 r f 00 2 L2jrj[1; 1]; f(1) + f(1) = 0; f 0(1) + f 0(1) = 0g: Here is the above mentioned observation concerning the eigenvalue problems (1.1) and (1.2). Lemma 2.1. Let  2 C and assume that f 2 dom A2D and g 2 dom A2a satisfy (AD )2f = 0 and (Aa )2g = 0: Then f and g already satisfy (AD )f = 0 and (Aa )g = 0: Proof. Note that L2jrj[1; 1] is a Kreı˘n space with the inner product [f; g] = Z 1 1 fgr dx; f; g 2 L2jrj[1; 1]; Acta Wasaensia 71 and that AD and Aa are self-adjoint and definitizable operators in this space; see (C´urgus & Langer, 1989). Furthermore, observe that the operator AD is nonnegative, since [ADf; f ] = Z 1 1 f 00 f dx = Z 1 1 jf 0j2 dx  0; f 2 dom AD; due to the boundary conditions. Likewise, one sees that Aa is nonnegative. Therefore, in both cases, p() =  is a definitizing polynomial. By (Langer, 1982: Proposition II, 2.1) a Jordan chain of length at least 2 can only appear for the zeros of the definitizing polynomial, i.e., for  = 0. However,  = 0 is neither an eigenvalue for AD nor for Aa. Note that the situation sketched in Lemma 2.1 is different in the case of periodic boundary condi- tions; see (C´urgus, Fleige & Kostenko, 2013: Example 4.12). For convenience we now recall some definitions and known results about the Riesz basis property which will be used in the subsequent examples. We start with some properties from the theory of regularly varying functions studied at 0 rather than at 1, see, e.g., (Bingham, Goldie & Teugels, 1987; Kostenko, 2013; C´urgus, Derkach & Trunk, 2020). Let I be a nondecreasing function defined on [0; b] with b > 0 such that I(x) > 0; x 2 (0; b]; 0 = I(0) = lim x&0 I(x): (2.1) Then the function I is said to be slowly varying if lim x&0 I(tx) I(x) = 1; for all t > 0; and is said to be positively increasing if lim sup x&0 I(t0x) I(x) < 1; for some t0 2 (0; 1): Of course, these properties exclude each other: a slowly varying function cannot be positively in- creasing, and vice versa. Below, these properties will be checked for the functions I+0 (x) := Z x 0 r dt and I1 (x) := Z 1 1x r dt; x 2 [0; 1]; (2.2) obviously satisfying (2.1). Furthermore, some local oddness conditions on the weight function will be used. A function r is called locally odd at the turning point if there exists " > 0 such that the restriction of r to ("; ") is odd. Similarly, a function r is called locally odd at the boundary if there exists " > 0 such that r(1 + x) = r(1 x) for a.a. x 2 (0; "). The following result, here formulated in the terminology of positively increasing functions as in C´urgus, Fleige & Kostenko (2013: Corollary 3.6), goes back to Parfenov (2003; 2005). Theorem 2.2 (Parfenov (2003; 2005)). The following statements hold: (i) If I+0 is positively increasing, then the eigenvalue problem (1.1) has the Riesz basis property. (ii) If r is locally odd at the turning point, then the eigenvalue problem (1.1) has the Riesz basis property if and only if I+0 is positively increasing. 72 Acta Wasaensia Originally, (ii) was formulated for the stronger case of odd weights. However, Pyatkov (2005: Theorem 4.2) observed that only the local behaviour at the turning point is relevant for the Riesz basis property of (1.1). Next we recall another aspect of Pyatkov’s result (Pyatkov, 2005: Theorem 4.2), again in the terminology of positively increasing functions as in C´urgus, Fleige & Kostenko (2013: Theorem 4.1). Theorem 2.3 (Pyatkov (2005)). If I1 is positively increasing, then the eigenvalue problem (1.2) has the Riesz basis property if and only if the eigenvalue problem (1.1) has the Riesz basis property. This result was improved in C´urgus, Fleige & Kostenko (2013: Theorem 4.10) in the case when certain oddness conditions are satisfied. Theorem 2.4 (C´urgus, Fleige & Kostenko (2013)). If r is locally odd at the turning point and at the boundary, then the eigenvalue problem (1.2) has the Riesz basis property if and only if I+0 and I 1 are both positively increasing. Note that originally the last two results were formulated in a more general setting and they are now applied in the present situation. Finally, we present a slight extension of these results, which will be proved in a similar way as (C´urgus, Fleige & Kostenko, 2013: Theorem 4.10). Theorem 2.5. If r is locally odd at the boundary, then the eigenvalue problem (1.2) has the Riesz basis property if and only if the eigenvalue problem (1.1) has the Riesz basis property and I1 is positively increasing. Proof. By (C´urgus, Fleige & Kostenko, 2013: Lemma 4.7) the Riesz basis property of (1.2) is equivalent to the Riesz basis property of the shifted problem f 00 = erf on [ea;eb]; f(ea) + f(eb) = 0; f 0(ea) + f 0(eb) = 0; (2.3) in L2jerj[ea;eb], where ea := 1 " and eb := 1 " for some " 2 (0; 12 ). Here the weight function er is given by er(x) := ( r(x+ 2); x 2 [ea;1); r(x); x 2 [1;eb]: This function has two turning points, i.e., sign changes, at 0 and at1. Thus we can apply (Pyatkov, 2005: Theorem 4.2) (using a criterion for an even number of turning points, see also (C´urgus, Fleige & Kostenko, 2013: Theorem 4.1)) and obtain the equivalence of the Riesz basis property of (2.3) to the Riesz basis properties of two local problems with er on [; ] and er on [1 ;1 + ], both with Dirichlet boundary conditions. Here,  2 (0; ") can be chosen arbitrarily. By assumption, the weight er is locally odd at the turning point 1; i.e., er(1 x) = er(1 + x) for a.a. x in a neighborhood of 0. Therefore, by Parfenov’s result, see Theorem 2.2 (ii), the Riesz basis property for the problem on [1 ;1 + ] is equivalent to the condition that the functionZ 1 1x er dt = I1 (x) is positively increasing. Furthermore, again by (Pyatkov, 2005: Theorem 4.2), the Riesz basis prop- erty for the problem on [; ] is equivalent to the Riesz basis property of (1.1). Acta Wasaensia 73 Note that the arguments above also apply in a more general setting: Theorem 2.5 remains true with (1.2) replaced by the eigenvalue problem f 00 + qf = rf; eitf(1) = f(1); f 0(1) = eitf 0(1) + d f(1); with a real potential q 2 L1[1; 1], t 2 [0; 2), and d 2 R, see also the proof of (C´urgus, Fleige & Kostenko, 2013: Proposition 4.9). However, in this case there may appear a Jordan chain as in C´urgus, Fleige & Kostenko (2013: Example 4.12). Therefore, as in C´urgus, Fleige & Kostenko (2013), for the Riesz basis property not only eigenfunctions but also root functions must be allowed. 3 Examples We start with an example which combines the negative properties of (C´urgus, Fleige & Kostenko, 2013: Example 3.17) (going back to (Parfenov, 2003)) and (C´urgus, Fleige & Kostenko, 2013: Example 4.12 (ii)). It is worse than each of these examples in the sense that the Riesz basis criteria at the turning point and at the boundary are both violated simultaneously. 3.1 A weight with "bad" behavior at the turning point and at the boundary First, consider the functions r0 and r1 belonging to L1[1; 1] defined as r0(x) := 1 x(1 log jxj)2 and r1(x) := sgn(x) (1 jxj)(1 log(1 jxj))2 ; (3.1) for x 2 (1; 1) n f0g. Furthermore, define the function r as r := r0 + r1: (3.2) Lemma 3.1. For the weight function r from (3.2), the functions I+0 and I 1 , as defined in (2.2), coincide: I+0 = I 1 . Moreover, this function I + 0 = I 1 is slowly varying. Proof. First, note that for x 2 (0; 1) we have that r1(x) = r0(1 x) and, hence, I1 (x) = Z 1 1x (r0 + r1) dt = Z x 0 (r0(1 s) + r1(1 s)) ds = Z x 0 (r1(s) + r0(s)) ds = I + 0 (x): Now, observe that for t > 0 the following limits exist lim x&0 t r0(tx) r0(x) = lim x&0 (1 log x)2 (1 log x log t)2 = 1 and limx&0 t r1(tx) r0(x) = 0: In order to see that I+0 is slowly varying use l’Hôpital’s rule: lim x&0 I+0 (tx) I+0 (x) = lim x&0 t(r0(tx) + r1(tx)) r0(x) + r1(x) = lim x&0 t( r0(tx)r0(x) + r1(tx) r0(x) ) 1 + r1(x)r0(x) = 1: 74 Acta Wasaensia Since r is odd here, we can now conclude the following result from Lemma 3.1, Theorem 2.2 (ii), and Theorem 2.4. Proposition 3.2. For the weight function r as in (3.2) neither the eigenvalue problem (1.1) nor the eigenvalue problem (1.2) has the Riesz basis property. In the following we "smoothen" the weight function in (3.2) in some sense. 3.2 "Relaxing" the weight on the right of the turning point Using again the functions r0 and r1 from (3.1), we now consider the weight function r(x) :=  1; x 2 [c; d]; r0(x) + r1(x); x 2 [1; 1] n [c; d]; (3.3) for some numbers 0  c < d  1. Obviously, this weight is not odd any more, but it is locally odd at the turning point if c > 0, and locally odd at the boundary if d < 1. Lemma 3.3. Let the weight function r be given by (3.3). Then the following statements hold: (i) If 0 < c < d < 1, then the functions I+0 and I 1 are slowly varying. (ii) If 0 < c < d = 1, then I+0 is slowly varying and I 1 is positively increasing. (iii) If 0 = c < d < 1, then I+0 is positively increasing and I 1 is slowly varying. (iv) If c = 0 and d = 1, then the functions I+0 and I 1 are positively increasing. Proof. In some cases the functions I+0 and I 1 coincide locally at 0 with the corresponding function in Lemma 3.1. In these cases the functions are slowly varying by Lemma 3.1. In all other cases these functions are linear near 0 and, hence, positively increasing. Finally, the theorems from Section 2 lead to the following three different Riesz basis results in the four cases of Lemma 3.3. Proposition 3.4. Let the weight function r be given by (3.3). Then the following statements hold: (i) If 0 < c < d < 1, then neither the eigenvalue problem (1.1) nor the eigenvalue problem (1.2) has the Riesz basis property. (ii) If 0 < c < d = 1, then neither the eigenvalue problem (1.1) nor the eigenvalue problem (1.2) has the Riesz basis property. (iii) If 0 = c < d < 1, then the eigenvalue problem (1.1) has the Riesz basis property but the eigenvalue problem (1.2) does not. (iv) If c = 0 and d = 1, then both eigenvalue problems (1.1) and (1.2) have the Riesz basis property. Acta Wasaensia 75 Proof. As in Proposition 3.2, statement (i) follows from Theorem 2.2 (ii) and Theorem 2.4, since r is in this case locally odd at the turning point and at the boundary. Similarly, we obtain statement (ii) for problem (1.1) from Theorem 2.2 (ii), and then also for problem (1.2) from Theorem 2.3. (Note that here we cannot apply Theorem 2.4, since r is no longer locally odd at the boundary.) Statement (iii) follows from Theorem 2.2 (i) and Theorem 2.5. Here, we use the fact that r is in this case locally odd at the boundary. Finally, statement (iv) can be obtained from Theorem 2.2 (i) and Theorem 2.3. References Binding, P. & C´urgus, B. (2004). A counterexample in Sturm-Liouville completeness theory. Proc. Roy. Soc. Edinburgh Sect. A 134, 244-248. Bingham, N.H., Goldie, C.M. & Teugels, J.I. (1987). Regular Variation. Cambridge: Cambridge University Press. C´urgus, B., Derkach, V. & Trunk, C. (2020). Indefinite Sturm–Liouville operators in polar form. arXiv:2101.00104v1. C´urgus, B., Fleige, A. & Kostenko, A. (2013). The Riesz basis property of an indefinite Sturm- Liouville problem with non-separated boundary conditions. Integral Equations Operator Theory 77, 533–557. C´urgus, B. & Langer, H. (1989). A Krein space approach to symmetric ordinary differential opera- tors with an indefinite weight function. J. Differential Equations 79, 31–61. Fleige, A. (2015). The critical point infinity associated with indefinite Sturm-Liouville problems. In Operator Theory, 395–429. Basel: Springer. Kostenko, A. (2013). The similarity problem for indefinite Sturm–Liouville operators and the HELP inequality. Adv. Math. 246, 368–413. Langer, H. (1982). Spectral functions of definitizable operators in Krein spaces. In: Functional Analysis. Lecture Notes in Mathematics, vol. 948, 1–46. Berlin, Heidelberg, New York: Springer- Verlag. Parfenov, A.I. (2003). On an embedding criterion for interpolation spaces and application to indefi- nite spectral problems. Siberian Math. J. 44, 638–644. Parfenov, A.I. (2005). The C´urgus condition in indefinite Sturm–Liouville problems. Siberian Adv. Math. 15, 68–103. Pyatkov, S.G. (2005). Some properties of eigenfunctions and associated functions of indefi- nite Sturm-Liouville problems. In Nonclassical Problems of Mathematical Physics. Novosibirsk. Sobolev Institute of Mathematics, 240–251. (In Russian.) Baroper Schulstrasse 27 a, D-44225 Dortmund, Germany E-mail address: fleige.andreas@gmx.de Acta Wasaensia 77 ON PARSEVAL J-FRAMES Alan Kamuda and Sergii Kuz˙el This paper is dedicated to Seppo Hassi, to whom we would like to send our warmest wishes for his 60th birthday. All the best! 1 Introduction Frame theory finds many applications in engineering, applied mathematics, and computer sciences. It turns out to be useful because of its properties, which are unavailable for bases, e.g., non-unique decompositions. For instance, frames have been shown to be useful in signal processing applications when noisy channels are involved, because a frame allows one to reconstruct vectors (signals) even if some of the frame coefficients are missing (or corrupted), see, e.g., (Christensen, 2016; Heil, 2011; Mallat, 1999). Usually, frames are defined in a Hilbert space setting. LetH be a Hilbert space with an inner product (; ) and let J be the index set (countable or finite). Then a set of vectors F' = f'j : j 2 Jg is called a frame forH if there exist constants (frame bounds) 0 < A  B <1 such that Akfk2  X j2J j(f; 'j)j2  Bkfk2; f 2 H: (1.1) Furthermore, the set F' is called a Parseval frame if the inequalities (1.1) hold for the constants A = B = 1. A frame F = f j : j 2 Jg satisfying the condition f = X j2J (f; j)'j ; f 2 H; (1.2) is called a dual frame of F'. In general, dual frames are not determined uniquely and their proper choice (i.e., one that fits the specific problem well) is of great importance. A description of dual frames based on the Naimark dilation theorem is discussed in Kamuda & Kuz˙el (2020). Each Parseval frame F' is dual to itself and the reconstruction formula (1.2) can be simplified f = X j2J (f; 'j)'j ; f 2 H: (1.3) There are several advantages in considering frames in the setting of Kreı˘n spaces, instead of Hilbert spaces. For example, noise can be extracted from the signal in an easy way. Indeed, consider a signal with two dominants (e.g., given by high and low frequency signals). Let the projection P inH be an ideal low pass filter and let the projection I P in H be an ideal high pass filter. A signal (vector) ' 2 H, for which the equality kP'k = k(I P )'k holds, is considered as a noise. In other words, 78 Acta Wasaensia ' is considered to be a noise if its high frequency level is equal to its low frequency level. From a practical perspective, it is convenient to fix " > 0, and to say that ' is considered to be a noise if kP'k2 k(I P )'k2 < ": Every difference between the norms for ' below " implies that ' is a noise. From a Hilbert space view, one has to look at the decomposition ' = P'+ (I P )'. Hence, it is necessary to consider two frames fP'g [ f(I P )'g. From a Kreı˘n space perspective one can, using P , introduce a fundamental symmetry J = P (I P ) = 2P I; and define an indefinite inner product [f; g] = (Jf; g) = (Pf; Pg) ((I P )f; (I Pg)); f; g 2 H: Thus ' is considered as a noise when j[';']j < " (for a fixed "). For more details about this example, we encourage the reader to look at Giribet et al. (2012: Section 3). In the present paper, we extend the concept of Parseval frames to the setting of Kreı˘n spaces, com- bining the simplicity of the reconstruction formula (1.3) with the additional possibilities for the reduction of noises offered by Kreı˘n spaces. In the last few years, several papers devoted to the development of frame theory in Kreı˘n space have been published, e.g., (Acosta-Humánez, Esmeral & Ferrer, 2015; Escobar, Esmeral & Ferrer, 2016; Esmeral, Ferrer & Wagner, 2015; Esmeral, Ferrer & Lora, 2016; Giribet et al., 2012; Giribet, Maestripieri & Martínez Pería, 2018). A review of the definitions for frames in Kreı˘n spaces is given in Kamuda & Kuz˙el (2019). Our definition of Parseval J-frames is based on the concept of dual quasi-maximal subspaces intro- duced in Kamuda, Kuzhel & Sudilovskaja (2019) and discussed in Section 2. In Section 3 we define Parseval J-frames and establish their principal properties. In Section 4 we show that eigenfunctions of the harmonic oscillator H = d2dx2 + x2 + 2iax, associated with a PT -symmetric potential, constitute a Parseval J-frame, where J is the space parity operator. 2 Dual quasi-maximal subspaces Let H be a Hilbert space with inner product (; ) and let J be a non-trivial fundamental symmetry, i.e., J = J, J2 = I , and J 6= I . The space H endowed with the indefinite inner product (indefinite metric) [f; g] = (Jf; g); f; g 2 H; is called a Kreı˘n space; it will be denoted by (H; [; ]). All of the following topological notions are considered in the Hilbert space topology. A closed subspaceL of the Kreı˘n space (H; [; ]) is called neutral, negative, or positive if all nonzero elements f 2 L are, respectively, neutral [f; f ] = 0, negative [f; f ] < 0, or positive [f; f ] > 0. A subspace L of H is called definite if it is either positive or negative. Subspaces L of H are called dual if L+ is positive, L is negative, and they are orthogonal with respect to the indefinite metric (J-orthogonal), the latter meaning that [f+; f] = 0 for all f 2 L. Acta Wasaensia 79 In each of the above mentioned classes we can define maximal subspaces. For instance, a closed positive subspace Lmax is called maximal positive if Lmax is not a proper subspace of a positive subspace in the Kreı˘n space (H; [; ]). Similarly, subspaces Lmax are called dual maximal definite if they are dual, Lmax+ is maximal positive, and Lmax is maximal negative. The pair of subspacesH+ = ker (I J) andH = ker (I + J) in the fundamental decomposition of the Kreı˘n space H = H+ []H (2.1) is an example of dual maximal definite subspaces; the brackets in (2.1) mean thatH are orthogonal with respect to [; ]. The next result follows from (Albeverio & Kuzhel, 2015; Kamuda, Kuzhel & Sudilovskaja, 2019). Lemma 2.1. The subspaces Lmax are dual maximal definite if and only if there exists a self-adjoint operator Q inH that anticommutes with J , i.e., QJf = JQf; f 2 D(Q); such that Lmax+ = (I + tanhQ=2)H+ and Lmax = (I + tanhQ=2)H: (2.2) The self-adjoint strong contraction tanhQ=2 in (2.2) anticommutes with J and it characterizes the "deviation" of the dual maximal definite subspaces Lmax with respect to H. By a strong contraction we mean an operator T such that kTfk < kfk for nonzero f . In view of Lemma 2.1, a self-adjoint operator Q can be considered as a parameter describing all possible pairs of dual maximal definite subspaces Lmax . This fact allows one to associate with Lmax a new Hilbert spaceHQ, which is determined as the completion of the direct sum Dmax = Lmax+ [ _+]Lmax (2.3) with respect to the norm k  kQ = p (; ), generated by the inner product (f; g)Q = (eQf; g); f; g 2 Dmax: If Q is bounded, then Dmax = H, the space HQ coincides with H (as the set of elements), and (; )Q is equivalent to the initial inner product (; ). Let L be a pair of dual definite subspaces such that their J-orthogonal sum L+ [ _+]L (2.4) is dense inH. Due to the Phillips result (Phillips, 1961: Theorem 2.1), the pair L can be extended to dual maximal definite subspaces Lmax . In general, this extension is not determined uniquely, see (Kamuda, Kuzhel & Sudilovskaja, 2019; Langer, 1970). Definition 2.2. The dual definite subspaces L are called quasi-maximal if their direct sum (2.4) is dense in H and there exists extensions L ! Lmax to dual maximal definite subspaces Lmax such that (2.4) is a dense set in the Hilbert spaceHQ associated with Lmax . The next technical result was proved in Kamuda, Kuzhel & Sudilovskaja (2019: Lemma 4.7). 80 Acta Wasaensia Lemma 2.3. Let L+ and L be a pair of dual definite subspaces and let Lmax+ and Lmax be their extensions to the dual maximal definite subspaces, respectively. Moreover, letM =M+ M be a subspace ofH, whereM determine L  Lmax in (2.2), i.e., L+ = (I + tanhQ=2)M+; L = (I + tanhQ=2)M; M+  H+; M  H: (2.5) Then the direct sum (2.4) is dense in the Hilbert spaceHQ, constructed by Lmax , if and only if R(cosh1Q=2) \ (H M) = f0g: The direct sum (2.4) allows one to define the operator Sf = S(f+ + f) = Jf+ Jf; f 2 L: with the domain D(S) = L+ [ _+]L: It follows from (Kamuda, Kuzhel & Sudilovskaja, 2019) that S is a densely defined symmetric operator inH and (S(f+ + f); f+ + f) = [f+; f+] [f; f] > 0: Similarly, the direct sum (2.3) of dual maximal definite subspaces Lmax determines a positive self- adjoint operator Af = A(f+ + f) = Jf+ Jf; f 2 Lmax ; D(A) = Dmax; (2.6) which is an extension of S when the pair Lmax is an extension of L, see (Albeverio & Kuzhel, 2015; Kamuda, Kuzhel & Sudilovskaja, 2019). From the construction it follows that L = (I  JS)D(S). Therefore, S determines the subspaces L and one can expect that the quasi-maximality of L can be characterized in terms of self-adjoint extensions of S. For this reason, we recall from Arlinskii˘ et al. (2001) that a nonnegative self-adjoint extension A of S is called an extremal extension if inf f2D(S) (A( f); ( f)) = 0 for all  2 D(A): (2.7) The Friedrichs extension and the Kreı˘n-von Neumann extension are examples of extremal extensions of S. Theorem 2.4. The dual definite subspaces L are quasi-maximal if and only if there exists an extremal extension A = eQ of S, where Q anticommutes with J . Proof. Let L be quasi-maximal subspaces and let Lmax be the corresponding dual maximal def- inite subspaces as in Definition 2.2. Due to Lemma 2.1, the spaces Lmax are defined by (2.2). Therefore, vectors f 2 Lmax have the form f = (I + tanhQ=2)x, with x 2 H, and the operator A defined by (2.6) acts as follows: A(f+ + f) = (I tanhQ=2)x+ + (I tanhQ=2)x = eQ(f+ + f): (2.8) Here we used the relations Jx = x, eQ(I + tanhQ=2) = I tanhQ=2, and the fact that tanhQ=2 anticommutes with J . Therefore A = eQ is a self-adjoint extension of S. Now we Acta Wasaensia 81 should prove that A is an extremal extension. To do that, we rewrite (2.7) with the use of k  kQ, inf f2D(S) (A( f); ( f)) = inf f2D(S) k fk2Q = 0 for all  2 D(eQ); and take into account that D(S) is dense in the Hilbert spaceHQ. The converse statement is obvious: the required dual maximal definite subspaces Lmax in Definition 2.2 are determined by the formula Lmax = (I  JA)D(A), where A = eQ and Q anticommutes with J . 3 Parseval J-frames Definition 3.1. A set of vectors F' = f'j : j 2 Jg is called a Parseval J-frame if there exists a pair of dual quasi-maximal subspaces L such that each vector 'j 2 F' belongs either to L+ or L, and j[f; f]j = X j2J j[f; 'j ]j2 for all f 2 L: (3.1) Theorem 3.2. For each Parseval J-frame F' there exists a self-adjoint operator Q which anticom- mutes with J , such that F' is a Parseval frame in a new Hilbert spaceHQ. Proof. Each Parseval J-frame is associated with certain dual quasi-maximal subspacesL. Accord- ing to Definition 2.2, there exists a self-adjoint operator Q which anticommutes with J and which is such that the direct sum L+ + L is dense in the Hilbert spaceHQ. In view of (2.6) and (2.8), eQf = eQ(f+ + f) = Jf+ Jf; f 2 Lmax : This means that the subspaces Lmax are orthogonal with respect to the inner product (; )Q of HQ. Moreover, kfk2Q = j[f; f]j; (f; g)Q = [f; g]; f; g 2 L: (3.2) Taking these relations into account, we rewrite (3.1) as kfk2Q = kf+k2Q + kfk2Q = X j2J j(f; 'j)Qj2 (3.3) for f = f+ + f 2 L+ [ _+]L. The relation (3.3) is extended to all f 2 HQ with the use of (Christensen, 2016: Lemma 5.1.7). By Theorem 3.2 every Parseval J-frame F' turns out to be a Parseval frame in a suitably chosen Hilbert spaceHQ and the following reconstruction formula holds f = X j2J (f; 'j)Q 'j ; f 2 HQ; (3.4) see (1.3), where the series converges inHQ. Assuming that f = f+ + f 2 L+ [ _+]L and using 82 Acta Wasaensia the second relation in (3.2) we obtain (f; 'j)Q = (f+; 'j)Q = [f+; 'j ] = [f; 'j ]; 'j 2 L+; (f; 'j)Q = (f; 'j)Q = [f; 'j ] = [f; 'j ]; 'j 2 L: Therefore, for f 2 L+ [ _+]L, the series (3.4) takes the form f = X j2J j [f; 'j ]'j ; j = sgn(['j ; 'j ]): The obtained relation leads to the conclusion that [f; f ] = X j2J j j[f; 'j ]j2: As was mentioned above, one of the characteristic properties of J-frames is the possibility to in- terpret the signal f as a noise if its indefinite metric [f; f ] is close to 0. Given that fact, a Parseval J-frame F' turns out to be useful. The above formula allows one to extract vectors f , which repre- sent a noise by evaluation of the indefinite metric coefficients [f; 'j ]. The next statement shows that Parseval J-frames can be easily constructed. Theorem 3.3. Let F' = f'j : j 2 Jg be a complete set in a Hilbert space H. Then the following statements are equivalent: (i) F' is a Parseval J-frame. (ii) There exist a, not necessarily bounded, self-adjoint operator Q in H, which anticommutes with J , and a Parseval frame Fe = fej : j 2 Jg consisting of eigenfunctions of the operator J , i.e., Jej = ej or Jej = ej , such that 'j = e Q=2ej ; j 2 J: (3.5) Proof. (i)) (ii) In view of Theorem 3.2, there exists a self-adjoint operator Q in H that anticom- mutes with J and F' is a Parseval frame in the Hilbert spaceHQ. Setting f 2 D(eQ=2)  HQ in (3.4) and taking into account that 'j 2 L+ [ _+]L  D(eQ) we arrive at the conclusion that k k2 = X j2J j( ; eQ=2'j)j2; for all = eQ=2f in the setR(eQ=2) which is dense inH. By Christensen (2016: Lemma 5.1.9) the obtained equality can be extended to the whole spaceH. Hence, Fe = fej = eQ=2'j : j 2 Jg is a Parseval frame inH and the relation (3.5) holds. The definition of ej gives us that ej 2 D(eQ=2) \ D(eQ=2), since 'j 2 D(eQ). Therefore, it belongs to the domain of definition of the operator coshQ=2 = 12 (e Q=2 + eQ=2). Moreover, (I + tanhQ=2) coshQ=2 ej = (coshQ=2 + sinhQ=2) ej = e Q=2 ej = 'j 2 L: (3.6) Comparing the above relation with (2.2) and taking into account that ker (I + tanhQ=2) = f0g (since tanhQ=2 is a strong contraction) we arrive at the conclusion that coshQ=2 ej 2 H+ or coshQ=2 ej 2 H: (3.7) Acta Wasaensia 83 Since coshQ=2 commutes with J , one has J coshQ=2 = J 1 2 eQ=2 + eQ=2  = 1 2 eQ=2 + eQ=2  J = coshQ=2 J: Therefore formula (3.7) implies that ej 2 H+ = ker (I J) or ej 2 H = ker (I + J). (ii)) (i) In view of (3.5), ['j ; 'i] = (Je Q=2ej ; e Q=2ei) = (e Q=2Jej ; eQ=2ei) = (Jej ; ei) = 8<: (ej ; ei); ej ; ei 2 H+; (ej ; ei); ej ; ei 2 H; 0; ej 2 H; ei 2 H: This means that each non-zero vector f 2 span f'n = eQ=2en : en 2 H+g is positive, because [f; f ] = hX j cj'j ; X k ck'k i = X j;k cjck(ej ; ek) = X j cjej ; X k ckek  = keQ=2fk2: Similarly, vectors g 2 span f'j = eQ=2ej : ej 2 Hg are negative. Obviously, [f; g] = 0. Denote by L the completion of span f'j = eQ=2ej : ej 2 Hg inH. By the construction, L are dual definite subspaces. Furthermore, for each f 2 L+, [f; 'j ] = (Jf; e Q=2ej) = (f; Je Q=2ej) = (f; e Q=2Jej) = (eQ=2f; ej): Taking into account that Fe = fej : j 2 Jg is a Parseval frame, we getX j2J j[f; 'j ]j2 = X j2J j(eQ=2f; ej)j2 = keQ=2fk2 = (f; f)Q = [f; f ]; f 2 L+: Similarly, for f 2 L, [f; 'j ] = (eQ=2f; ej) andX j2J j[f; 'j ]j2 = X j2J j(eQ=2f; ej)j2 = keQ=2fk2 = (f; f)Q = [f; f ]: This analysis leads to the conclusion that (3.1) holds. To complete the proof, it suffices to verify that the spaces L are quasi-maximal. To this end, we show that L+ [ _+]L is a dense set inHQ. Taking into account that the spaces L are defined as completions of spanf'jg, as above, and com- paring the formulas (2.5) and (3.6) we arrive at the conclusion thatM coincides with the completion of span fcoshQ=2 ej : j 2 Jg. Hence, H M = H span fcoshQ=2 ejg: Assume that L+ [ _+]L is not dense in HQ. Then, in view of Lemma 2.3, there exists a vector p = cosh1Q=2u 6= 0, such that p 2 H M, i.e., 0 = (p; coshQ=2 ej) = (cosh 1Q=2u; coshQ=2 ej) = (u; ej) for all ej 2 Fe: Since Fe is a Parseval frame in H, this means that u = 0 and ,hence, p = 0; a contradiction. Consequently, L+ [ _+]L is a dense set inHQ. 84 Acta Wasaensia 4 Harmonic oscillator with PT -symmetric potential In the spaceH = L2(R) we consider the fundamental symmetry J = P , where Pf(x) = f(x) is the space parity operator. The subspaces H of the fundamental decomposition (2.1) coincide with the subspaces of even and odd functions of L2(R). The Hermite functions ej(x) = 1p 2jj! p  Hj(x)e x2=2; Hj(x) = ex 2=2  x d dx j ex 2=2; j 2 J = N [ f0g; are eigenfunctions, H0ej = (2j + 1)ej , of the harmonic oscillator H0 = d 2 dx2 + x2; D(H0) = ff 2W 22 (R) : x2f 2 L2(R)g and they form an orthonormal basis Fe = fej : j 2 Jg of L2(R). The functions ej are even or odd for j being even or odd, respectively. This means that ej 2 H+ or ej 2 H. Since the functions ej(x) are entire functions on C, their complex shift can be defined: 'j(x) := ej(x+ ia); a 2 R n f0g; j = 0; 1; 2; : : : The set F' = f'j : j 2 Jg is complete in L2(R), see (Mityagin, Siegl & Viola, 2017: Lemma 2.5); in the following the dependence on a will not be explicitly indicated. Applying the Fourier transform Ff = 1p 2 Z 1 1 eixf(x) dx to 'j , we obtain F'j = eaFej . Therefore, 'j = F1eaFej : The last relation can be rewritten as 'j = e Q=2ej ; where Q = 2ai ddx ; D(Q) = W 12 (R), is a self-adjoint operator in L2(R) that anticommutes with J = P in L2(R). By virtue of Theorem 3.3, the set F' is a Parseval J-frame. The set F' cannot be a Schauder basis in L2(R). Indeed, assume that F' is a Schauder basis. Then, by Heil (2011: Theorem 4.13), 1  k'jk k jk  C; j 2 J; where F = f j : j 2 Jg is the bi-orthogonal sequence for F'. It is easy to see that j = sgn(['j ; 'j ])J'j : Hence, the last inequalities take the form 1  k'jk2  C: On the other hand, by (Mityagin, Siegl & Viola, 2017: Theorem 2.6) lim j!1 1p j log k'jk2 = 23=2jaj; which contradicts the inequality k'jk2  C. Hence, F' cannot be a Schauder basis. The functions of the Parseval J-frame F' are simple eigenfunctions, H'j = (2j + 1 + a2)'j , of Acta Wasaensia 85 the non-self-adjoint operator: H = d 2 dx2 + x2 + 2iax; D(H) = D(H0); see (Mityagin, Siegl & Viola, 2017: Lemma 2.4). The operatorH can be considered as a perturbation of the harmonic oscillator H0 by a PT -symmetric potential V (x) = 2iax, i.e., H = H0 + V . The PT -symmetry of V (x) means that PT V (x) = V (x)PT , where T is the complex conjugation operator, i.e., T f = f . References Acosta-Humánez, P., Esmeral, K. & Ferrer, O. (2015). Frames of subspaces in Hilbert spaces with W -metrics. An. S¸tiint¸. Univ. "Ovidius" Constant¸a Ser. Mat. 23, 5–22. Albeverio, S. & Kuzhel, S. (2015). PT -symmetric operators in quantum mechanics: Krein spaces methods. In Non-Selfadjoint Operators in Quantum Physics. Hoboken, N.J.: Wiley. Arlinskiı˘, Y.M., Hassi, S., Sebestyén, Z. & de Snoo, H.S.V. (2001). On the class of extremal exten- sions of a nonnegative operator. Oper. Theory Adv. Appl. 127, 41–81. Christensen, O. (2016). An Introduction to Frames and Riesz Bases. Second edition. Boston: Birkhäuser. Escobar, G., Esmeral, K. & Ferrer, O. (2016). Construction and coupling of frames in Hilbert spaces with W -metrics. Rev. Integr. Temas Mat. 34, 81–93. Esmeral, K., Ferrer, O. & Lora, B. (2016). Dual and similar frames in Krein spaces. I. J. Math. Anal. 10, 932–952. Esmeral, K., Ferrer, O. & Wagner, E. (2015). Frames in Krein spaces arising from a non-regular W -metric. Banach J. Math. Anal. 9, 1–16. Giribet, J.I., Maestripieri, A. & Martínez Pería, F. (2018). Duality for frames in Krein spaces. Math. Nachr. 291, 879–896. Giribet, J. I., Maestripieri, A., Martínez Pería, F. & Massey, P. G. (2012). On frames for Krein spaces. J. Math. Anal. Appl. 393, 122–137. Heil, C. (2011). A Basis Theory Primer. Boston, Mass.: Birkhäuser. Kamuda, A. & Kuz˙el, S. (2019). On J-frames related to maximal definite subspaces. Ann. Funct. Anal. 10, 106–121. Kamuda, A. & Kuz˙el, S. (2020). On description of dual frames. arXiv:2007.09216. Kamuda, A., Kuzhel, S. & Sudilovskaja, V. (2019). On dual definite subspaces in Krein space. Complex Anal. Oper. Theory 13, 1011–1032. Langer, H. (1970). Maximal dual pairs of invariant subspaces of J-self-adjoint operators. Mat. Za- metki 7, 443–447. (In Russian: English translation in Math. Notes 7 (1970), 269–271.) Mallat, S. (1999). A Wavelet Tour of Signal Processing. San Diego, CA: Academic Press. 86 Acta Wasaensia Mityagin, B., Siegl, P. & Viola, J. (2017). Differential operators admitting various rates of spectral projection growth. J. Funct. Anal. 272, 3129–3175. Phillips, R.S. (1961). The extension of dual subspaces invariant under an algebra. Proc. of the Inter- national Symposium on Linear Spaces, Jerusalem 1960, 366–398. Oxford: Pergamon Press. Department of Applied Mathematics, AGH University of Science and Technology, 30-059 Kraków, Poland E-mail address: kuzhel@agh.edu.pl, kamuda@agh.edu.pl Acta Wasaensia 87 IDEMPOTENT RELATIONS, SEMI-PROJECTIONS, AND GENERALIZED INVERSES Jean-Philippe Labrousse, Adrian Sandovici, Henk de Snoo, and Henrik Winkler Dedicated to our friend Seppo Hassi on the occasion of his sixtieth birthday 1 Introduction As a motivation for this paper, consider the identities ABA = A and BAB = B; (1.1) where A 2 B(H;K) and B 2 B(K;H) for Hilbert spaces H and K, which have been studied in the literature in much detail. Clearly, if (1.1) holds, then the operatorsBA 2 B(H) andAB 2 B(K) are both idempotent. IfA is invertible, i.e.,A1 2 B(K;H), thenB = A1 makes (1.1) valid. However, given any A 2 B(H;K) with ran A closed in K, there are also candidates for B 2 B(K;H), so that (1.1) is satisfied and so that, in addition, (BA) = BA and (AB) = AB; (1.2) thus BA and AB are orthogonal projections. In the matrix case, all this goes back to E.H. Moore (1920), A. Bjerhammar (1951), and R. Penrose (1955); see also, for instance, (Ben-Israel & Greville, 2003; Campbell & Meyer, 1991; Nashed, 1976; Rao & Mitra, 1971). An extension to the case that the operators A andB are unbounded can be found in Labrousse & M’Bekhta (1992) and Labrousse (1992). The purpose of the present paper is to look at the formal aspects of the identities in (1.1) and (1.2) in the wider context of linear relations, but in the absence of any topology, and to give a survey of the characteristic results. In order to consider the identities (1.1) in an algebraic setting, let H and K be linear spaces, let A be a linear relation from H to K, and let B be a linear relation from K to H. Products of linear relations will be in the sense of relations. When the identities (1.1) are satisfied by linear relations A and B, then it is clear that the products BA and AB are idempotent relations in H and K, respectively; cf. Definition 5.1. In the present general context the conditions in (1.2) are replaced by ran BA  dom BA and ran AB  dom AB; (1.3) i.e., BA and AB are semi-projections; cf. Definition 5.3. The notions of idempotent relation and semi-projection go back to Labrousse (2003). Note, in particular, that if B = A1, the formal inverse ofA, then the linear relationsA1A andAA1 are indeed semi-projections; cf. (2.1). There are various other possibilities to satisfy (1.1) and (1.3) by considering, instead of A1, specific choices for an algebraic operator part of A1. Here is an outline of the contents of this paper. Some useful properties are recalled in Section 2. As a preparation there are several modifications of the notion of idempotent relation that will be 88 Acta Wasaensia considered in Section 3 and Section 4. Idempotent relations and semi-projections are discussed in Section 5. Also the geometric meaning of semi-projections is explained there. Section 6 contains a resolvent identity for semi-projections. In Sections 7, 8, and 9 the cases A  ABA, ABA  A, and A = ABA are considered, respectively; they serve as illustrations for the earlier sections. In Section 10 the previous cases are characterized in terms of set inclusions between A and B1. The notion of generalized inverse for linear relations is introduced in Section 11. It is shown in Section 12 that one may choose an operator part for A1, so that it serves as a generalized inverse. In Section 13 the results from Sections 3 and Section 4 are augmented. 2 Preliminaries Let H and K be linear spaces and let A be a linear relation from H to K. Recall that a linear relation A is defined as a linear subspace of the product space HK, with dom A, ran A, kerA, and mul A being the domain, range, kernel, and multivalued part of A; cf. (Arens, 1961; Sandovici, de Snoo & Winkler, 2007); see also (Behrndt, Hassi & de Snoo, 2020). The inverse of a relation A is given by A1 := ffg; fg : ff; gg 2 Ag; which is a linear relation from K to H. It is not difficult to see that the following componentwise sum decompositions hold A1A = IdomA b+ (f0g  kerA) = IdomA b+ (kerA f0g); AA1 = IranA b+ (mul A f0g) = IranA b+ (f0g mul A); (2.1) cf. (Behrndt, Hassi & de Snoo, 2020). Here, and in the following, all such products are meant in the sense of linear relations. In particular, it is useful to observe that for any linear relation A one has the identities AA1A = A and A1AA1 = A1; (2.2) as can be easily verified by means of the identities in (2.1). Note that (2.1) shows that the terminology of inverse relation is a rather formal way of speaking. Let L and R be linear relations from H to K such that L  R. If C is a linear relation from K to K0, then CL  CR, while if C is a linear relation from H0 to H, then LC  RC; cf. (Arens, 1961). Next a number of useful statements will follow. Lemma 2.1. Let H and K be linear spaces, and let L and R be linear relations from H to K. Then the following statements are equivalent: (i) L  R and dom L  dom R; (ii) R = L b+ (f0g mul R). Moreover, the following statements are equivalent: (iii) L  R and ran L  ran R; (iv) R = L b+ (kerR f0g). Acta Wasaensia 89 Proof. By symmetry it suffices to show the equivalence between (i) and (ii). (i) ) (ii) It suffices to show that R  L b+ (f0g  mul R). For this purpose, let fh; h0g 2 R. Since h 2 dom R  dom L, there exists an element k0 2 K such that fh; k0g 2 L. Hence, with '0 = h0 k0, it follows that fh; h0g = fh; k0g+ f0; '0g; and thus f0; '0g 2 R or '0 2 mul R. Hence (ii) follows. (ii)) (i) This implication is trivial. Corollary 2.2. Let H and K be linear spaces, and let L and R be linear relations from H to K. Then the following statements are equivalent: (i) L = R; (ii) L  R, dom L  dom R, and mul L  mul R; (iii) L  R, ran L  ran R, and kerL  kerR. Corollary 2.3. Let H and K be linear spaces, and let L and R be linear relations from H to K. Then the following statements hold: (a) If L  R, dom L = H, and mul R = f0g, then L = R. (b) If L  R, ran L = K, and kerR = f0g, then L = R. Definition 2.4. Let H and K be linear spaces, and let L and R be linear relations from H to K. The relation L is said to be an algebraic operator part of R if (a) L  R; (b) dom L = dom R; (c) mul L = f0g. As a consequence of Lemma 2.1 (ii), the following characterization of the algebraic operator part holds. Corollary 2.5. Let H and K be linear spaces, and let L and R be linear relations from H to K. Then L is an algebraic operator part of R if and only if R = L b+ (f0g mul R); direct sum: Definition 2.6. Let H and K be linear spaces, and letR be a linear relation from H to K. The relation R is said to be decomposable if there exists an algebraic operator part L of R. In order to characterize the notion of a decomposable relation, it is helpful to introduce projections. 90 Acta Wasaensia Definition 2.7. Let H be a linear space with linear subspaces X, Y, and Z, and assume the direct sum decomposition X = Y+ Z: The projection P from X onto Z, parallel to Y, associated with this decomposition, is defined by Px := z, when x = y + z, x 2 X, y 2 Y, and z 2 Z. Clearly, P is a well-defined linear operator in H, which is idempotent. Moreover, one sees that dom P = X; ran P = Z; and kerP = Y: It is clear that a relation R is decomposable if and only if there exists a projection P such that P : ran R! mul R: Let L be an algebraic operator part of R, in other words, R = L b+ (f0gmul R), direct sum. Then the projection P from R onto f0g mul R has the property kerP = L. Conversely, any projection P from R onto f0g mul R leads to an algebraic operator part of R. Lemma 2.8. Let H and K be linear spaces, let R be a linear relation from H to K, and let P be a projection from ran R onto mul R. Then Pfx; yg = f0; Pyg; fx; yg 2 R; (2.3) defines a projection P from R onto f0g mul R. Lemma 2.9. Let H and K be linear spaces, and let P be a projection from R onto f0g  mul R. Then P is of the form (2.3), for some projection P from ran R to mul R, if and only if Pfx; 0g = f0; 0g; fx; 0g 2 R: (2.4) Proof. Assume that the projection P is of the form (2.3). Then the property (2.4) follows from property (2.3). For the converse, let P be the projection from R onto f0g mul R. Then for fx; yg 2 R one has fx; yg = fu; vg+ f0; 'g with fu; vg 2 kerP and f0; 'g = Pfx; yg: It is clear that with this decomposition P = ffy; 'g : fu; vg 2 R; f0; 'g 2 f0g mul Rg defines a linear relation from ran R onto mul R. Moreover, P is the graph of a linear operator, in fact of a projection, if the projection P satisfies (2.4). In this case ' = Py and P is of the form (2.3). The discussion of operator parts in the presence of topologies is interesting. For the case of Hilbert spaces, see, for instance, (Hassi, de Snoo & Szafraniec, 2009). Acta Wasaensia 91 3 Linear relations included in a product Let H and K be linear spaces and let P be a linear relation from H to K. If Q is a linear relation in H, then one can consider the product relation PQ, while if Q is a linear relation in K one can consider the product relation QP . First consider the case that Q is a linear relation in H. Then PQ is a linear relation from H to K and it is clear that dom PQ  dom Q and ran PQ  ran P . Hence, the following implication holds P  PQ )  dom P  dom PQ  dom Q; ran P = ran PQ: (3.1) Therefore, Lemma 2.1 shows that P  PQ , PQ = P b+ (kerPQ f0g): (3.2) Such relations have some useful properties. Lemma 3.1. Let H and K be linear spaces, and let P be a linear relation from H to K. Assume that Q is a linear relation in H such that P  PQ. Then dom P  kerP + ran Q: (3.3) As a consequence the following equivalences hold: kerP  ran Q , dom P  ran Q; (3.4) and dom P = kerP + ran Q , ran Q  dom P: (3.5) Proof. To see (3.3), let x 2 dom P . Then fx; yg 2 P for some y 2 K. Since P  PQ there exists an element z 2 H such that fx; zg 2 Q and fz; yg 2 P . Hence fx z; 0g = fx; yg fz; yg 2 P; which shows that x z 2 kerP . Thus one concludes x = (x z) + z 2 kerP + ran Q. This shows (3.3). Clearly, the equivalences in (3.4) and (3.5) are consequences of (3.3). Next consider the case that Q is a linear relation in K. Then QP is a linear relation from H to K and it is clear that dom QP  dom P and ran QP  ran Q. Hence, it follows that P  QP )  ran P  ran QP  ran Q; dom P = dom QP: Therefore, Lemma 2.1 shows that P  QP , QP = P b+ (f0g mul QP ): (3.6) Such relations have some useful properties; cf. Lemma 3.1. 92 Acta Wasaensia Lemma 3.2. Let H and K be linear spaces, and let P be a linear relation from H to K. Assume that Q is a linear relation in K such that P  QP . Then ran P  mul P + dom Q: As a consequence the following equivalences hold: mul P  dom Q , ran P  dom Q; and ran P = mul P + dom Q , dom Q  ran P: 4 Linear relations containing a product Let H and K be linear spaces and let P be a linear relation from H to K. As in the previous section the interplay between P and another linear relation Q is considered. First consider the case that Q is a linear relation in H. Then PQ is a linear relation from H to K and it is clear that kerQ  kerPQ and mul P  mul PQ. Hence it follows that PQ  P )  kerQ  kerPQ  kerP; mul PQ = mul P: (4.1) The relations that satisfy PQ  P have some useful properties. Lemma 4.1. Let H and K be linear spaces, and let P be a linear relation from H to K. Assume that Q is a linear relation in H such that PQ  P . Then dom P \mul Q  kerP: (4.2) As a consequence the following equivalences hold: mul Q  dom P , mul Q  kerP; (4.3) and dom P \mul Q = kerP , kerP  mul Q: (4.4) Proof. To see (4.2), let x 2 dom P \mul Q. Then fx; yg 2 P for some y 2 K, and f0; xg 2 Q. Hence f0; yg 2 PQ  P and therefore fx; 0g = fx; ygf0; yg 2 P , which shows that x 2 kerP . This shows (4.2). Clearly, the equivalences in (4.3) and (4.4) are consequences of (4.2). Next consider the case that Q is a linear relation in K. Then QP is a linear relation from H to K and it is clear that kerP  kerQP and mul Q  mul QP . Hence it follows that QP  P )  mul Q  mul QP  mul P; kerQP = kerP: The linear relations that satisfy QP  P have some useful properties; cf. Lemma 4.1. Acta Wasaensia 93 Lemma 4.2. Let H and K be linear spaces, and let P be a linear relation from H to K. Assume that Q is a linear relation in K such that QP  P . Then ran P \ kerQ  mul P: As a consequence the following equivalences hold: kerQ  ran P , kerQ  mul P; and ran P \ kerQ = mul P , mul P  kerQ: 5 Idempotent linear relations and semi-projections In this section the notions of idempotent linear relation and semi-projection are introduced. These definitions and corresponding lemmas go back to Labrousse (2003). The main aim of this section is to characterize semi-projections in terms of various equivalent conditions. Definition 5.1. Let P be a linear relation in a linear space H. Then P is said to be idempotent if P 2 = P . Lemma 5.2. Let P be a linear relation in a linear space H. Then P is idempotent if and only if I P is idempotent. Proof. It suffices to assume that P is idempotent and to prove that IP is idempotent, i.e., to show that (I P ) = (I P )2. () Let fx; yg 2 I P , then fx; x yg 2 P = P 2, so that fx; zg 2 P and fz; x yg 2 P for some z 2 H . Hence fx+ z; x+ z yg 2 P or fx+ z; yg 2 I P , so that fx z; yg = f2x; 2yg fx+ z; yg 2 I P: Together with fx; x zg 2 I P , this gives fx; yg 2 (I P )2. Thus I P  (I P )2. () Let fx; yg 2 (IP )2, then fx; zg 2 IP and fz; yg 2 IP for some z 2 H . Consequently, fx; x zg 2 P and fz; z yg 2 P , so that fx z; x + y 2zg 2 P . Thus one also has fx; x+ y 2zg 2 P 2 = P , so that fx; 2z yg 2 I P . This leads to fx; yg = f2x; 2zg fx; 2z yg 2 I P: Thus (I P )2  I P . Definition 5.3. Let P be a linear relation in a linear space H. Then P is said to be a semi-projection if P is idempotent and ran P  dom P . Lemma 5.4. Let P be a linear relation in a linear space H. Then P is a semi-projection if and only if I P is a semi-projection. 94 Acta Wasaensia Proof. It suffices to show that with P also I P is a semi-projection. For this purpose, assume that P is a semi-projection; thus P is idempotent and satisfies ran P  dom P . Consequently I P is idempotent by Lemma 5.2 and from the inclusion ran P  dom P follows that ran (I P )  dom P + ran P = dom P = dom (I P ): Thus I P is a semi-projection. Observe that P is idempotent if and only if P  P 2 and P 2  P . Hence the results from Section 3 and Section 4 (with Q = P ) may be applied. Proposition 5.5. Let P be an idempotent relation in a linear spaceH. Then the following statements are equivalent: (i) dom P = kerP + ran P ; (ii) ran P  dom P ; (iii) mul P  kerP ; (iv) kerP \ ran P = mul P . Consequently, P is a semi-projection if and only if one of the preceding conditions holds. Proof. (i), (ii) This follows from Lemma 3.1 with K = H and Q = P . (ii)) (iii) The assumption ran P  dom P implies mul P  dom P . Consequently, the equiva- lence in (4.3) of Lemma 4.1 with K = H and Q = P yields that mul P  kerP . (iii) ) (ii) The assumption mul P  kerP implies mul P  dom P . Consequently, the first equivalence in Lemma 3.2 with K = H and Q = P yields that ran P  dom P . (iii), (iv) This follows from Lemma 4.2 with K = H and Q = P . Furthermore, it is useful to note that with P also P1 is idempotent, which leads to the following proposition. Proposition 5.6. Let P be an idempotent relation in a linear spaceH. Then the following statements are equivalent: (i) ran P = mul P + dom P ; (ii) dom P  ran P ; (iii) kerP  mul P ; (iv) dom P \mul P = kerP . Consequently, P1 is a semi-projection if and only if one of the preceding conditions holds. Acta Wasaensia 95 Proof. Since with P also the inverse P1 is idempotent, the equivalence of the items (i)–(iv) follows from applying Proposition 5.5 to the relation P1. The notion of semi-projection has a simple geometric interpretation; cf. Definition 2.7. This geo- metric explanation goes back to discussions with Seppo Hassi. Proposition 5.7. Let H be a linear space. Let X, Y, and Z be linear subspaces of H, and assume that X = Y+ Z: (5.1) Then the linear relation P in H, defined by P = ffx; zg : x = y + z; x 2 X; y 2 Y; z 2 Zg; (5.2) is a semi-projection with the properties dom P = X; ran P = Z; kerP = Y; and mul P = Y \ Z: (5.3) Moreover, every semi-projection in H is of this form. Proof. Assume that (5.1) is satisfied. Then it is clear thatP in (5.2) is a well-defined linear relation in H. Furthermore, P is idempotent. To see thatP  P 2, observe that fx; zg 2 P implies fx; zg 2 P 2, since fz; zg 2 P . Likewise, to see that P 2  P , let fx; zg 2 P 2. Then fx; g 2 P and f ; zg 2 P for some 2 Z, and consequently, with some ' 2 Y and  2 Y one has x = '+ and = + z: Therefore x = '+ + z with '+  2 Y and, hence, fx; zg 2 P . Consequently, P is idempotent. In addition, one sees that (5.3) is satisfied. Thus, it follows from Proposition 5.5 that P in (5.2) is a semi-projection. Now let P be any semi-projection in H, so that P is idempotent and dom P = kerP + ran P ; cf. Proposition 5.5. Let fx; zg 2 P . Then, by assumption, x = + with 2 kerP and 2 ran P , i.e., f ; 0g 2 P and f ; g 2 P for some 2 H. Note that fx; zg = f ; 0g+ f ; zg; which implies that f ; zg 2 P . As f ; g 2 P , one also sees that f ; zg 2 P 2 = P , so that z 2 mul P . Since mul P = kerP \ ran P  kerP by Proposition 5.5, it follows that x = + = + z + z; where + z 2 kerP: Thus the assertion follows with X = dom P , Y = kerP , and Z = ran P . In the context of Proposition 5.7 one can view P as a multivalued projection from X onto Z, parallel to Y, associated with the not necessarily direct decomposition (5.1). 96 Acta Wasaensia 6 A resolvent formula for semi-projections This section contains a formula for the resolvent relation of a semi-projection in H. It may be convenient to first remember that for any linear relation P in H and any  2 C the linear relation (I P )1 is called the resolvent relation of P (I P )1 = ffx y; xg : fx; yg 2 Pg: Then it is clear that fx; yg 2 P , fx y; xg 2 (I P )1; (6.1) and setting y = x in (6.1) gives the equivalence fx; xg 2 P , f0; xg 2 (I P )1: (6.2) Now let P be a semi-projection in a linear space H, so that ran P  dom P or, equivalently, mul P  kerP , see Proposition 5.5. Then the following observations are straightforward. Lemma 6.1. Let P be a semi-projection in a linear space H. Then for  2 C n f0; 1g fx; xg 2 P ) x 2 mul P; (6.3) while for all  2 C x 2 mul P ) fx; xg 2 P: (6.4) Proof. Assume that fx; xg 2 P for some  2 C. Then, since P is idempotent, fx; 2xg 2 P , and hence f0; ( 1)xg 2 P . Thus (6.3) has been verified. Since mul P  kerP , it is clear that x 2 mul P implies that fx; xg 2 P . Thus (6.4) has been verified. As a consequence of (6.4), it can be noted that semi-projections, that are not operators, have non- trivial singular chains; cf. (Sandovici, de Snoo & Winkler, 2004) and (Berger, de Snoo, Trunk & Winkler, 2021). In the general case of semi-projections, the resolvent identity in the following proposition is an identity between linear relations. Proposition 6.2. Let P be a semi-projection in a linear space H and let  2 C n f0; 1g. Then (I P )1 =  1  I + 1 ( 1)P  b+ (f0g mul P ): (6.5) Proof. Assume that P is a semi-projection and that  2 C. If x 2 mul P , then fx; xg 2 P by Lemma 6.1. Hence, by (6.2), it follows that f0; xg 2 (I P )1. Consequently, one sees that mul P  mul (I P )1;  2 C: (6.6) Moreover, by (6.1), every element in (I P )1 is of the form fx y; xg for some fx; yg 2 P . Thus, thanks to ran P  dom P , it therefore follows that dom (I P )1  dom P;  2 C: (6.7) Acta Wasaensia 97 Now the restriction  2 C n f0; 1g will be assumed and the following inclusion will be established: (( 1)I + P ) b+ (f0g mul P )  ( 1)(I P )1: (6.8) Observe that (6.6) gives mul P  mul ( 1)(I P )1, and thus (f0g mul P )  ( 1)(I P )1: Hence, in order to establish (6.8), it remains to show that (( 1)I + P )  ( 1)(I P )1: For this, observe that every element in ( 1)I + P is of the form fx; ( 1)x+ yg for some fx; yg 2 P . Hence, the required inclusion follows, once it is recalled that fx; yg 2 P implies that x y 2 kerP . Thus the inclusion (6.8) has been established. The inclusion in (6.7) guarantees that in (6.8) the domain of the right-hand side is contained in the domain of the left-hand side when  2 C n f0; 1g. Thanks to the equivalence (i), (ii) in Lemma 2.1, one concludes that there is equality in (6.8). It is clear that equality in (6.8) is equivalent to (6.5). 7 The inclusion A  ABA Let H and K be linear spaces, let A be a linear relation from H to K, and let B be a linear relation from K to H. Observe that the inclusion A  ABA can be written as A  A(BA) or A  (AB)A: (7.1) Hence, the inclusion A  ABA leads to some automatic identities. Lemma 7.1. Assume that A  ABA. Then (a) dom A = dom ABA = dom BA; (b) ran A = ran ABA = ran AB. Proof. To see (a) apply the first conclusion in (3.1) with P = A and Q = BA, then dom A  dom ABA  dom BA; and note that dom BA  dom A. To see (b) apply the second conclusion in (3.1) with P = A and Q = BA, then ran A = ran ABA; and note that ran ABA  ran AB  ran A. Due to the first inclusion in (7.1), Lemma 3.1 implies the following lemma. 98 Acta Wasaensia Lemma 7.2. Assume that A  ABA. Then dom A  kerA+ ran BA: As a consequence the following equivalences hold: kerA  ran BA , dom A  ran BA; and dom A = kerA+ ran BA , ran BA  dom A: Due to the second inclusion in (7.1), Lemma 3.2 implies the following lemma. Lemma 7.3. Assume that A  ABA. Then ran A  mul A+ dom AB: As a consequence the following equivalences hold: mul A  dom AB , ran A  dom AB; and ran A = mul A+ dom AB , dom AB  ran A: 8 The inclusion ABA  A Let H and K be linear spaces, let A be a linear relation from H to K, and let B be a linear relation from K to H. Observe that the inclusion ABA  A can be written as A(BA)  A or (AB)A  A: (8.1) Hence, the inclusion ABA  A leads to some automatic identities. Lemma 8.1. Assume that ABA  A. Then (a) kerBA = kerA = kerABA; (b) mul AB = mul A = mul ABA. Proof. To see (a) apply the first conclusion in (4.1) with P = A and Q = BA, then kerBA  kerABA  kerA: Hence, (a) holds, because kerA  kerBA. To see (b) apply the second conclusion in (4.1) with P = A and Q = BA, then mul ABA = mul A; and note that mul A  mul AB  mul ABA. Acta Wasaensia 99 Due to the first inclusion in (8.1), Lemma 4.1 implies the following lemma. Lemma 8.2. Assume that ABA  A. Then dom A \mul BA  kerA: As a consequence the following equivalences hold: mul BA  dom A , mul BA  kerA; and dom A \mul BA = kerA , kerA  mul BA: Due to the second inclusion in (7.1), Lemma 4.2 implies the following lemma. Lemma 8.3. Assume that ABA  A. Then ran A \ kerAB  mul A: As a consequence the following equivalences hold: kerAB  ran A , kerAB  mul A; and ran A \ kerAB = mul A , mul A  kerAB: 9 The identity A = ABA Let H and K be linear spaces, let A be a linear relation from H to K, and let B be a linear relation from K to H. The identity A = ABA leads to some automatic identities, which can be seen by combining Lemma 7.1 and Lemma 8.1 for the inclusions A  ABA and ABA  A, respectively. Lemma 9.1. Assume that A = ABA. Then (a) dom BA = dom A; (b) ran AB = ran A; (c) kerBA = kerA; (d) mul AB = mul A. Furthermore, note that A = ABA implies that the products AB and BA are automatically idempo- tent relations in H and K, respectively; cf. Definition 5.1. To see this, recall that an identity between relations may be multiplied from the left or from the right remaining an identity; cf. (Arens, 1961). An application of the equivalences in Proposition 5.5 and Proposition 5.6 for the products AB and BA leads to the two following propositions giving necessary and sufficient conditions for AB and BA to be semi-projections; cf. Definition 5.3. Note that the identities from Lemma 9.1 have been used in the formulation of the following descriptions. 100 Acta Wasaensia Proposition 9.2. Assume that ABA = A. Then the following statements are equivalent: (i) dom AB = kerAB + ran A; (ii) ran A  dom AB; (iii) mul A  kerAB; (iv) kerAB \ ran A = mul A. Consequently, the idempotent relation AB is a semi-projection if and only if one of the conditions (i)-(iv) holds. Moreover, the following statements are equivalent: (v) ran A = dom AB + mul A; (vi) dom AB  ran A; (vii) kerAB  mul A; (viii) dom AB \mul A = kerAB. Consequently, the idempotent relation (AB)1 is a semi-projection if and only if one of the condi- tions (v)-(viii) holds. Proposition 9.3. Assume that ABA = A. Then the following statements are equivalent: (i) dom A = kerA+ ran BA; (ii) ran BA  dom A; (iii) mul BA  kerA; (iv) kerA \ ran BA = mul BA. Consequently, the idempotent relation BA is a semi-projection if and only if one of the conditions (i)-(iv) holds. Moreover, the following statements are equivalent: (v) ran BA = dom A+ mul BA; (vi) dom A  ran BA; (vii) kerA  mul BA; (viii) dom A \mul BA = kerA. Consequently, the idempotent relation (BA)1 is a semi-projection if and only if one of the condi- tions (v)-(viii) holds. Acta Wasaensia 101 10 Characterizations of inclusions Let H and K be linear spaces, let A be a linear relation from H to K, and let B be a linear relation from K to H. First a simple but useful observation is presented. Lemma 10.1. Let H and K be linear spaces, let A be a linear relation from H to K, and let B be a linear relation from K to H. Then the following equivalences hold: f0g mul A  (kerA f0g) b+B1 , mul A  kerAB; (10.1) and kerA f0g  B1 b+ (f0g mul A) , kerA  mul BA: (10.2) Proof. By symmetry it suffices to show (10.1). ()) Let  2 mul A. Then f0; g = fx; 0g+ fx; g with x 2 kerA and f;xg 2 B. Hence, it follows that  2 kerAB. (() Let  2 mul A. Then  2 kerAB implies f; xg 2 B and fx; 0g 2 A, which gives that f0; g = fx; 0g+ fx; g. Therefore f0g mul A  (kerA f0g) b+B1. Next it will be shown that each of the inclusions A  ABA and A  ABA gives a certain interplay between A and B1. Lemma 10.2. Let H and K be linear spaces, let A be a linear relation from H to K, and let B be a linear relation from K to H. Then the following statements are equivalent: (i) A  ABA; (ii) A  (kerA f0g) b+B1 b+ (f0g mul A). Moreover, the following statements are equivalent: (iii) A  ABA and mul A  kerAB; (iv) A  (kerA f0g) b+B1. Finally, the following statements are equivalent: (v) A  ABA and kerA  mul BA; (vi) A  B1 b+ (f0g mul A). Proof. (i) ) (ii) Let fu; vg 2 A. Then, by assumption, fu; vg 2 ABA, and there exist elements s 2 K and t 2 H such that fu; sg 2 A; fs; tg 2 B; ft; vg 2 A; 102 Acta Wasaensia which, since fu; vg 2 A, implies that fu t; 0g 2 A and f0; v sg 2 A: Hence, it follows that fu; vg = fu t; 0g+ ft; sg+ f0; v sg 2 (kerA f0g) b+B1 b+ (f0g mul A): Thus (ii) has been shown. (ii) ) (i) Let fu; vg 2 A. Then, by assumption, there exist elements 2 kerA and 2 mul A, such that fu; vg = f ; 0g+ fu ; v g+ f0; g; where fv ; u g 2 B. Since fu; v g 2 A and fu ; vg 2 A, it follows that fu; vg 2 ABA. (iii) , (iv) This follows from the equivalence (i) , (ii) and the equivalence (10.1) contained in Lemma 10.1. (v) , (vi) This follows from the equivalence (i) , (ii) and the equivalence (10.2) contained in Lemma 10.1. As a direct corollary of the equivalences of (i) and (ii) in Lemma 10.2 one obtains the following characterization. Corollary 10.3. Let H and K be linear spaces, let A be a linear relation from H to K, and let B be a linear relation from K to H. Then the following statements are equivalent: (i) A  ABA, mul A  kerB, and kerA  mul B; (ii) A  B1. The converse inclusions lead to a similar result. Lemma 10.4. Let H and K be linear spaces, let A be a linear relation from H to K, and let B be a linear relation from K to H. Then the following statements are equivalent: (i) ABA  A, ran B  dom A, and dom B  ran A; (ii) B1  A. Proof. (i) ) (ii) Assume the inclusion in (i) and let fu; vg 2 B. Then it follows by assumption that there exist elements ' 2 H and 2 K such that f'; ug 2 A and fv; g 2 A. Therefore one sees that f'; g 2 ABA  A. As a consequence it follows that f ;'g 2 A1, which leads to fv; ug 2 AA1A  A by (2.2), so that fu; vg 2 A1. Thus B  A1 or B1  A. (ii)) (i) Assume the inclusion in (ii). Then also B  A1 which implies that ABA  AA1A = A; by means of (2.1). Acta Wasaensia 103 Combining Corollary 10.3 and Lemma 10.4 gives the following result. Corollary 10.5. Let H and K be linear spaces, let A be a linear relation from H to K, and let B be a linear relation from K to H. Then the following statements are equivalent: (i) A = ABA, mul A  kerB, kerA  mul B, ran B  dom A, and dom B  ran A; (ii) A = B1. 11 Characterization of generalized inverses Let H and K be linear spaces, let A be a linear relation from H to K, and let B be a linear relation from K to H. Recall from Lemma 10.2 that A  B1 b+ (kerA f0g) , A  ABA and mul A  kerAB; (11.1) and, by symmetry, one obtains B  A1 b+ (kerB  f0g) , B  BAB and mul B  kerBA: (11.2) It is convenient to have extra conditions in (11.1) and (11.2) that guarantee identities A = ABA and B = BAB, respectively, instead of inclusions. Lemma 11.1. Let H and K be linear spaces, let A be a linear relation from H to K, and let B be a linear relation from K to H. Then the following statements are equivalent: (i) A  B1 b+ (kerA f0g) and dom A \mul B  kerA; (ii) A = ABA and mul A  kerAB. Similarly, the following statements are equivalent: (iii) B  A1 b+ (kerB  f0g) and dom B \mul A  kerB; (iv) B = BAB and mul B  kerBA. Proof. By symmetry, only the first equivalence needs to be verified. (i) ) (ii) Due to the equivalence in (11.1), the first assumption in (i) implies that A  ABA and mul A  kerAB. To conclude A = ABA, it suffices by Corollary 2.2 to show that dom ABA  dom A and mul ABA  mul A: The first inclusion follows from Lemma 7.1. For the second inclusion note that the established inclusion mul A  kerAB implies the inclusion mul ABA  mul AB, while the assumption dom A \mul B  kerA implies the inclusion mul AB  mul A. (ii) ) (i) The first statement in (i) is a consequence of A = ABA and mul A  kerAB by the equivalence in (11.1). The assumption A = ABA implies by Lemma 8.2 the following inclusion dom A \mul BA  kerA. This gives the second statement since mul B  mul BA. 104 Acta Wasaensia Note that the consequences of the identity A = ABA can be found in Section 9. In case B = BAB one gets similar results by interchanging A and B. Let it suffice to mention that if A = ABA and B = BAB, then dom BA = dom A; ran AB = ran A; kerBA = kerA; mul AB = mul A; dom AB = dom B; ran BA = ran B; kerAB = kerB; mul BA = mul B; (11.3) as follows from Lemma 9.1; cf. (Labrousse, 1992). One can proceed with Lemma 9.2 and Lemma 9.3 in a similar way. Also recall that both AB and BA are idempotent when A = ABA and B = BAB. This leads to the following definition. Definition 11.2. Let H and K be linear spaces, let A be a linear relation from H to K, and let B be a linear relation from K to H. Then A and B are said to be generalized inverses (of each other) if ABA = A; BAB = B; and, in addition, BA and AB are semi-projections in H and K, respectively. In the above definition the linear relations A and B play a symmetric role. One also uses the termi- nology that B is a generalized inverse of A (or vice versa). Note that if B = A1, then A and B are generalized inverses. To see this, recall that AA1A = A and A1AA1 = A1, while A1A and AA1 are semi-projections; cf. (2.1) and (2.2). In the above mentioned terminology one can say that B = A1 is a generalized inverse of A. The following theorem incorporates this special situation; see also (Labrousse, 2021). Theorem 11.3. Let H and K be linear spaces, let A be a linear relation from H to K, and let B be a linear relation from K to H. Then A and B are generalized inverses of each other if and only if the following statements hold: A b+ (f0g  kerB) = B1 b+ (kerA f0g); (11.4) mul B  kerA and mul A  kerB: (11.5) Proof. Assume that A and B are generalized inverses of each other. Then the assumptions that ABA = A and that BA is a semi-projection imply by Proposition 9.3 (iii) that mul BA  kerA. Likewise, the assumptions that BAB = B and that AB is a semi-projection give mul AB  kerB. Hence by (11.3) one sees that (11.5) is satisfied, which can also be written as mul B  kerAB and mul A  kerAB: Together with the assumption ABA = A this gives via Lemma 11.1 that A  B1 b+ (kerA f0g) and B  A1 b+ (kerB  f0g); so that also A b+ (f0g  kerB)  B1 b+ (kerA f0g) and B b+ (f0g  kerA)  A1 b+ (kerB  f0g); which gives (11.4). Acta Wasaensia 105 Conversely, assume that (11.4) and (11.5) hold. Then A  B1 b+ (kerA f0g) and mul B  kerA; so that by Lemma 11.1 one has A = ABA and mul A  kerAB. Likewise, one concludes that B = BAB and mul B  kerBA. Finally, Proposition 9.2 (iii) implies that AB and BA are semi-projections. As to Definition 11.2, the question arises if for a linear relation A from H to K one can choose a generalized inverse with special properties. For instance, does there exist a generalized inverse of A which is an operator? 12 Special generalized inverses Let A be a linear relation from H to K. Then, in general, its (formal) inverse relation A1 is not the graph of an operator, since mul A1 = kerA. However, any projection from dom A onto kerA leads to an algebraic operator part of the relation A1; cf. Section 2. It will be shown that any such algebraic operator part will serve as a generalized inverse. Let Q be a projection from dom A onto kerA. Then the following identity holds A(I Q) = A: (12.1) To see this, observe that A(I Q) = ff(I Q)f; gg : ff; gg 2 Ag and that fQf; 0g 2 A when ff; gg 2 A. As a consequence of (12.1) one obtains A1 = ffg; (I Q)fg : ff; gg 2 Ag: (12.2) Note that if g = 0 in (12.2), then f = Qf 2 kerA and (I Q)f = 0. In light of these facts, the following definition is natural. Definition 12.1. Let A be a linear relation from H to K and let Q be a projection from dom A onto kerA. Then the linear relation (A1)s = ffg; (I Q)fg : ff; gg 2 Ag is called the algebraic operator part of A1 (relative to the projection Q). Lemma 12.2. Let A be a linear relation from H to K and let (A1)s be the algebraic operator part of A1. Then (A1)sA = I Q (12.3) and A(A1)s = IranA b+ (f0g mul A): (12.4) Proof. In order to verify (12.3) two inclusions will be shown. 106 Acta Wasaensia () Let fh; lg 2 (A1)sA. Then fh; 'g 2 A and f'; lg 2 (A1)s for some ' 2 K. Note that f'; lg = fg; (I Q)fg for some ff; gg 2 A; so that ' = g and l = (I Q)f . In particular, it follows that fh; gg = fh; 'g 2 A and thus Q(f h) = 0. Since l = (I Q)f , one obtains l = (I Q)h. Therefore it follows that fh; lg = fh; (I Q)hg 2 I Q. This shows (A1)sA  I Q. () Let fh; lg 2 I Q, then h 2 dom A and l = (I Q)h. Thus there exists some g 2 K so that fh; gg 2 A and also f(I Q)h; gg 2 A, i.e., fg; (I Q)hg 2 (A1)s. Since l = (I Q)h this implies that fh; lg 2 (A1)sA. This shows the inclusion I Q  (A1)sA. In order to verify (12.4) two inclusions will be shown. () It follows from (12.2) and (2.1) that A(A1)s  AA1 = IranA b+ (f0g mul A): () First observe that f0g mul A  A(A1)s. Next it will be shown that IranA  A(A1)s. To see this, let k 2 ran A. Then there exists h 2 H such that fh; kg 2 A and also f(I Q)h; kg 2 A. Since fk; (I Q)hg 2 (A1)s, it follows that fk; kg 2 A(A1)s. Hence IranA  A(A1)s. Corollary 12.3. Let A be a linear relation from H to K and let (A1)s be the algebraic operator part of A1. Then A(A1)sA = A; (12.5) and (A1)sA(A1)s = (A1)s: (12.6) Proof. It follows from Lemma 12.2 that A(A1)sA = A(I Q). The statement in (12.5) now follows from (12.1). Likewise, it follows from Lemma 12.2 that (A1)sA(A1)s = (I Q)(A1)s. In order to show (12.6), it suffices to show that (I Q)(A1)s = (A1)s: (12.7) However, the identity (12.7) is clear, as it is a direct consequence of Definition 12.1 and the fact that I Q is an idempotent operator. A combination of Definition 12.1, Lemma 12.2, and Corollary 12.3 leads to the following theorem. Theorem 12.4. LetA be a linear relation fromH toK and letB = (A1)s be the algebraic operator part of its inverse. Then A and the operator B are generalized inverses. In the presence of topologies and under additional condition there exist generalized inverses as in Definition 12.1; this goes beyond the context of this survey. Acta Wasaensia 107 13 Further characterizations Let H and K be linear spaces, and let P be a linear relation from H to K. In Section 3 and Section 4 above, one can find results for the interplay with a second relation Q, either in H or K. Recall, in particular, that the inclusions P  PQ and P  QP were characterized in terms of identities in (3.2) and (3.6). In this section the various inclusions P  PQ, P  QP , PQ  P , and QP  P will be characterized in terms of inclusions. Lemma 13.1. Let H and K be linear spaces, let P be a linear relation from H to K, and let Q be a linear relation in H. Then the following statements are equivalent: (i) P  PQ; (ii) dom P  dom Q and Q domP  IH b+ (kerP  f0g) b+ (f0g mul Q) : Proof. (i)) (ii) Assume that (i) holds. Then the first inclusion in (ii) is clear. To prove the second inclusion let fx; tg 2 Q domP . Since x 2 dom P it follows that fx; yg 2 P for some y 2 ran P and thus fx; yg 2 PQ by (i). Hence fx; zg 2 Q and fz; yg 2 P for some z 2 H. Consequently, x z 2 kerP and t z 2 mul Q. Therefore one sees that fx; tg = fz; zg+ fx z; 0g+ f0; t zg 2 IH b+ (kerP  f0g) b+ (f0g mul Q) : Hence the second inclusion in (ii) has been shown. (ii) ) (i) Assume that (ii) holds and let fx; yg 2 P . Since dom P  dom Q one observes that fx; tg 2 Q domP for some t 2 H. Then fx; tg = fv; vg+ fu; 0g+ f0;mg; for some v 2 H, u 2 kerP , and m 2 mul Q. Then x = u+ v, t = v +m so that fv; yg = fx u; yg = fx; yg fu; 0g 2 P; and also, fx; vg = fx; tmg = fx; tg f0;mg 2 Q: Consequently, fx; yg 2 PQ. Hence (i) has been shown. Lemma 13.2. Let H and K be linear spaces, let P be a linear relation from H to K, and let Q be a linear relation in K. Then the following statements are equivalent: (i) P  QP ; (ii) dom P = dom QP , mul P  mul QP , and Q ranP  IK b+ (f0g mul QP ) : 108 Acta Wasaensia Proof. (i)) (ii) Assume that (i) holds. Then, clearly, dom P = dom QP and mul P  mul QP . It remains to show the last inclusion in (ii). Let fx; yg 2 Q ranP \domQ; so that x 2 ran P \ dom Q. Hence fz; xg 2 P for some z 2 dom P and thus fz; yg 2 QP . Furthermore observe that fz; xg 2 P  QP by (i). Therefore, f0; y xg = fz; yg fz; xg 2 QP , which further shows that fx; yg = fx; xg+ f0; y xg 2 IK b+ (f0g mul QP ) : Hence the last inclusion in (ii) holds. (ii)) (i) Assume that (ii) holds. Let fx; yg 2 P so that x 2 dom P = dom QP by the identity in (ii). Then fx; zg 2 QP for some z 2 ran QP ; and thus fx; tg 2 P and ft; zg 2 Q for some t 2 ran P \ dom Q. It follows from ft; zg 2 Q and the last inclusion in (ii) that t z 2 mul QP . This leads to fx; tg = fx; zg+ f0; t zg 2 QP: It follows from fx; yg, fx; tg 2 P that y t 2 mul P  mul QP , due to the first inclusion in (ii). Consequently, fx; yg = fx; tg+ f0; y tg 2 QP: Thus P  QP and (i) has been shown. Lemma 13.3. Let H and K be linear spaces, let P be a linear relation from H to K, and let Q be a linear relation in H. Then the following statements are equivalent: (i) PQ  P , (ii) mul PQ = mul P , dom PQ  dom P , and Q domPQ IH b+ (kerP  f0g) b+ (f0g mul Q) : Proof. (i)) (ii) Assume that (i) holds. The identity and the first inclusion in (ii) are clear. Now let fx; yg 2 Q domPQ, so that fx; tg 2 PQ for some t 2 ran PQ and fx; tg 2 P by (i). Note that fx; zg 2 Q and fz; tg 2 P for some z 2 H. Consequently, x z 2 kerP and y z 2 mul Q. Therefore one sees that fx; yg = fz; zg+ fx z; 0g+ f0; y zg 2 IH b+ (kerP  f0g) b+ (f0g mul Q) : Hence (ii) has been shown. (ii) ) (i) Assume that (ii) holds and let fx; yg 2 PQ. Then x 2 dom PQ, fx; g 2 Q, and f ; yg 2 P for some 2 H. Since dom PQ  dom P it follows that fx; g 2 P for some 2 K. By the second inclusion in (ii) one sees that fx; g = fu; ug+ fp; 0g+ f0;mg; for some u 2 H, p 2 kerP and m 2 mul Q. Then x = p+ u, = u+m, so that fm; y g = fp+ u; g+ fu+m; yg+ fp; 0g 2 P: Acta Wasaensia 109 Since f0;mg 2 Q it follows that f0; y g 2 PQ, so that y 2 mul PQ = mul P by the identity in (ii). This implies that fx; yg = fx; g+ f0; y g 2 P: Hence (i) has been shown. Lemma 13.4. Let H and K be linear spaces, let P be a linear relation from H to K, and let Q be a linear relation in K. Then the following statements are equivalent: (i) QP  P ; (ii) Q ranP  IK b+ (f0g mul P ). Proof. (i) ) (ii) Assume that (i) holds. Let fx; yg 2 Q ranP . Since x 2 ran P it follows that fz; xg 2 P for some z 2 dom P . Thus, fz; yg 2 QP  P by (i). This implies that f0; y xg = fz; yg fz; xg 2 P; so that y x 2 mul P . Consequently, fx; yg = fx; xg+ f0; y xg 2 IK b+ (f0g mul P ) : Hence (ii) has been shown. (ii)) (i) Now assume that (ii) holds. Let fx; yg 2 QP so that fx; zg 2 P and fz; yg 2 Q for some z 2 ran P \ dom Q. It follows from (ii) that fz; yg = fz; zg+ f0; y zg; with y z 2 mul P , which further leads to fx; yg = fx; zg+ f0; y zg 2 P: Hence QP  P which shows that (i) Acknowledgement: The authors thank Rudi Wietsma for improvements in the exposition of various results in this paper. References Arens, R. (1961). Operational calculus of linear relations. Pacific J. Math. 11, 9–23. Behrndt, J., Hassi, S. & de Snoo, H.S.V. (2020). Boundary Value Problems, Weyl Functions, and Differential Operators. Monographs in Mathematics, vol. 108. Cham: Birkhäuser. Ben-Israel, A. & Greville, T.N.E. (2003). Generalized Inverses: Theory and Applications. New York: Springer Verlag. 110 Acta Wasaensia Berger, T., de Snoo, H.S.V., Trunk, C. & Winkler, H. (2021). Linear relations and their singular chains. To appear in Methods Funct. Anal. Topology. Bjerhammar, A. (1951). Application of calculus of matrices to method of least squares; with special references to geodetic calculations. Trans. Roy. Inst. Tech. Stockholm 49, 1–86. Campbell, S.L. & Meyer, C.D., Jr. (1991). Generalized Inverses of Linear Transformations. New York: Dover Publications. Hassi, S., de Snoo, H.S.V. & Szafraniec, F.H. (2009). Componentwise and Cartesian decompositions of linear relations. Dissertationes Math. 465, 59 pp. Labrousse, J.-Ph. (1992). Inverses généralisés d’opérateurs non bornés. Proc. Amer. Math. Soc. 115, 125–129. Labrousse, J.-Ph. (2003). Idempotent linear relations. Theta Ser. Adv. Math. 2, 129–149. Labrousse, J.-Ph. (2021). Representable projections and semi-projections in a Hilbert space. To ap- pear in Complex Anal. Oper. Theory Labrousse, J.-Ph. & M’Bekhta, M. (1992). Les opérateurs points de continuité pour la conorme et l’inverse de Moore-Penrose. Houston J. Math. 18, 7–23. Moore, E.H. (1920). On the reciprocal of the general algebraic matrix. Bull. Amer. Math. Soc. 26, 394–95. Nashed, M.Z. (1976). Generalized Inverses and Applications. New York - London: Academic Press. Penrose, R. (1955). A generalized inverse for matrices. Proc. Cambridge Phil. Soc. 51, 406–13. Rao, C.R. & Mitra, S.K. (1971). Generalized Inverse of Matrices and its Applications. New York: John Wiley & Sons. Sandovici, A., de Snoo, H.S.V. & Winkler, H. (2004). The structure of linear relations in Euclidean spaces. Lin. Alg. Appl. 397, 141–169. Sandovici, A., de Snoo, H.S.V. & Winkler, H. (2007). Ascent, descent, nullity, and defect for linear relations in linear spaces. Lin. Alg. Appl. 423, 456–497. 63 Avenue Cap de Croix, 06100 Nice, France E-mail address: labro@math.unice.fr Department of Mathematics, "Gheorghe Asachi" Technical University, 700506 Ias¸i, Romania E-mail address: adrian.sandovici@luminis.ro Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, 9700 AK Groningen, The Netherlands E-mail address: hsvdesnoo@gmail.com Institut für Mathematik, Technische Universität Ilmenau, 98693 Ilmenau, Germany E-mail address: henrik.winkler@tu-ilmenau.de Acta Wasaensia 111 LIPSCHITZ PROPERTY OF EIGENVALUES AND EIGENVECTORS OF 2 2 DIRAC-TYPE OPERATORS Anton Lunyov and Mark Malamud Dedicated to our friend and colleague Seppo Hassi on the occasion of his sixtieth birthday 1 Introduction Continuing our investigation (Lunyov & Malamud, 2016), this paper is concerned with the sta- bility properties of different spectral characteristics of a boundary value problem associated in L2([0; 1];C2) with the following first order system of differential equations Ly = iB1y0 +Q(x)y = y; y = col(y1; y2); x 2 [0; 1]; (1.1) where B =  b1 0 0 b2  ; b1 < 0 < b2; and Q =  0 Q12 Q21 0  2 L1([0; 1];C22): If B = 1 0 0 1  , then the system (1.1) is equivalent to the Dirac system, see the classical mono- graphs (Levitan & Sargsyan, 1991; Marchenko, 1986). Let us associate with the system (1.1) the following linearly independent boundary conditions (BCs) Uj(y) := aj1y1(0) + aj2y2(0) + aj3y1(1) + aj4y2(1) = 0; j 2 f1; 2g: (1.2) Moreover, denote by L(Q) := LU (Q) the operator in L2([0; 1];C2) associated with the boundary value problem (BVP) (1.1)–(1.2); its action is defined by the differential expression L in (1.1) and its domain is given by dom (LU (Q)) = ff 2 AC([0; 1];C2) : Lf 2 L2([0; 1];C2); U1(f) = U2(f) = 0g: (1.3) The above-mentioned stability properties refer to a perturbation of the potential Q! eQ. The completeness property of the system of root vectors (SRV) of BVPs for general n n systems of the form (1.1) with a nonsingular diagonal n  n matrix B with complex entries and a potential matrix Q() of the form B = diag(b1; b2; : : : ; bn) 2 Cnn and Q() =: (qjk())nj;k=1 2 L1([0; 1];Cnn) was established in Malamud & Oridoroga (2012) for a wide class of BVPs; note that for 2  2 Dirac systems with Q 2 C([0; 1];C22) it was proved earlier in Marchenko (1986: Chapter 1.3). In Malamud & Oridoroga (2012); Lunyov & Malamud (2014a; 2015) the authors also found com- pleteness conditions for non-regular and even degenerate BCs. In Lunyov & Malamud (2015) the Riesz basis property (with and without parentheses) of SRV was also established for different classes of BVPs for nn systems with arbitraryB andQ 2 L1([0; 1];Cnn). Note also that BVPs for the 112 Acta Wasaensia 2m 2m Dirac equation, i.e., the case that B = diag(Im; Im), were investigated in Mykytyuk & Puyda (2013) (Bari-Markus property for Dirichlet BVP with Q 2 L2([0; 1];C2m2m) and in Kur- banov & Abdullayeva (2018); Kurbanov & Gadzhieva (2020) (Bessel and Riesz basis properties on an abstract level). The Riesz basis property in L2([0; 1];C2) of BVP (1.1)–(1.2), i.e., of the operator LU (Q) defined above, was investigated with various assumptions on the potential matrix Q in numerous papers, see (Trooshin & Yamamoto, 2001; 2002; Hassi & Oridoroga, 2009; Djakov & Mityagin, 2010; Baskakov, Derbushev & Shcherbakov, 2011; Djakov & Mityagin, 2012a;b;c; 2013; Lunyov & Mala- mud, 2014b; Savchuk & Shkalikov, 2014; Lunyov & Malamud, 2016; Uskova, 2019) and references therein. At that time the strongest result was obtained by P. Djakov and B. Mityagin (2010; 2012c), and A. Baskakov, A. Derbushev, and A. Shcherbakov (2011). They proved under the assumption Q 2 L2([0; 1];C22) that SRV of the BVP (1.1)–(1.2) with strictly regular BCs forms a Riesz basis, and with BCs that are only regular forms a Riesz basis with parentheses. Note, however, that the methods of these papers substantially rely on L2-techniques (such as Parseval’s equality, Hilbert- Schmidt operators, etc.) and cannot be applied to L1-potentials. Later the case Q 2 L1([0; 1];C22) was treated independently and with different methods by the authors (Lunyov & Malamud, 2014b; 2016) on the one hand, and by A.M. Savchuk and A.A. Shka- likov (2014) on the other hand. It was proved that a BVP (1.1)–(1.2) with Q 2 L1([0; 1];C22) and strictly regular boundary conditions has the Riesz basis property, while a BVP whose BCs are only regular has the property of Riesz basis with parentheses. Recall in this connection that the boundary conditions (1.2) are called regular if and only if they are equivalent to the following conditions bU1(y) = y1(0) + by2(0) + ay1(1) = 0; bU2(y) = dy2(0) + cy1(1) + y2(1) = 0; (1.4) for certain a; b; c; d 2 C satisfying ad bc 6= 0. Recall also that regular BCs (1.2) are called strictly regular if the sequence 0 = f0ngn2Z of the eigenvalues of the unperturbed BVP (1.1)–(1.2) (of the operator LU (0), i.e., Q = 0) is asymptotically separated. In particular, the eigenvalues f0ngjnj>n0 are geometrically and algebraically simple. It is well known that non-degenerate separated BCs are always strictly regular. Moreover, the con- ditions (1.4) are strictly regular for the Dirac operator if and only if (a d)2 6= 4bc. In particular, antiperiodic (periodic) BC are regular but not strictly regular for Dirac systems, while they become strictly regular for Dirac-type systems if b1; b2 2 N and b2 b1 is odd. To describe our approach to the Riesz basis property used in Lunyov & Malamud (2014b; 2016), let us denote by e(; ) the solutions of the system (1.1) satisfying the initial conditions e(0; ) =  1 1  : Our investigation in Lunyov & Malamud (2014b; 2016) substantially relies on the following repre- sentation of the solutions e(; ) by means of triangular transformation operators: e(x; ) = (I +KQ)e0(x; ) = e0(x; ) + Z x 0 KQ (x; t)e 0 (t; ) dt; (1.5) Acta Wasaensia 113 where e0(x; ) = col eib1x;eib2x and KQ = Kjk2j;k=1 2 X0;21;1 \X0;21;1; see (2.1) and (2.2) for the definitions of these spaces. Let us denote by Q := U;Q = fQ;ngn2Z := fngn2Z the spectrum of the operator LU (Q). Our main tool in the investigation of the asymptotic behavior of the eigenvalues is the characteristic determinant Q() = Q;U (). This function is an entire function whose zeros coincide with the sequence Q of eigenvalues counting multiplicities, see the formulas (4.1)–(4.4). The representa- tion (1.5) immediately leads to the following key formula for the characteristic determinant Q() of the problem (1.1)–(1.2): Q() = 0() + Z 1 0 g1;Q(t)e ib1tdt+ Z 1 0 g2;Q(t)e ib2t dt; (1.6) where gk;Q() 2 L1[0; 1]; k 2 f1; 2g, are expressed via Kjk(1; ), see (4.12) and (4.7). Recall that 0() = 0;U () is the characteristic determinant of the problem (1.1)–(1.2) with Q = 0, see the identity (4.5). Formula (1.6) immediately yields an estimate of the difference Q()0() from above. Com- bining this estimate with the classical estimate of 0() from below and applying the Rouché theo- rem one arrives at the asymptotic formula n =  0 n + o(1); as n!1; (1.7) relating the eigenvalues  = fngn2Z and 0 = f0ngn2Z of the operators LU (Q) and LU (0) (with regular BCs), respectively; see (Lunyov & Malamud, 2014b; 2016) for details and also (Savchuk & Shkalikov, 2014), where the formula (1.7) was obtained by another method. Note also that the representation (1.6) for the determinant Q() was substantially used in papers by A.S. Makin (2020; 2021). In Lunyov & Malamud (2014b; 2016) we also applied the representation (1.5) to obtain asymptotic formulas for the solutions of the equation (1.1) as well as for eigenfunctions of the BVP (1.1)–(1.2). In the present paper we continue the investigation from (Lunyov & Malamud, 2014b; 2016) of the BVP (1.1)–(1.2) and the transformation operators for the system (1.1). In Section 2 we prove the Lipschitz property of the mappings Q! K on the balls U22p;r :=  F 2 Lp([0; 1];C22) : kFkp := kFkLp([0;1];C22)  r ; r > 0; (1.8) in Lp([0; 1];C22). Namely, our first main result reads as follows. Theorem 1.1. For any p 2 [1;1) and r > 0, there exists C = C(B; p; r) > 0 such that the following uniform estimate holds kKQ KeQkX21;p + kKQ KeQkX21;p  C kQ eQkp; Q; eQ 2 U22p;r : (1.9) Here KeQ are the kernels from the representation (1.5) for the solutions e of (1.1), with eQ in place of Q, and the spaces X21;p and X 2 1;p are as introduced in (2.1) and (2.2), respectively. Combining the uniform estimate (1.9) with the representation (1.6) we obtain the following state- ment concerning the Lipschitz property of the map Q! gl;Q on Lp-balls. It will play a crucial role in our approach to subsequent estimates. 114 Acta Wasaensia Proposition 1.2. Let Q; eQ 2 U22p;r with p 2 [1;1], and let gk := gk;Q gk; eQ, k 2 f1; 2g, see (4.12). Then gk 2 Lp[0; 1], k 2 f1; 2g, and the difference of characteristic determinants of the problem (1.1)–(1.2) admits the following representation Q() eQ() = Z 1 0 g1(t)e ib1t dt+ Z 1 0 g2(t)e ib2t dt: (1.10) Moreover, there exists a constant C = C(B; p; r) > 0 such that kg1kp + kg2kp = kg1;Q g1; eQkp + kg2;Q g2; eQkp  C kQ eQkp: (1.11) As an immediate application of Proposition 1.2 we complete the formula (1.7) by establishing the c0-Lipschitz property of the spectrum Q = fQ;ngn2Z on compact sets: if the boundary condi- tions (1.2) are regular, then for each compact K ( U221;r ) and any " > 0 there exists N" > 0, not dependent on Q 2 K, such that the following uniform relation holds supjnj>N" Q;n 0n  "; Q 2 K: (1.12) In the case of Dirac systems this result was established by Sadovnichaya (2016: Theorem 3). As is evident from the representations (1.5) and (1.10), the stability of eigenvalues and eigenvec- tors of the operator L(Q) reduces to certain properties of the Fourier transform and the "maximal" Fourier transform Fg() := Z 1 0 g(t)eit dt and Fg() := supx2[0;1] Z x 0 g(t)eit dt ;  2 C: To this end, we generalize the classical Hausdorff-Young and Hardy-Littlewood theorems for Fourier coefficients, see (Zigmund, 1959: Theorems XII.2.3 & XII.3.19). Throughout the paper p0 will denote p=(p 1), and the following notations with h > 0 and n 2 Z will be used for strips in the complex plane: h := fz 2 C : jIm zj  hg; h;n := fz 2 C : n  Re z  n+ 1; jIm zj  hg: (1.13) Moreover, note that the concept of an incompressible sequence with density d will be defined in Definition 4.6. Proposition 1.3. Let p 2 (1; 2]. Then there exists a constant C = C(p; h; d) > 0 such that the following estimates hold uniformly for g 2 Lp[0; 1] and for incompressible sequences  = fngn2Z with density d contained in the strip h with h > 0:X n2Z jFg(n)jp0  X n2Z F p 0 g (n)  C kgkp 0 p ; (1.14) X n2Z (1 + jnj)p2jFg(n)jp  X n2Z (1 + jnj)p2F pg (n)  C kgkpp: (1.15) The proof of the inequalities in (1.14)–(1.15) involving the "maximal" Fourier transformFg relies on the deep Carleson-Hunt theorem, while the estimates of ordinary Fourier transforms are elementary in character. Inequality (1.15) generalizes the Hardy-Littlewood theorem and coincides with it for the ordinary Fourier transform when n = 2n. In turn, this inequality is an important ingredient in Acta Wasaensia 115 proving the estimate (1.19) below. By combining Propositions 1.2 and 1.3 we establish the Lipschitz property of the mapping Q! Q in different norms. Theorem 1.4. LetQ; eQ 2 U22p;r for some p 2 (1; 2] and r > 0, and let the boundary conditions (1.2) be strictly regular. Then there exists an enumeration of the spectra fQ;ngn2Z and f eQ;ngn2Z of the operators LU (Q) and LU ( eQ), respectively, and a set IQ; eQ  Z, such that with certain constants C;C1; C2; N > 0, not dependent on Q and eQ, the following uniform estimates hold: card  Z n IQ; eQ   N; (1.16) C1  eQQ;n  jQ;n  eQ;nj  C2  eQQ;n ; n 2 IQ; eQ; (1.17)X n2IQ; eQ Q;n  eQ;n p0  C kQ eQkp0p ; (1.18) X n2IQ; eQ (1 + jnj)p2 Q;n  eQ;n p  C kQ eQkpp: (1.19) On a compact set K in Lp([0; 1];C22) the subsets IQ; eQ  Z can be chosen independent of the pair fQ; eQg and, in view of (1.16), the summation in (1.18)–(1.19) takes the form PjnjN1. Here N1 2 N does not depend on Q; eQ 2 K. Note also that the two-sided estimate (1.17) plays a crucial role in the proof of the estimates in (1.18)–(1.19). On account of the representation (1.10), it reduces the Lipschitz property of the map Q ! Q to the property that the generalized Fourier coefficients of g1 and g2 belong to certain weighted `p-spaces. Observe that in proving (1.18)–(1.19) we use only the evaluation of the ordinary Fourier transform and we do not use the deep Carleson-Hunt result. In particular, the proof of (1.19) relies only on the uniform estimate between the first and third terms in (1.15), i.e., it concerns only the ordinary F and not the "maximal" F . This fact makes the proof of the estimates (1.18)–(1.19) elementary in character. Relation (1.12) is also valid for regular BCs and extends Theorem 3 from Sadovnichaya (2016) to the case of Dirac-type systems (b1 6= b2). When eQ = 0, then the estimates (1.18)–(1.19) give `p-estimates (uniform on balls) of the remainder in the asymptotic formula (1.7) for the eigenvalues of the strictly regular problem (1.2) for Dirac-type systems. For the Dirac operator (b1 = b2 = 1) the estimate (1.18) with eQ = 0 generalizes the corresponding result obtained first by Savchuk & Shkalikov (2014: Theorem 4.3 & Theorem 4.5) with a constant C that depends on Q (i.e., for the two points compact set K = fQ; 0g) and later in Savchuk & Sadovnichaya (2018) for arbitrary compact sets K in L1([0; 1];C22). Note in this connection that A. Gomilko and L. Rzepnicki (2020), and A. Gomilko (2020) ob- tained new, sharp, asymptotic formulas for eigenfunctions of Sturm–Liouville operators with singu- lar potentials, and for eigenvalues and eigenfunctions of Dirichlet BVPs for the Dirac system with Q 2 Lp([0; 1];C22), 1  p < 2. The weighted estimate (1.19) is new even for the Dirac system with Q 2 U22p;r and eQ = 0, and even for the trivial compact set K = fQ; 0g. 116 Acta Wasaensia Turning to the stability of the eigenvectors of the operator L(Q), we first investigate the Fourier transform of the kernels KQ of the transformation operators in the representation (1.5). Namely, in Theorem 7.3 we estimate deviationsZ x 0 KQ KeQ  jk (x; t)eibkt dt via "maximal" Fourier transforms of the deviations Q eQ and kQ eQk1 eQ. Furthermore, the representation (1.5) leads to similar estimates for the fundamental matrix solution Q(x; ) of the system (1.1), which in the case of eQ = 0 reads as follows. Proposition 1.5. LetQ 2 U221;r for some r > 0. Then there exists C = C(B; r) > 0, not dependent on Q, such that the following uniform estimate holds for x 2 [0; 1] and  2 C, jQ(x; ) 0(x; )jC22  2 2X j;k=1 Z x 0 K+jk(x; t)e ibkt dt + 2 2X j;k=1 Z x 0 Kjk(x; t)e ibkt dt  C e2(b2b1)jImjx X j 6=k sups2[0;x] Z s 0 Qjk(t)e i(bkbj)t dt : Now we are ready to state `p-stability properties of eigenfunctions of the operators LU (Q). Assume the spectrum U;Q = fQ;ngn2Z of LU (Q) to be asymptotically simple, and introduce a sequence ffQ;ngjnj>N of the corresponding normalized eigenfunctions: LU (Q)fQ;n = Q;nfQ;n. Theorem 1.6. Let Q; eQ 2 U22p;r , p 2 (1; 2], p0 := p=(p 1), and r > 0. Moreover, assume the BCs fUjg21 of the form (1.2) to be strictly regular. Then there exist enumerations of the spectra fQ;ngn2Z and f eQ;ngn2Z of the operators LU (Q) and LU ( eQ), respectively, and a set IQ; eQ  Z, such that for some constants C;N > 0, not dependent on Q and eQ, the following estimates holdX n2IQ; eQ fQ;n f eQ;n p01  C kQ eQkp0p ; (1.20) X n2IQ; eQ (1 + jnj)p2 fQ;n f eQ;n p1  C kQ eQkpp: (1.21) On compact sets K in Lp the estimates (1.20)–(1.21) are simplified, since the subsets IQ; eQ  Z can then be chosen to be independent of the pair fQ; eQg. Moreover, in view of (1.16), the summation in (1.18)–(1.21) can in that case be replaced by P jnjN1. Here N1 2 N does not depend on Q andeQ. Inequality (1.21) generalizes the classical Hardy-Littlewood inequality for Fourier coefficients (Zigmund, 1959: Theorem XII.3.19), see Remark 7.9. Recall that antiperiodic boundary conditions could be strictly regular for Dirac-type operators as op- posed to the Dirac case. Therefore, all the previous results imply the following surprising statement. Corollary 1.7. Let Q; eQ 2 U22p;r , p 2 (1; 2], and let b1; b2 2 N and b2 b1 be odd. Then antiperiodic BCs are strictly regular and, hence, the operator LU (Q) has the Riesz basis prop- erty. Moreover, the corresponding eigenvalues and eigenvectors satisfy the uniform Lipschitz type estimates (1.18)–(1.19) and (1.20)–(1.21). This result demonstrates a substantial difference between Dirac and Dirac-type operators. Acta Wasaensia 117 Observe in conclusion that periodic and antiperiodic (necessarily non-strictly regular) BVPs for 2  2 Dirac and Sturm-Liouville equations have also attracted attention during the last decade. For instance, a criterion for SRV of the periodic BVP for 2  2 Dirac equation to contain a Riesz basis (without parentheses!) was obtained by P. Djakov and B. Mityagin in (2012b), see also the recent survey (Djakov & Mityagin, 2020) and the recent papers by A.S. Makin (2021; 2020), and the references therein. It is also worth mentioning that F. Gesztesy and V.A. Tkachenko (2009; 2012), for q 2 L2[0; ], and P. Djakov and B.S. Mityagin (2012b), for q 2 W1;2[0; ], established by different methods a criterion for SRV to contain a Riesz basis for the Sturm-Liouville operator d2dx2 + q(x) on [0; ], see also the survey (Makin, 2012). The contents of the paper will now be briefly described. In Section 2 the Banach spaces X1;p and X1;p are studied. Section 3 is concerned with triangular transformation operators. The general properties of a 2  2 Dirac-type BVP are discussed in Section 4. In Section 5 one can find Fourier transform estimates. The stability properties of eigenvalues are discussed in Section 6 and the sta- bility properties of eigenfunctions are discussed in Section 7. 2 The Banach spaces X1;p and X1;p Let p 2 [1;1]. Following (Malamud, 1994) denote by X1;p := X1;p( ) and X1;p := X1;p( ) the linear spaces composed of (equivalent classes of) measurable functions defined on the triangular set := f(x; t) : 0  t  x  1g satisfying kfkpX1;p := ess sup t2[0;1] Z 1 t jf(x; t)jp dx <1; p <1; (2.1) kfkpX1;p := ess sup x2[0;1] Z x 0 jf(x; t)jp dt <1; p <1; (2.2) respectively, and kfkX1;1 = kfkX1;1 := ess sup(x;t)2 jf(x; t)j. It can easily be shown that the spaces X1;p and X1;p equipped with the norms (2.1) and (2.2) form Banach spaces that are not separable. Denote by X01;p := X 0 1;p( ) and X 0 1;p := X 0 1;p( ) the closures of the subspace of continuous functionsC( ) inX1;p( ) andX1;p( ), respectively. Clearly, the setC1( ) of smooth functions is also dense in both spacesX01;p andX 0 1;p. Note also that the following embeddings hold and are continuous X1;p1  X1;p2  X1;1 and X1;p1  X1;p2  X1;1; p1 > p2  1: The following simple property of the spaces X01;p and X 0 1;p will be important in the sequel. Lemma 2.1. Let p  1. For each a 2 [0; 1] the trace mappings ia;1 : C( )! C[0; a]; ia;1 N(x; t)  := N(a; t); ia;1 : C( )! C[a; 1]; ia;1 N(x; t)  := N(x; a); admit continuous extensions, which are also denoted by ia;1 and ia;1, to mappings from X01;p( ) onto Lp[0; a] and X01;p( ) onto L p[a; 1], respectively. 118 Acta Wasaensia Going over to the vector case we denote for u = col(u1; : : : ; un) 2 Cn juj := ju1j + : : :+ junj ; 0 < <1; juj1 = maxfju1j; : : : ; junjg: Furthermore, for A = (ajk)nj;k=1 2 Cnn we define jAj ! := sup fjAuj : u 2 Cn; juj = 1g; ; 2 (0;1]: Now we are ready to introduce the Banach spaces Xn1;p := X1;p( ;Cnn) and Xn1;p := X1;p( ;Cnn); consisting of n  n matrix-functions F = (Fjk)nj;k=1 with X1;p- and X1;p-entries, respectively, equipped with the norms kFkpXn1;p := ess sup t2[0;1] Z 1 t jF (x; t)jp1!p dx <1; p 2 [1;1); kFkpXn1;p := ess sup x2[0;1] Z x 0 jF (x; t)jpp0!1 dt <1; p 2 [1;1): Moreover, kFkXn1;1 = kFkXn1;1 := ess sup(x;t)2 jF (x; t)j1!1. Besides, we introduce the sub- spaces X0;n1;p := X 0 1;p( ;Cnn) and X0;n1;p := X01;p( ;Cnn); which are separable parts of Xn1;p and X n 1;p, respectively. Furthermore, for brevity, throughout the section we use the following notation Ls := Ls([0; 1];Cn); s 2 [1;1]: With each measurable matrix kernel N(; ) = Njk(; )nj;k=1 on one associates a Volterra type operator N as follows N : f 7! Z x 0 N(x; t)f(t) dt: (2.3) Denote by kNk ! := kNkL !L , ; 2 [1;1], the norm for bounded operators N acting from L to L . The following result demonstrates the natural occurrence of the spaces Xn1;p and X n 1;p in the study of the integral operators acting from L to L for special and . In particular, the third statement sheds light on the interpolation role of these spaces, cf. (Malamud, 1994). This result substantially complements Lemma 2.3 from (Lunyov & Malamud, 2016). Recall that a Volterra operator on a Banach space is a compact operator with zero spectrum. Proposition 2.2. LetN be a Volterra type operator given by (2.3) for a measurable matrix-function N(; ) and let p 2 [1;1]. Then the following statements hold: (i) The inclusion N 2 B(L1; Lp) holds if and only if N 2 Xn1;p, in which case kNk1!p = kNkXn1;p : Acta Wasaensia 119 Moreover, if N 2 X0;n1;p , then the operator N is compact from L1 to Lp and the following relation holds N (I +N )1 =: S 2 B(L1; Lp); where S : f 7! Z x 0 S(x; t)f(t) dt with S 2 X0;n1;p ; here (I +N )1 is treated as an operator from B(L1; L1). (ii) The inclusion N 2 B(Lp0 ; L1) holds if and only if N 2 Xn1;p , in which case kNkp0!1 = kNkXn1;p : Moreover, if N 2 X0;n1;p, then N maps Lp 0 to C := C([0; 1];Cn) and is compact. Let NC : C ! C be a restriction ofN to C, then (I+NC)1 2 B(C; C) and the following relation holds (I +NC)1N =: S 2 B(Lp0 ; C); where S : f 7! Z x 0 S(x; t)f(t) dt with S 2 X0;n1;p: (iii) Let N 2 Xn1;1 \Xn1;1  Xn1;p \Xn1;p. Then N 2 B(Ls; Ls) for each s 2 [1;1], and kNks!s  kNk1=sXn1;1 kNk 11=s Xn1;1 : (iv) Let N 2 X0;n1;p \ X0;n1;p. Then N is a Volterra operator in Ls for each s 2 [1;1], and the inverse operator (I +N )1 is given by (I +N )1 =: I + S; where S : f 7! f + Z x 0 S(x; t)f(t) dt with S 2 X0;n1;p \X0;n1;p: Remark 2.3. In connection with Proposition 2.2, let us recall Theorems XI.1.5 and XI.1.6 from Kan- torovich & Akilov (1977) concerning integral representations of bounded linear operators. Namely, let p 2 (1;1], and let R and S be bounded linear operators from L1[0; 1] to Lp[0; 1] and Lp0 [0; 1] to C[0; 1], respectively. Then they admit the following integral representations: (Rf)(x) = Z 1 0 R(x; t)f(t) dt; kRkp1!p = ess sup t2[0;1] Z 1 0 jR(x; t)jp dx <1; (Sf)(x) = Z 1 0 S(x; t)f(t) dt; kSkpp0!1 = ess sup x2[0;1] Z 1 0 jS(x; t)jp dt <1: 3 Triangular transformation operators The existence of a triangular transformation operator for the system (1.1) with summable potential matrix Q 2 L1([0; 1];C22) was established in our previous paper (Lunyov & Malamud, 2016). Moreover, the case B = B 2 Cnn and Q 2 L1([0; 1];Cnn) was treated earlier in Malamud (1999). The purpose of this section is to prove the Lipschitz property (in respective norms) for the kernels of the transformation operators of Q 2 Lp([0; 1];C22). We start with the following result from Lun- yov & Malamud (2016). 120 Acta Wasaensia Theorem 3.1 (Lunyov & Malamud (2016: Theorem 2.5)). Let Q = codiag(Q12; Q21) belong to L1([0; 1];C22). Assume that e(; ) are the solutions of the system (1.1) satisfying the initial conditions e(0; ) = 1 1  . Then the solutions e(; ) admit the following representation by means of the triangular transformation operators KQ = Kjk 2 j;k=1 2 X0;21;1 \X0;21;1 e(x; ) = e0(x; ) + Z x 0 KQ (x; t)e 0 (t; ) dt; e 0 (x; ) =  eib1x eib2x  : (3.1) It was shown in Malamud (1999) that if Q = codiag(Q12; Q21) 2 C1([0; 1];C22), then the matrix kernel in the triangular representation (3.1) is smooth, K = KQ = Kjk 2 j;k=1 2 C1( ;C22), and it is the unique solution of the following boundary value problem B1DxK(x; t) +DtK(x; t)B1 + iQ(x)K(x; t) = 0; (3.2) K(x; x)B1 B1K(x; x) = iQ(x); x 2 [0; 1]; (3.3) K(x; 0)B1 11  = 0; x 2 [0; 1]: (3.4) The proof of this result in Malamud (1999) was divided into two steps. First it was proved that there exists the smooth unique solution RQ = Rjk 2 j;k=1 2 C1( ;C22) of the problem (3.2)–(3.3), satisfying, instead of (3.4), the following conditions: R11(x; 0) = R22(x; 0) = 0; x 2 [0; 1]: (3.5) As the second step we defined the kernelsKQ via the auxiliary matrix functionRQ by formula (3.7) and showed that they have the required properties. By means of this result, the following proposition was proved in Lunyov & Malamud (2016); it is the starting point of our investigation here. Note also that for smooth kernels relation (3.7) was already exploited in Malamud (1999). Proposition 3.2 (Lunyov & Malamud (2016)). Let Q 2 L1([0; 1];C22) and let KQ be the kernels of the corresponding transformation operators from the representation (3.1). Then there exist RQ = (Rjk) 2 j;k=1 2 X0;21;1 \X0;21;1 and PQ = diag(P1 ; P2 ) 2 L1([0; 1];C22); (3.6) such that KQ (x; t) = RQ(x; t) + P  Q (x t) + Z x t RQ(x; s)P  Q (s t) ds; 0  t  x  1: (3.7) Moreover, RQ(; ) is the unique solution of the following system for 0  t  x  1, Rkk(x; t) = ibk Z x xt Qkj()Rjk ;  x+ t d; (3.8) Rjk(x; t) = ibj jQjk( kx+ jt) ibj Z x kx+ jt Qjk()Rkk  ; bj bk ( x) + t  d; (3.9) where k := bj bjbk with j = 2=k for k 2 f1; 2g. Note that for smooth Q, i.e., for Q 2 C1([0; 1];C22), the system (3.8)–(3.9) is equivalent to the system (3.2)–(3.3), (3.5). To refine Theorem 3.1 in the Lp-case, we start by refining properties of the auxiliary kernel RQ appearing in Proposition 3.2. In the following result we show that the (non- Acta Wasaensia 121 linear) mapping Q ! R = RQ is Lipschitz in X21;p and X21;p on each ball of radius r in Lp. Proposition 2.2 can be used to establish part (ii) thereof. Proposition 3.3. Let Q; eQ 2 U22p;r for some p  1 and r > 0. Then the following statements hold: (i) The solutions RQ and R eQ of the system of integral equations (3.8)–(3.9) for Q and eQ respec- tively, unique in X0;21;1 \ X0;21;1, belong to X0;21;p \ X0;21;p and the following uniform estimate holds kRQ R eQkX21;p + kRQ R eQkX21;p  C0 kQ eQkp; where the constant C0 > 0 does not depend on Q; eQ 2 U22p;r . (ii) The operator RQ : f 7! Z x 0 RQ(x; t)f(t) dt; f 2 Ls([0; 1];C2); is a Volterra operator in every space Ls([0; 1];C2), s 2 [1;1]. Moreover, there exists a constant C1 = C1(B; p; r) > 0, not dependent on s 2 [1;1] and Q; eQ 2 U22p;r , such that the following uniform estimate holds k(I +RQ)1 (I +R eQ)1ks!s  C1 kQ eQkp; Combining these properties of the kernel RQ(; ) with the convolution identity (3.7) allows us to prove the main result of this section: the Lipschitz property of the mapping Q 7! KQ on the balls in Lp [0; 1];C22  . Theorem 3.4. Let Q; eQ 2 U22p;r for some p 2 [1;1) and r > 0. Moreover, let KQ and KeQ be the kernels of the corresponding transformation operators from the representation (3.1) for Q and eQ, respectively. Then KQ ;K eQ 2 X0;21;p \X0;21;p; and there exists a constantC = C(B; p; r) that does not depend onQ and eQ, such that the following estimate holds kKQ KeQkX21;p + kKQ KeQkX21;p  C kQ eQkp: Remark 3.5. (i) For 2 2 Dirac systems (B = diag(1; 1)) with continuous potential Q the trian- gular transformation operators have been constructed in Levitan & Sargsyan (1991: Chapter 10.3) and Marchenko (1986: Chapter 1.2). For Q 2 L1 these transformation operators were constructed in Albeverio, Hryniv & Mykytyuk (2005) by an appropriate generalization of Marchenko’s method. (ii) Let J : f ! R x 0 f(t) dt denote the Volterra integration operator on Lp[0; 1]. Note that the sim- ilarity of the integral Volterra operators given by (2.3) to the simplest Volterra operator of the form B J acting in the spaces Lp([0; 1];C2) has been investigated in Malamud (1999); Romaschenko (2008). The technique of investigating integral equations for the kernels of the transformation oper- ators in the spaces X1;1( ) and X1;1( ) goes back to the paper (Malamud, 1994). 4 General properties of a 2 2 Dirac-type BVP Consider the 22 Dirac-type equation (1.1) subject to the general boundary conditions (1.2) and the corresponding operator L(Q) defined in (1.3). In this section we recall and extend some properties 122 Acta Wasaensia of this BVP from (Lunyov & Malamud, 2016). Let us set A :=  a11 a12 a13 a14 a21 a22 a23 a24  ; Ajk :=  a1j a1k a2j a2k  ; Jjk := det(Ajk); (4.1) where j; k 2 f1; : : : ; 4g. Moreover, let (; ) =  '11(; ) '12(; ) '21(; ) '22(; )  =: 1(; ) 2(; )  ; (0; ) =  1 0 0 1  ; (4.2) be a fundamental matrix solution of the system (1.1). The eigenvalues of the problem (1.1)–(1.2) (counting multiplicity) are the zeros (counting multiplicity) of the characteristic determinant Q() := det  U1(1(; )) U1(2(; )) U2(1(; )) U2(2(; ))  : (4.3) Inserting (4.2) and (1.2) into (4.3), setting 'jk() := 'jk(1; ), and taking the notations in (4.1) into account, we arrive at the following expression for the characteristic determinant Q() = J12 + J34e i(b1+b2) + J32'11() + J13'12() + J42'21() + J14'22(): (4.4) Alongside the problem (1.1)–(1.2) we consider the same problem with eQ in place of Q. Denote the corresponding fundamental matrix solution, its entries, and the corresponding characteristic de- terminant as e(; ), e'jk(; ), j; k 2 f1; 2g, and e(), respectively. If Q = 0, then we denote a fundamental matrix solution as 0(; ). Clearly 0(x; ) =  eib1x 0 0 eib2x  =: 01(x; )  0 2(x; )  ; x 2 [0; 1];  2 C: Here 0k(; ) is the kth-column of 0(; ). In particular, the characteristic determinant 0() becomes 0() = J12 + J34e i(b1+b2) + J32e ib1 + J14e ib2: (4.5) In the case of Dirac systems, i.e., when B = diag(1; 1), this formula simplifies to 0() = J12 + J34 + J32e i + J14ei: 4.1 Representation of the characteristic determinant Our investigation of the perturbation determinant relies on the following result, clarifying our Propo- sition 3.1 from Lunyov & Malamud (2016) and coinciding with it for Q 2 L1([0; 1];C22). Lemma 4.1. Let Q 2 Lp([0; 1];C22) for some p 2 [1;1). Then the functions 'jk(; ), with j; k 2 f1; 2g, admit the following representations for x 2 [0; 1] and  2 C 'jk(x; ) = jke ibkx + Z x 0 Kj1;k(x; t)e ib1t dt+ Z x 0 Kj2;k(x; t)e ib2t dt; (4.6) where Kjl;k := 2 1  K+jl + (1)l+kKjl  2 X01;p( ) \X01;p( ); j; k; l 2 f1; 2g: (4.7) Acta Wasaensia 123 Our study of the Lipschitz property of the eigenvalues and eigenfunctions is based on the following simple corollary of Theorem 3.4. Lemma 4.2. Let Q; eQ 2 U22p;r for some p 2 [1;1) and r > 0. Then the following representation holds for x 2 [0; 1] and  2 C 'jk(x; ) e'jk(x; ) = Z x 0 bKj1;k(x; t)eib1t dt+ Z x 0 bKj2;k(x; t)eib2t dt; (4.8) where bKjl;k := Kjl;k eKjl;k 2 X01;p( ) \X01;p( ); j; k; l 2 f1; 2g: (4.9) Moreover, for some C = C(p; r;B) the following uniform estimate holds k bKjl;kkX1;p( ) + k bKjl;kkX1;p( )  C kQ eQkp; j; k; l 2 f1; 2g: (4.10) Considering next the properties of the characteristic determinant, we first refine (Lunyov & Mala- mud, 2016: Lemma 4.1) in the Lp case. Note that the existence of trace values Kjk(1; ) is implied by Lemma 2.1 and the inclusions Kjk 2 X01;p \X01;p: The later inclusions are important, because the weaker inclusions Kjk 2 (X1;p \X1;p) n (X01;p \X01;p) do not ensure the existence of such traces. Lemma 4.3. Let Q 2 Lp([0; 1];C22) for some p 2 [1;1). Then the characteristic determinant Q() of the problem (1.1)–(1.2) is an entire function of exponential type and admits the following representation Q() = 0() + Z 1 0 g1;Q(t)e ib1t dt+ Z 1 0 g2;Q(t)e ib2t dt; (4.11) where for l 2 f1; 2g gl;Q() = J32K1l;1(1; ) + J42K2l;1(1; ) + J13K1l;2(1; ) + J14K2l;2(1; ) 2 Lp[0; 1]: (4.12) The next result is immediate by combining Lemma 4.3 with the estimate (4.10). Lemma 4.4. Let Q; eQ 2 U22p;r for some p 2 [1;1) and r > 0. Then the following representation holds Q() eQ() = Z 1 0 bg1(t)eib1t dt+ Z 1 0 bg2(t)eib2t dt; (4.13) where bgl := gQ;lg eQ;l 2 Lp[0; 1], l 2 f1; 2g. Moreover, for some bC = bC(p; r;B;A), the following uniform estimate holds kbg1kp + kbg2kp = kgQ;1 g eQ;1kp + kgQ;2 g eQ;2kp  bC kQ eQkp; Q; eQ 2 U22p;r : 124 Acta Wasaensia 4.2 Regular and strictly regular boundary conditions Recall that Jjk = detAjk, see (4.1), and recall the following definitions. Definition 4.5. The boundary conditions (1.2) are called regular if J14J32 6= 0: Definition 4.6 (cf. (Katsnel’son, 1971)). The sequence  := fngn2Z  C is called an incompress- ible sequence of density d 2 N, if every rectangle [t 1; t+ 1] R  C contains at most d entries of the sequence, i.e., if cardfn 2 Z : jRen tj  1g  d; t 2 R: Let us recall certain important properties of the characteristic determinant () in the case of regular boundary conditions from Lunyov & Malamud (2016). Recall that Dr(z)  C denotes the disc of radius r with center z. Proposition 4.7 (Lunyov & Malamud (2016: Proposition 4.6)). Let the boundary conditions (1.2) be regular and let Q() be the characteristic determinant of the problem (1.1)–(1.2), given by (4.4). Then the following statements hold: (i) The characteristic determinant Q() is a sine-type function with h(=2) = b1 and h(=2) = b2. In particular, the function Q() has infinitely many zeros  := fngn2Z counting multiplicities and   h for some h  0, see (1.13). (ii) The sequence  is incompressible. (iii) For any " > 0 there exists C" > 0 such that the determinant Q() admits the following estimate from below jQ()j  C"(eb1Im + eb2Im);  2 C n [ n2Z D"(n): Clearly, the conclusions of Proposition 4.7 are valid for the characteristic determinant 0() given by (4.5). Let 0 = f0ngn2Z be the sequence of its zeros counting multiplicity. From now on, let us order 0 in a (possibly non-unique) way such that Re0n  Re0n+1; n 2 Z: Let us recall an important result from Lunyov & Malamud (2014b; 2016) and Savchuk & Shkalikov (2014) concerning the asymptotic behavior of eigenvalues. Proposition 4.8 (Lunyov & Malamud (2016: Proposition 4.7)). Let Q 2 L1([0; 1];C22) and let the boundary conditions (1.2) be regular. Then the sequence  = fngn2Z of zeros of Q() can be ordered in such a way that the following asymptotic formula holds n =  0 n + o(1); as jnj ! 1; n 2 Z: (4.14) Let us refine this ordering to have some additional important properties. Acta Wasaensia 125 Proposition 4.9. Let Q 2 L1([0; 1];C22) and let the boundary conditions (1.2) be regular. Then the following statements hold: (i) For any " > 0 there exist M" = M"(Q;B;A) > 0 and C" = C"(B;A) > 0, such that jQ()0()j < j0()j;  =2 e "; (4.15) jQ()j > C" eb1Im + eb2Im  ;  =2 e "; (4.16) where e " := DM"(0) [ 0"; 0" := [ n2Z D"(0n): (4.17) (ii) The sequence  = fngn2Z can be ordered such that for any " > 0 and n 2 Z the values n and 0n belong to the same connected component of e ". In addition, the relation (4.14) also holds for this ordering. Definition 4.10. Let  = fngn2Z be the sequence of zeros of the characteristic determinant Q() of the Dirac-type operator LU (Q) with summable potential and regular boundary conditions. Lete " be defined in (4.17). The ordering of  for which n and 0n belong to the same connected component of e " for all " > 0 and n 2 Z, is called a canonical ordering. Observe that Proposition 4.9 implies the existence of a canonical ordering for each sequence of zeros of the characteristic determinant Q() of the Dirac-type operator LU (Q) with summable potential and regular boundary conditions. In the sequel we need the following definitions. Definition 4.11. A sequence  := fngn2Z of complex numbers is said to be separated if for some  > 0 the inequality jj kj > 2 holds whenever j 6= k: In particular, all entries of a separated sequence are distinct. Furthermore, the sequence  is said to be asymptotically separated if for some N 2 N the subsequence fngjnj>N is separated. Definition 4.12. The boundary conditions (1.2) are called strictly regular if they are regular, i.e., J14J32 6= 0, and the sequence of zeros 0 = f0ngn2Z of the characteristic determinant 0() is asymptotically separated. In particular, if the boundary conditions (1.2) are strictly regular, then there exists n0 2 N such that zeros f0ngjnj>n0 of its characteristic determinant are geometrically and algebraically simple. Observe that it follows from Proposition 4.8 that the sequence  = fngn2Z of zeros of Q() is asymptotically separated if the boundary conditions are strictly regular. Assuming the boundary conditions (1.2) to be regular, let us rewrite them in a more convenient form. Since J14 6= 0, the inverse matrixA114 exists. Therefore writing down the boundary conditions (1.2) as the vector equation U1(y) U2(y)  = 0 and multiplying it by the matrix A114 , they are transformed into the following conditions ( bU1(y) = y1(0) + by2(0) + ay1(1) = 0;bU2(y) = dy2(0) + cy1(1) + y2(1) = 0; (4.18) 126 Acta Wasaensia for some a; b; c; d 2 C. Now J14 = 1 and the boundary conditions (4.18) are regular if and only if J32 = ad bc 6= 0. Thus, the characteristic determinants 0() and () take the form 0() = d+ ae i(b1+b2) + (ad bc)eib1 + eib2; () = d+ aei(b1+b2) + (ad bc)'11() + '22() + c'12() + b'21(): Remark 4.13. Let us list some types of strictly regular boundary conditions (4.18). In all of these cases, except 4 (b), the set of zeros of 0 is a union of a finite number of arithmetic progressions. 1. Regular BCs (4.18) for the Dirac operator (b1 = b2 = 1) are strictly regular if and only if (a d)2 6= 4bc. 2. Separated BCs (a = d = 0, bc 6= 0) are always strictly regular. 3. Let b1=b2 2 Q. Without loss of generality we can assume that b1; b2 2 N and that gcd (b1; b2) = 1. It is clear that the BCs (4.18) are strictly regular if and only if a cer- tain polynomial of degree b2 b1 does not have multiple roots. In addition, if ad 6= 0 and bc = 0, then the BCs (4.18) are strictly regular if and only if b1 ln jdj+ b2 ln jaj 6= 0 or b1 arg(d) + b2 arg(a) =2 2Z: In particular, antiperiodic BCs (a = d = 1, b = c = 0) are strictly regular if and only if b1 b2 is odd. Note that these BCs are not strictly regular in the case of a Dirac system. 4. Let := b1=b2 =2 Q. Then the problem of strict regularity of BCs is generally much more complicated. Let us list some known cases: (a) Let ad 6= 0 and bc = 0. Then the BCs (4.18) are strictly regular if and only if b1 ln jdj+ b2 ln jaj 6= 0: (b) Let a = 0 and bc; d 2 R n f0g. Then the BCs (4.18) are strictly regular if and only if d 6= ( + 1) jbcj  1 +1 ; see (Lunyov & Malamud, 2016: Proposition 5.6). 5 Fourier transform estimates 5.1 Generalizations of the Hausdorff-Young and Hardy-Littlewood theorems To evaluate deviations of eigenvalues of the operators L(Q) and L( eQ), we extend here the classical Hausdorff-Young and Hardy-Littlewood interpolation theorems for Fourier coefficients (see (Zig- mund, 1959: Theorem XII.2.3) and (Zigmund, 1959: Theorem XII.3.19), respectively) to the case of arbitrary incompressible sequences  = fngn2Z instead of  = f2ngn2Z. Acta Wasaensia 127 For an efficient estimate of eigenvectors deviations in Section 7 we will use the following (sublinear) Carleson transform (the maximal version of the classical Fourier transform) Ef () := supN>0 Z N N Ff (t)e it dt ;  2 R; where Ff denotes the classical Fourier transform, Ff () = lim N!1 Z N N f(t)eit dt: (5.1) Its most important property is contained in the following Carleson-Hunt theorem, see (Grafakos, 2009: Theorems 6.2.1 & 6.3.3). Theorem 5.1. For any p 2 (1;1) the Carleson operator E is a bounded operator from Lp(R) to itself, i.e., there exists a constant Cp > 0 such that kEfkLp  CpkfkLp ; f 2 Lp(R): For our considerations it is more convenient to consider the following version of E Fg() := supx2[0;1] Z x 0 g(t)eit dt ; g 2 Lp[0; 1];  2 C: For brevity we putF g () := (Fg()) . Also recall that p0 = p=(p 1). Combining the Carleson-Hunt theorem (Theorem 5.1) and the Hausdorff-Young theorem leads to the following result, see, e.g., (Savchuk, 2019) for details. Proposition 5.2. For any p 2 (1; 2] the maximal Fourier transformF maps Lp[0; 1] boundedly into Lp 0 [0; 1], i.e., the following estimate holdsZ 1 1 F p 0 g (x) dx  p kgkp 0 p ; g 2 Lp[0; 1]; (5.2) where p > 0 does not depend on g 2 Lp[0; 1]. In the sequel we will need the following lemma whose proof substantially relies on the estimate (5.2). Recall the definition of h;n in (1.13). Lemma 5.3. Let g 2 Lp[0; 1] for some p 2 (1; 2] and h  0, and let gn := sup  Fg() :  2 h;n : Then the following inequality holdsX n2Z gp 0 n  Cp;h kgkp 0 p ; Cp;h := p e p0(h+1): The proof of Lemma 5.3 extends the classical reasoning about estimates of Hardy space func- tions and Lp-classes of entire functions, see (Levin, 1996: Lectures 20-21) and (Katsnel’son, 1971: Lemma 2) for the case of the maximal Fourier transform. 128 Acta Wasaensia Now we are ready to state the main result of this section which is a generalization of the Hausdorff- Young and Hardy-Littlewood theorems to the case of non-harmonic series with exponents forming an incompressible sequence  = fngn2Z instead of  = f2ngn2Z. Theorem 5.4. Let p 2 (1; 2]. Let  = fngn2Z be an incompressible sequence of density d 2 N lying in the strip h, and let g 2 Lp[0; 1]. Then there exists C = C(p; h; d) > 0 that does not depend on  and g, such that the following estimates hold uniformly with respect to g and :X n2Z jFg(n)jp0  X n2Z F p 0 g (n)  C kgkp 0 p ; (5.3) X n2Z (1 + jnj)p2jFg(n)jp  X n2Z (1 + jnj)p2F pg (n)  C kgkpp: (5.4) Estimate (5.3) is an immediate consequence of Lemma 5.3. The proof of (5.4) is based on the Marcinkiewicz theorem (Zigmund, 1959: Theorem XII.4.6). Note also that the parts of the inequali- ties (5.3)–(5.4) involving the classical Fourier transform Fg defined in (5.1) can be proved in a direct way, which is elementary in character, because it does not involve the Carleson-Hunt theorem. Corollary 5.5. Let  = fngn2Z be a sequence of zeros of a sine-type function (). Then for any p 2 (1; 2] the estimates (5.3) and (5.4) hold uniformly in g 2 Lp[0; 1]. Proof. The proof is immediate from Theorem 5.4 if one notes that the null set of a sine-type function () is always incompressible, see (Levin, 1961), (Katsnel’son, 1971), and Proposition 4.7(ii). Inverse statements for the Hausdorff-Young and Hardy-Littlewood theorems also hold in the case of non-harmonic exponential series with exponents  = fngn2Z forming the null set of a sine-type entire function instead of  = f2ngn2Z. 5.2 Uniform versions of the Riemann-Lebesgue lemma Lemma 5.6, needed in the sequel, easily follows by combining Lemma 5.3 with Chebyshev’s in- equality. It can be understood as a uniform version of the classical Riemann-Lebesgue lemma. Lemma 5.6. Let g 2 Up;r for some p 2 (1; 2] and r > 0. Moreover, let b 2 R n f0g, let h  0, and let p0 be such that 1=p0 + 1=p = 1. Then for any  > 0 there exists a set Ig;  Z such that the following inequalities hold uniformly with respect to g 2 Up;r card(Z n Ig;)  N := C (r=)p0 ; (5.5) Z 1 0 g(t)eibt dt  supx2[0;1] Z x 0 g(t)eibt dt < ;  2 [ n2Ig; h;n; (5.6) where h;n is given by (1.13). Here C = C(p; h; b) > 0 does not depend on g, r, and . Let us emphasize that "uniformity" in Lemma 5.6 does not relate to the set Ig; , but only to the "size" of its complement, see (5.5). Note also that the part of estimate (5.6) involving the regular Fourier transform R 1 0 g(t)eibt dt can be proved in an easier way without using the Carleson-Hunt theorem. Acta Wasaensia 129 Next we investigate the "maximal" Fourier transform defined on the space X1;1( ) by F [G]() := supx2[0;1] Z x 0 G(x; t)eibt dt ;  2 C; G 2 X1;1( ): (5.7) The results in the rest of this section do not use the deep Carleson-Hunt theorem. First we present the following "uniform" version of the Riemann-Lebesgue lemma for the space X01;1( ). To this end for any h  0 we set C0(h) := f' 2 C(h) : lim t!1'(t iy) = 0 uniformly in y 2 [h; h]g: Proposition 5.7. Let h  0 and let F be given by (5.7). Then the following statements hold: (i) The nonlinear mapping F : X01;1( )! C(h) is well-defined and is Lipschitz kF [G]F [ eG]kC(h)  ejbjh kG eGkX1;1( ); G; eG 2 X01;1( ): (ii) For any h  0 the mapping F maps X01;1( ) continuously into C0(h). (iii) For any compact set X in X01;1( ) the following relation holds lim !1 F [G]() = 0; (5.8) uniformly in G 2 X and  2 h. Proposition 5.7 (iii) contains as a special case the following "uniform" version of the classical Riemann-Lebesgue lemma: for any compact set K in L1[0; 1] one has supg2K Z 1 0 g(t)eit dt = o(1) as !1; uniformly in g 2 K and  2 h: Next we complete Proposition 5.7 by evaluating the "maximal" Fourier transform F [G]() in the plane instead of a strip. Lemma 5.8. Let X be a compact set in X01;1( ), let b 2 R n f0g, and let  > 0. Then there exists a constant C = C(X ; b; ) > 0, such that the following estimate holds F [G]()  (ebIm + 1); jj > C; G 2 X ; uniformly in G 2 X . Finally, we apply Proposition 5.7 (i), Theorem 3.4, and Lemma 5.8 to transformation operators. Corollary 5.9. LetKQ be the kernel of the transformation operator from representation (3.1). Then the composition Q! KQ ! F [KQ ] maps Lp([0; 1];C22) continuously into C0(h;C22), h  0, and it is a Lipschitz mapping on balls in Lp([0; 1];C22), p 2 [1;1) kF [KQ ]F [KeQ ]kC(h)  ejbjh C(B; p; r) kQ eQkLp ; Q; eQ 2 U22p;r : (5.9) 130 Acta Wasaensia The following statement will be useful in Section 6 when applying Rouché’s theorem; it is an im- mediate consequence of Theorem 3.4 and Lemma 5.8. Lemma 5.10. Let K be a compact set in L1([0; 1];C22), let Q 2 K, and let KQ = (Kjk)2j;k=1 be the kernel of the transformation operator from representation (3.1). Then for any  > 0 there exists a constant M = M(K; B; ) > 0 such that the following estimate holds uniformly in Q 2 K F [Kjk]() = supx2[0;1] Z x 0 Kjk(x; t)e ibkt dt  (ebkIm + 1); jj > M; (5.10) where j; k 2 f1; 2g. In particular, for any h  0, one has supQ2K F [Kjk]()! 0 as jj ! 1 and  2 h: Let us demonstrate Corollary 5.9 and Lemma 5.10 for concrete examples of compacts. Corollary 5.11. Let K be a ball either in the Sobolev spaces W s1 [0; 1] with s 2 R+, in the Lipschitz space  [0; 1] with 2 (0; 1], or in the space V [0; 1] of functions of bounded variation. Then the relations (5.9) and (5.10) hold true uniformly in Q 2 K. Remark 5.12. Let us present a simple example of a non-compact set in Lp[0; 1] for which the uniform relation (5.8) is violated. Define the following set of functions G := fg(x) := g0(x)eix :  2 Rg; where g0 2 Lp[0; 1] is such that c0 := R 1 0 g0(t) dt > 0. It is clear that F [g]()  Z 1 0 g0(t)e iteit dt = c0 6= 0 and limjj!1F [g]() = 0: (5.11) The last relation in (5.11) is satisfied not uniformly on G. Moreover, inequality (5.6) holds on sets Ig; = Z n ( N;  + N) that depend on g, and their complements "tend to infinity" when  ! 1, but have uniformly bounded "sizes": card(Z n Ig;)  2N . We are indebted to V.P. Zastavnyi who informed us about this example. 6 Stability property of eigenvalues 6.1 Uniform localization of spectrum In this subsection we will obtain a version of the asymptotic formula (4.14) which is uniform with respect to Q 2 K, where K is either a compact set in L1([0; 1];C22) or K = U22p;r for p 2 (1; 2]. Recall that A is the matrix defined in (4.1) and composed from the coefficients of the linear forms U1 and U2 as in (1.2) and that B = diag(b1; b2). First, we enhance Proposition 4.8 to obtain uniform estimates for Q 2 K, where K is compact in L1([0; 1];C22). The following result generalizes (Sadovnichaya, 2016: Theorem 3) to the case of Dirac-type systems with regular boundary conditions. Its proof is substantially based on the representation (4.11), Lemma 5.10, and Rouché’s theorem. Acta Wasaensia 131 Proposition 6.1. Let K be compact in L1([0; 1];C22) and let Q 2 K. Let the boundary con- ditions (1.2) be regular, let () := Q() be the corresponding characteristic determinant, and let  := Q := fQ;ngn2Z be the canonically ordered sequence of its zeros. Moreover, let 0 = f0ngn2Z be the sequence of zeros of 0. Then the following estimates hold: (i) There exists M = M(K; B;A) > 0, that does not depend on Q, such that supn2Z Q;n 0n M; Q 2 K: In particular, there exist h = h(K; B;A) > 0 and d = d(K; B;A), that do not depend on Q, such that Q is an incompressible sequence of density d and lying in the strip h. (ii) For any " > 0 there exists a constant N" = N"(K; B;A) 2 N, such that supjnj>N" Q;n 0n  "; Q 2 K: If, in addition, the boundary conditions (1.2) are strictly regular, then there exists a constant "0 = "0(B;A), such that for any " 2 (0; "0] the discs D2"(Q;n), jnj > N", are disjoint and there exists a constant eC" = eC"(B;A) > 0, such that min jQ;nj=2" jQ()j  eC"; jnj > N"; Q 2 K: Next we extend Proposition 6.1 to the case K = U22p;r , p 2 (1; 2]. Part (i) remains valid but the assumption p > 1 is important. Part (ii) is based on Lemma 5.6 that involves only the classical Fourier transform Fg without the use of the deep Carleson-Hunt theorem. It remains valid if we replace the inequality jnj > N" by an inclusion n 2 IQ;", assuming that the complements of the sets IQ;" have uniformly bounded cardinalities for Q 2 U22p;r . Proposition 6.2. Let Q 2 U22p;r for some p 2 (1; 2] and some r > 0. Moreover, let the boundary conditions (1.2) be regular, let () := Q() be the corresponding characteristic determinant, and let  := Q = fQ;ngn2Z be a canonically ordered sequence of its zeros. Then the following statements hold: (i) There exists a constant M = M(p; r; B;A) > 0, not dependent on Q, such that supn2Z Q;n 0n M; Q 2 U22p;r : In particular, there exist h = h(p; r;B;A)  0 and d = d(p; r;B;A) > 0, not dependent on Q, such that Q is an incompressible sequence of density d lying in the strip h. (ii) For any " > 0 there exists N" = N"(p; r;B;A) 2 N, that does not depend on Q, and a set IQ;"  Z, such that jn 0nj < "; n 2 IQ;"; and card (Z n IQ;")  N": Proposition 6.1 combined with the maximum and minimum principles imply the following uniform version of the relation jn enj  je(n)j that is pivotal for establishing the stability property of the mapping Q! Q := fQ;ngn2Z. 132 Acta Wasaensia Proposition 6.3. Let K be a compact set in L1([0; 1];C22) and let Q; eQ 2 K. Moreover, let the boundary conditions (1.2) be strictly regular, and let Q = fQ;ngn2Z and  eQ = f eQ;ngn2Z be canonically ordered sequences of zeros of characteristic determinants  := Q and e :=  eQ, respectively. Then there exist constants N = N(K; A;B) 2 N and C = C(K; A;B)  1, that do not depend on Q and eQ, such that the following uniform estimate holds C1 j eQ(Q;n)j  jQ;n  eQ;nj  C j eQ(Q;n)j; jnj > N: 6.2 Stability property of eigenvalues for Q 2 Lp In this section we apply the abstract results from Section 5 to establish the stability of the mapping Q ! Q := fQ;ngn2Z in different norms. Proposition 6.3 shows that to this end, it suffices to evaluate the sequences fe(n)gn2Z = f eQQ;ngn2Z when Q runs through either the ball U22p;r or a compact K in L1([0; 1];C22). In turn, these sequences can be easily evaluated by combining the representation (4.13) and the results of Section 5. For example, the estimate (5.3) implies that X n2Z  eQQ;n p0  Cp;r;B kQ eQkp0p ; Q; eQ 2 U22p;r : Next we enhance and complete Proposition 4.8 in the case of Q 2 Lp([0; 1];C22) with p 2 [1; 2]. Our first result restricts the set K of potential matrices to be a compact. Theorem 6.4. Let K be compact in Lp([0; 1];C22) for some p 2 [1; 2], and let Q; eQ 2 K. Moreover, let the boundary conditions (1.2) be strictly regular, and let Q := fQ;ngn2Z and  eQ := f eQ;ngn2Z be canonically ordered sequences of zeros of the characteristic determinants () := Q() and e() :=  eQ(), respectively. Then there exist constants N = N(K; A;B) 2 N and C = C(p;K; A;B) > 0, not dependent on Q and eQ, such that the following estimates hold:X jnj>N jQ;n  eQ;njp0  C kQ eQkp0p ; p 2 (1; 2]; (6.1) X jnj>N (1 + jnj)p2 jQ;n  eQ;njp  C kQ eQkpp; p 2 (1; 2]: (6.2) If p = 1, then supQ; eQ2K jQ;n  eQ;nj ! 0 as n!1: In other words, the set of sequences jQ;n  eQ;nj n2Z Q; eQ2K forms a compact set in c0(Z). Applying Theorem 6.4 with a compact set K = fQ; 0g, we can complete Proposition 4.8. Corollary 6.5. Let Q 2 Lp([0; 1];C22) for some p 2 (1; 2]. Moreover, let the boundary condi- tions (1.2) be strictly regular and let () be the corresponding characteristic determinant. Then the sequence  = fngn2Z of its zeros can be ordered such that the following inequality holdsX n2Z n 0n p0 +X n2Z (1 + jnj)p2 n 0n p <1: Note that the inclusion fn 0ngn2Z 2 `p 0 (Z) in the case of 2 2 Dirac systems (b1 = b2 = 1) was first obtained in Savchuk & Shkalikov (2014: Theorems 4.3 & 4.5). Acta Wasaensia 133 Remark 6.6. Let p = 3=2 and p0 = 3. Moreover, assume that eQ = 0 and let Q 2 Lp be fixed. Note that n := Q;n  eQ;n = (1 + jnj) ln2(1 + jnj)1=3 ; n 2 Z; satisfies (6.1), but does not satisfy (6.2). Hence, this sequence cannot be a sequence of eigenvalue deviations of operators L(Q) and L(0) for some Q 2 Lp. Note that the results of Savchuk & Shkalikov (2014) do allow it. In fact, inequality (6.2) is generally more restrictive and, hence, leads to a sharper estimate of the sequence fQ;n  eQ;ngn2Z than (6.1). For example, inequality (6.2) implies (6.1) under the general assumption n = o(n1=p 0 ) as n!1. Next we extend Theorem 6.4 to the case K = U22p;r . As in Proposition 6.2, we cannot select a universal constant N serving all potentials. Instead, we need to sum over the sets of integers whose complements have uniformly bounded cardinality. Theorem 6.7. LetQ; eQ 2 U22p;r for some p 2 (1; 2] and r > 0. Let the boundary conditions (1.2) be strictly regular, and let Q = fQ;ngn2Z and  eQ = f eQ;ngn2Z be canonically ordered sequences of zeros of characteristic determinants  := Q and e :=  eQ, respectively. Then there exist constants N 2 N, C1; C2; C > 0, not dependent on Q and eQ, and a set I := IQ; eQ  Z, such that the following estimates hold: card  Z n IQ; eQ   N; (6.3) C1  eQQ;n  jQ;n  eQ;nj  C2  eQQ;n ; n 2 IQ; eQ; (6.4)X n2IQ; eQ Q;n  eQ;n p0  C kQ eQkp0p ; (6.5) X n2IQ; eQ (1 + jnj)p2 Q;n  eQ;n p  C kQ eQkpp: (6.6) Remark 6.8. (i) Observe that the proofs of all results in this section, including the proofs of Theo- rems 6.4 and 6.7, rely on the Bessel type inequalities (5.3)–(5.4) for the ordinary Fourier transform, not for its maximal version described in Theorem 5.4, whose proof relies on Theorem 5.1. (ii) The case of Dirac systems (b1 = b2 = 1) and eQ = 0 has extensively been studied in many re- cent papers of Sadovnichaya, Savchuk, and Shkalikov by applying a different method. In particular, the estimate (6.1) was established earlier in Savchuk & Shkalikov (2014: Theorems 4.3 & 4.5) with a constant C that depends on Q, while estimate (6.5) of Theorem 6.4 with eQ = 0 was established in Savchuk & Sadovnichaya (2018). (iii) The weighted estimates (6.2) and (6.6), as well as the estimate (6.1), which establish stability properties of the spectrum under the perturbation Q! eQ, are new even for Dirac system. (iv) L. Rzepnicki (2020) obtained sharp asymptotic formulas for deviations n 0n = n + n in the case of Dirichlet BVPs for the Dirac system with Q 2 Lp([0; 1];C22), 1  p < 2. Namely, n is explicitly expressed via Fourier coefficients and Fourier transforms of Q12 and Q21, while fngn2Z 2 `p0=2(Z), i.e., the convergence to zero is "twice" better than what formula (6.1) guaran- tees for n 0n. A similar result was obtained for eigenfunctions. (v) We mention also the papers (Cascaval et al., 2004), (Clark & Gesztesy, 2006), and (Brown et al., 2019), where different spectral properties of j-selfadjoint Dirac operators were investigated. 134 Acta Wasaensia 7 Stability property of eigenfunctions 7.1 Estimates of Fourier transforms of transformation operators In this subsection we study "Fourier" transforms of the kernels of the corresponding transformation operators from the representation (3.1) of the formZ x 0 Kjk(x; t)e ibkt dt: Our investigation is motivated by the representation (4.6) for the entries of the fundamental matrix (; ). As distinguished from the considerations of Section 6, here our proofs substantially involve the deep Carleson-Hunt theorem via the corresponding results of Section 5. As a first step we study "Fourier" transforms of the auxiliary kernelsRQ from the representation (3.7) for the kernels of the transformation operators KQ . Below, we first estimate generalized "Fourier" transforms with an arbitrary bounded function, instead of the exponential function in the integral. Recall that k := bj bjbk , j = 2=k, k 2 f1; 2g. Proposition 7.1. Let Q 2 L1([0; 1];C22) and let RQ = (Rjk) 2 j;k=1 2 X01;1( ) \X01;1( )  C22 be the (unique) solution of the system of integral equations (3.8)–(3.9). Moreover, let x 2 [0; 1] be fixed, let f 2 L1(R) be such that f(t) = 0 for t =2 [0; x], and set Fjk(s; f) := sup u2[0;s] v2[u;x] Z u 0 Rjk(s; t)f(t+ v) dt ; s 2 [0; x]; j; k 2 f1; 2g: Then the following estimates hold for s 2 [0; x], k 2 f1; 2g, and j = 2=k: Fkk(s; f)  jbkj Z s 0 jQkj(t)jFjk(t; f) dt; Fjk(s; f)  jbj j sup u2[0;s] v2[u;x] j Z u 0 Qjk( ks+ jt) f(t+ v) dt + 2jbjbkj kQjkkL1[0;s] Z s 0 jQkj(t)jFjk(t; f) dt: In particular, one has that the following uniform estimate holds for Q 2 U221;r , x 2 [0; 1],  2 C, and j; k 2 f1; 2g sups2[0;x] Z s 0 Rjk(x; t)e ibkt dt  Ce(b2b1)jImjx sups2[0;x] Z s 0 Qjk(t)e i(bkbj)t dt ; (7.1) where j = 2=k and C = C(B; r) > 0 does not depend on Q, x, and . Note that (7.1) follows by taking f(t) = eibkt, t 2 [0; x], and f(t) = 0, t =2 [0; x], in the formulas preceding (7.1). The estimate (7.1) allows us to obtain a similar estimate for the Fourier transforms of the auxiliary functions Pk from the representation (3.6)–(3.7). Combining them, we arrive at the following important estimate of the Fourier transforms of the kernels KQ . Acta Wasaensia 135 Theorem 7.2. Let Q 2 U221;r for some r > 0 and let KQ be the kernels of the corresponding transformation operators from representation (3.1). Then the following uniform estimate holds for x 2 [0; 1] and  2 C 2X j;k=1 Z x 0 Kjk(x; t)e ibkt dt  C e2(b2b1)jImjxX j 6=k sups2[0;x] Z s 0 Qjk(t)e i(bkbj)t dt : This estimate is uniform in the sense that C = C(B; r) > 0 does not depend on Q, x, and . 7.2 Stability property of the fundamental matrix Alongside equation (1.1) we consider similar Dirac-type equations with the same matrix B but with a different potential matrix eQ 2 L1([0; 1];C22). Recall that Q(x; ) and  eQ(x; ) denote the fundamental matrix solutions of the system (1.1) for Q and eQ, that satisfy the initial conditions Q(0; ) =  eQ(0; ) = I . We can extend Theorem 7.2 to obtain the stability of "Fourier" transforms of the kernel differences of the corresponding transformation operators from the representation (3.1). Combining it with the representations (4.8)–(4.9) in Lemma 4.2 for entries of the deviation Q(; )  eQ(; ) of the fundamental matrices, we obtain the following uniform estimate that plays an important role in studying deviations of root vectors and which is of independent interest. Theorem 7.3. Let Q; eQ 2 U221;r for some r > 0, and let KQ and KeQ be the kernels of the corresponding transformation operators from representation (3.1) for Q and eQ, respectively. Then with some C = C(B; r) > 0 the following uniform estimate holds for x 2 [0; 1] and  2 C Q(x; )  eQ(x; )  2 2X j;k=1 X  Z x 0 KQ KeQ  jk (x; t)eibkt dt  C e2(b2b1)jImjx X j 6=k  sups2[0;x] Z s 0 (Qjk(t) eQjk(t))ei(bkbj)t dt +kQ eQk1 sups2[0;x] Z s 0 eQjk(t)ei(bkbj)t dt  : Combining Theorems 7.3 and 5.4, we arrive at an important stability (Lipschitz) property of the fundamental matrix. Proposition 7.4. Let Q; eQ 2 U22p;r for some p 2 (1; 2] and r > 0, and let  = fngn2Z be an incompressible sequence of density d lying in the strip h. Then for some C = C(p; r; B; h; d) > 0, not dependent on Q, eQ, and , the following uniform estimates hold: X n2Z Q(; n)  eQ(; n) p01  C kQ eQkp0p ; X n2Z (1 + jnj)p2 Q(; n)  eQ(; n) p1  C kQ eQkpp: 136 Acta Wasaensia 7.3 Stability property of the eigenfunctions Now the main results of this section are formulated. The following result for p = 1 general- izes (Sadovnichaya, 2016: Theorem 4) to the case of Dirac-type systems and extends it for p 2 (1; 2]. It can be proved by combining results of the previous subsection with Theorem 6.4. Theorem 7.5. Let K be compact in Lp([0; 1];C22) for some p 2 [1; 2] and let Q; eQ 2 K. Let the BCs (1.2) be strictly regular and let s 2 (0;1]. Then there exist SRVs ffQ;ngn2Z and ff eQ;ngn2Z of the operators L(Q) and L( eQ), respectively, such that kfQ;nks = kf eQ;nks = 1, jnj > N , and that the following relations hold uniformly for Q; eQ 2 K: supQ; eQ2K fQ;n f eQ;n 1 ! 0 as jnj ! 1; (7.2)X jnj>N fQ;n f eQ;n p01  C kQ eQkp0p ; p 2 (1; 2]; (7.3)X jnj>N (1 + jnj)p2 fQ;n f eQ;n p1  C kQ eQkpp; p 2 (1; 2]: (7.4) Here the constants N 2 N and C > 0 do not depend on Q, eQ, and s. Next we extend Theorem 7.5 to the case K = U22p;r . As in Theorem 6.7, we cannot select a uni- versal constant N serving all potentials. Instead, we need to sum over the sets of integers, whose complements have uniformly bounded cardinality. Theorem 7.6. Let Q; eQ 2 U22p;r for some p 2 (1; 2] and some r > 0. Let the BCs (1.2) be strictly regular and let s 2 (0;1]. Then there exist SRVs ffQ;ngn2Z and ff eQ;ngn2Z of the operators L(Q) and L( eQ), respectively, and a set IQ; eQ  Z, such that the following uniform relations hold for Q; eQ 2 U22p;r : kfQ;nks = kf eQ;nks = 1; n 2 IQ; eQ; and card  Z n IQ; eQ   N; X n2IQ; eQ fQ;n f eQ;n p01  C kQ eQkp0p ; X n2IQ; eQ (1 + jnj)p2 fQ;n f eQ;n p1  C kQ eQkpp: Here the constants N 2 N and C > 0 do not depend on Q, eQ, and s. Applying Theorem 7.5 with a two-point compact K = fQ; 0g, we arrive at the following stability property of eigenfunctions demonstrating the core of both Theorems 7.5 and 7.6. Corollary 7.7. Let Q 2 Lp([0; 1];C22), p 2 (1; 2], and let the BCs (1.2) be strictly regular. Then SRVs ffngn2Z and ff0ngn2Z of the operators L(Q) and L(0) can be chosen asymptotically normalized in Lp 0 ([0; 1];C2) and satisfying the following uniform estimatesX n2Z fn f0n p01 +X n2Z (1 + jnj)p2 fn f0n p1 <1: Acta Wasaensia 137 The following case shows that in some cases we can relax the compactness condition and even boundedness of K and sum over all n 2 Z in (7.3)–(7.4). Proposition 7.8. Let Q12 = eQ12 = 0 and Q21; eQ21 2 Lp[0; 1] for some p 2 (1; 2]. Let the BCs (4.18) be strictly regular with b = 0. Then the eigenvalues of the operators L(Q) and L( eQ) are simple and separated, and there exist systems ffngn2Z and f efngn2Z of their eigenfunctions, both normalized in C([0; 1];C2), such that the following uniform estimates hold: X n2Z fn efn p01  C kQ eQkpp; (7.5) X n2Z (1 + jnj)p2 fn efn p1  C kQ eQkp0p : (7.6) These estimates are uniform in the sense that C = C(p;B;A) > 0 does not depend on Q and eQ. Remark 7.9. If Q12 = 0, b = 0, and a = 1 in the BCs (4.18), then the sequence of eigenvalues of the operator L(Q) is the union of two arithmetic progressions with one of them being the sequence n = 2(1 b2=b1)n, n 2 Z. The corresponding eigenfunctions can be expressed explicitly via the Fourier coefficients Z x 0 Q21(t)e int dt and Z 1 0 Q21(t)e int dt: Hence Proposition 7.8 shows that the stability properties (7.5)–(7.6) of the eigenfunctions of the operator L(Q) are equivalent to the abstract inequalities (5.3)–(5.4) from Theorem 5.4 with the sequence fngn2Z being an arithmetic progression. Acknowledgement: This paper has been supported by the RUDN University Strategic Academic Leadership Program. References Albeverio, S., Hryniv, R. & Mykytyuk, Ya. (2005). Inverse spectral problems for Dirac operators with summable potentials. Russian J. Math. Physics 12, 406–423. Baskakov, A.G., Derbushev, A.V. & Shcherbakov, A.O. (2011). 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Facebook, Inc., 1 Hacker Way, Menlo Park, CA 94025, United States of America E-mail address: A.A.Lunyov@gmail.com Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St., Moscow, 117198, Rus- sian Federation E-mail address: malamud3m@gmail.com Acta Wasaensia 141 COMPLETENESS AND MINIMALITY OF EIGENFUNCTIONS AND ASSOCIATED FUNCTIONS OF ORDINARY DIFFERENTIAL OPERATORS Manfred Möller Dedicated to Seppo Hassi on the occasion of his 60th birthday 1 Introduction One of the oldest and most important results in operator theory is that a selfadjoint operator with compact resolvent in a Hilbert gives rise to an orthogonal basis of eigenvectors. This property is extensively used in the Sturm-Liouville theory. If the operator is not selfadjoint, then there may be biorthogonal bases of the operator and its adjoint, but such bases are not guaranteed. There is some substantial literature especially regarding Sturm-Liouville problems, including operators with eigenvalue parameter dependent boundary conditions. Often, properties of the spectra of quite special cases are considered in detail, see, e.g., (Aliyev & Guliyeva, 2018; Aliyev & Namazov, 2017; Binding & Browne, 1995; Guliyev, 2019). Such proper- ties of spectra are then used to prove minimality or basisness of a subsystem of eigenfunctions and associated functions, see, e.g., (Aliyev, 2007; Aliev & Dun’yamalieva, 2015; Kerimov & Mirzoev, 2003; Namazov, 2017). Expansion theorems predate these results, see Schneider (1974), Walter (1973), or the more recent result on completeness in Allahverdiev (2005). Shkalikov (1983) de- vised a linearization method for differential operator polynomials in the eigenvalue parameter and he proved completeness and minimality of the eigenvectors and associated vectors of the linearized system. However, the degree of this operator polynomial equals the order of the differential equation and his method is therefore not applicable to the differential operators studied in the current note. For a Banach space H , Shkalikov (2019) starts with bases or complete and minimal systems in H  CN to give conditions when co-finite subsystems of projections onto H are bases or complete and minimal systems in H . In Möller (2020) and in this note B-biorthogonality for a bounded linear operator from a Banach space E to a Banach space F will be used, where E is densely and continuously embedded in the space H , see Section 2 for more details. This has the advantage that the general results of Mennicken & Möller (2003) for arbitrary Birkhoff regular n-th order differential operators can be applied. In particular, Birkhoff regularity guarantees the existence of co-finite systems which are minimal and complete in H , and more or less explicit criteria for the choice of such systems can be given. For the sake of completeness, the preparation needed and presented in Möller (2020) will be repeated here. The outline of the contents is as follows. In Section 2 notation is introduced and the abstract functional analytic theorem on completeness and minimality is proved. In Section 3 it is shown how the main result, Theorem 2.6, can be applied to an n-th order ordinary Birkhoff regular differential operator with boundary conditions which may depend on  linearly. In Section 4 the case of second order differential equations with separable boundary conditions is discussed. A general result is ob- tained when exactly one boundary condition depends on , whereas a positive result is also provided for a special case when both boundary conditions depend on . 142 Acta Wasaensia 2 Complete and minimal systems 2.1 Minimality Let E, F , and H be infinite-dimensional Banach spaces such that E is a dense subset of H with continuous embedding E ,! H . The dual spaces of E, F , H are denoted by E0, F 0, H 0 with corresponding bilinear forms h ; i. The notation h ; iE for the bilinear form in EE0, for example, may also be used. Definition 2.1. A sequence (wi)1i=1 in the Banach space H is called minimal in H if wj 62 span fwi : i 2 N; i 6= jg for all j 2 N: The sequence (wi)ni=1 in H is called complete in H if span fwi : i 2 Ng = H: Definition 2.2. Let B 2 L(E;F ). Two sequences (yi)1i=1 in E and (vi)1i=1 in F 0 are called B- biorthogonal if hByi; vjiF = ij ; i; j 2 N: (2.1) When the operator B is clear from the context, the notion B-biorthogonal may be shortened to biorthogonal. Let F be decomposed as F1  F2 where F1 is a Banach space and F2 is a finite-dimensional space of dimension , where  = 0 is allowed. Then the operator B from E to F can be decomposed as B =  B1 B2  ; where Bl 2 L(E;Fl), l = 1; 2. Furthermore it is assumed that B1 has a continuous extension B0 2 L(H;F1). With respect to the decomposition F 0 = F 01  F 02, let vi =: (vi;1; vi;2) for i 2 N. Theorem 2.3 (Möller (2020: Theorem 1)). Assume that the sequences (yi)1i=1, yi 2 E, and (vi)1i=1, vi 2 F 0, are B-biorthogonal. If span fv1;2; : : : ; v;2g = F 02, then (yi)1i=+1 is minimal in H . 2.2 Completeness For an operator S in a Banach space, its adjoint will be denoted by S. Proposition 2.4. Assume that span fv1;2; : : : ; v;2g = F 02 and that (yi)1i=1 is complete in H . Then there is an integer N with 1  N  + 1 such that, after possibly permutating the indices 1; : : : ; , the system (yi)1i=N is minimal and complete in H . Proof. If (yi)1i=+1 is complete in H , then the statement holds for N =  + 1, since (yi) 1 i=+1 is also minimal in H by Theorem 2.3. If (yi)1i=+1 is not complete in H , then span fyi : i 2 N; i  + 1gH $ H = span fyi : i 2 NgH : Acta Wasaensia 143 Therefore it may be assumed without loss of generality that y 62 span fyi : i 2 N; i  + 1gH : It is clear that also (yi)1i= is minimal in H . If (yi) 1 i= is not complete in H , then this procedure is repeated up to an index N with 2  N  + 1 which gives that (yi)1i=N is minimal and complete in H , or up to N = 1 which gives that (yi)1i=N is minimal in H . But for N = 1 the system (yi) 1 i=1 is complete in H by assumption, and the proof is complete. Proposition 2.4 states a sufficient condition under which the removal of at most  terms from the sequence results in the remaining sequence to be minimal and complete. However, the number of terms to be removed is not known, and therefore sufficient conditions will be found such that this number is exactly . Let eE := span fyi : i 2 NgE ; eB1 := B1j eE ; eB2 := B2j eE ; and eB := eB1eB2 ! : Proposition 2.5. Let the assumptions of Proposition 2.4 be satisfied and let N be the number in the statement of that proposition. Then eB2 is injective on span fv1;2; : : : ; vN1;2g. Proof. Of course, we only need to consider the case N > 1. Assume the statement to be false. Replacing y1; : : : ; yN1 and v1; : : : ; vN1 with suitable linear combinations thereof such that (2.1) remains true, it may be assumed without loss of generality that eB2v1;2 = 0. Then h eB2yi; v1;2i = hyi; eB2vi;2i = 0; i 2 N; and (2.1) would give i;1 = h eByi; v1i = hyi; eB0v1;1i eE = hyi; eB0v1;1iH ; i 2 N: But, since (yi)1i=N is complete in H , this would lead to eB0v1;1 = 0 and 1 = hy1; eB0v1;1iH , a contradiction. Consequently, under the assumptions of Proposition 2.4, for N in Proposition 2.4 to be  + 1 it is necessary that eB2 is injective or, equivalently, that eB2 is surjective. Since (yi)1i=N is minimal and complete in H , there exists a unique sequence (wi) 1 i=N in H 0 such that hyi; wjiH = i;j ; i; j  N: (2.2) Theorem 2.6. Assume that span fv1;2; : : : ; v;2g = F 02, that the sequence (yi)1i=1 is complete in H , that B2(span fyi : i 2 Ng) = F2, and that R(B2) \H 0 = f0g. Then (yi)1i=+1 is complete in H . Proof. By Proposition 2.4, the system (yi)1i=N is minimal and complete in H . In particular, eE is dense in H . Then it is clear that with B1 also eB1 has a continuous extension eB0 to H and thateB0 = B0. Furthermore, eB2 andB2 are surjective, which implies that eB2 andB2 are injective. Hence there is a one-to-one identification of elements in R( eB2) with elements in R(B2). But since H 0 is a subspace of eE0 as well as E0, the assumption R(B2) \H 0 = f0g gives that R( eB2) \H 0 = f0g. 144 Acta Wasaensia From (2.1) and (2.2) it follows that hyi; eBvj wji = 0; i; j  N: The completeness of (yi)1i=1 in eE and the biorthogonality (2.1) imply that eBvj wj is a linear combination of eBvl, l = 1; : : : ; N 1. Therefore eBvj wj = N1X l=1 hyl; eBvj wji eBvl = N1X l=1 hyl; wji eBvl; j  N; which gives eB2vj;2 + N1X l=1 hyl; wji eB2vl;2 = wj eB0vj;1 N1X l=1 hyl; wji eB0vl;1 2 H 0; j  N: Then R( eB2) \H 0 = f0g and the injectivity of eB2 show that vj;2 + N1X l=1 hyl; wjivl;2 = 0; which is impossible when N  j   because v1;2; : : : ; v;2 are linearly independent. Therefore N = + 1. 3 The differential operator Let a < b be real numbers. Recall the Sobolev space W kp (a; b) = fy 2 Lp(a; b) : y(i) 2 Lp(a; b); i = 1; : : : ; kg; p 2 (1;1); k 2 N; where the derivative is taken in the sense of distributions, see, e.g., (Mennicken & Möller, 2003: Section 2.1). The vector space W kp (a; b) becomes a Banach space when equipped with the norm k kp;k, defined by kykp;k = kX i=0 ky(i)kp; y 2W kp (a; b); where k kp is the norm in Lp(a; b). The dual of the Banach space W kp (a; b) will be identified with a space of distributions Wkp0 [a; b], where 1=p+ 1=p 0 = 1, and the corresponding bilinear form will be denoted by h ; ip;k, see, e.g., (Mennicken & Möller, 2003: Section 2.1). Now let n  2 and define Ky = nX i=0 kiy (i) with ki 2 W ip0(a; b) for i = 1; : : : ; n 1 and k0 2 Lmin(p;p0)(a; b). It will always be assumed that kn is a non-zero constant. The differential operator LD()y := K y; y 2Wnp (a; b); (3.1) Acta Wasaensia 145 satisfies LD() 2 L(Wnp (a; b); Lp(a; b)). Together with (3.1) two-point boundary conditions LR()y := n1X i=0 w (0) ki ()y (i1)(a) + n1X i=0 w (1) ki ()y (i1)(b) n k=1 = 0 (3.2) are considered, where the w(j)ki are polynomials of degree at most 1, i. e., they are constant or poly- nomials of degree 1. It is assumed that for each k = 1; : : : ; n at least one of the 2n polynomials w (0) ki ; w (1) ki , i = 1; : : : ; n, is not the zero polynomial. Clearly, L R() 2 L(Wnp (a; b);Cn). For  2 C define L() := (LD(); LR()): (3.3) It is clear that there are bounded operators A;B 2 L(Wnp (a; b); Lp(a; b) Cn) such that L() = A+ B;  2 C: (3.4) Since the dual spaces are defined via sesquilinear forms, the adjoint operator has the representation L() = A + B;  2 C: In order to define Birkhoff regularity some notation is introduced first. For positive integers r let Mr(C) be the set of r  r matrices with entries in C and put W (j)() = w (j) ki () n k;i=1 ; j = 0; 1; W () = (W (0)();W (1)()); C0() = diag(1; ; : : : ;  n1) 2Mn(C); l = deg  eT W (0)(n)C0();W (1)(n)C0()  ;  = 1; : : : ; n; (3.5) where deg denotes the degree as a vector polynomial in the variable  and e is the -th standard basis vector in Cn. Furthermore, let C1 = 0BBBB@ 1 : : : 1 k 1=n n !1 : : : k 1=n n !n ... ... k (n1)=n n ! n1 1 : : : k (n1)=n n !n1n 1CCCCA ; (3.6) where !j = exp  2i(j 1) n  ; j = 1; : : : ; n: Then diag(l1 ; : : : ; ln)W (j)(l)C0()C1 = W (j) 0 +O( 1); j = 0; 1: Clearly, the constants l defined in (3.5) satisfy 0  l  2n 1. Without loss of generality it will be assumed that l  l+1 for  = 1; : : : ; n 1. Let  := #f 2 f1; : : : ; ng : l  ng: (3.7) This means that for  > n  at least one entry of the -th row of the n  2n matrix W () is not constant, whereas for   n , all entries of the -th row of W () are constant. Let 146 Acta Wasaensia l^ 2 f0; : : : ; n 1g be such that l = l^ mod (n);  = 1; : : : ; n: Clearly, l = l^ for   n  and l = l^ + n for  > n . If n = 2m is even, then let L be the set of all n  n diagonal matrices with m consecutive entries 1 in the diagonal, in a cyclic arrangement, whereas the remaining entries are 0. If n = 2m + 1 is odd, then let L be the set of all n  n diagonal matrices with m or m + 1 consecutive entries 1 in the diagonal, in a cyclic arrangement, whereas the remaining entries are 0. Let B := fW (0)0  +W (1)0 (In ) :  2 Lg: (3.8) The problem (3.1)-(3.2) is called Birkhoff regular if det  6= 0 for all  2 B. This definition of Birkhoff regularity is a special case of the characterization of Birkhoff regularity in Mennicken & Möller (2003: Theorem 7.3.2). Then it follows from Mennicken & Möller (2003: Theorem 4.3.9) that the resolvent set of L is not empty, which implies the following result. Proposition 3.1. If the problem (3.1)-(3.2) is Birkhoff regular, then the spectrum (L) of L consists of a sequence (k)1k=1 of eigenvalues of finite multiplicity. Definition 3.2. (i) An ordered set fy0; y1; : : : ; yhg in E is called a chain of an eigenvector and associated vectors (CEAV) of L at 0 2 C if y0 6= 0 and if the vector polynomial y := hX l=0 ( 0)lyl is such that the vector polynomial Ly has a zero at 0 of multiplicity (y)  h+ 1. (ii) Let y0 2 N(L(0)) n f0g. Then (y0) denotes the maximum of all multiplicities (y), where y is as in part (i) with y(0) = y0. (iii) A system fy(j)l : 1  j  r; 0  l  mj 1g is called a canonical system of eigenvectors and associated vectors (CSEAV) of L at 0 if (a) fy(1)0 ; : : : ; y(r)0 g is a basis of N(L(0)); (b) fy(j)0 ; : : : ; y(j)mj1g is a CEAV of L at 0, j = 1; : : : ; r; (c) mj = maxf(y) : y 2 N(L(0)) n span fy(k)0 : k < jgg, j = 1; : : : ; r. The following two propositions are special cases of results in Mennicken & Möller (2003: Section 1.10). Proposition 3.3. Assume that L is Birkhoff regular and let 0 2 (L). Let y0; : : : ; yk be a CEAV of L at 0. Then (A+ 0B)y0 = 0; (A+ 0B)yl+1 = Byl; l = 0; : : : ; k 1: For each eigenvalue there exist biorthogonal canonical systems of eigenvectors and associated vec- tors of L and L. Acta Wasaensia 147 Proposition 3.4. Assume that the differential operator L is Birkhoff regular and let the CSEAV fy(j)l : 1  j  r; 0  l  mj 1g of L at 0 be given. Then there exists a CSEAV fv(j)l : 1  j  r; 0  l  mj 1g of L at 0 such that these two systems are biorthogonal, i.e., hBy(i)l ; v(j)mj1ki = ij lk; 1  i  r; 0  l  mi 1; 1  j  r; 0  k  mj 1: Before formulating the main result for differential operators, more notation is needed. Writing W () =: W0 + W1; (3.9) see (3.5), let cW1 be the   2n submatrix of W1 consisting of the last  rows of W1. Furthermore, for y 2Wnp (a; b) let bY be the 2n-vector bY := (y(a); : : : ; y(n1)(a); y(b); : : : ; y(n1)(b)): The obvious modification for indexed functions applies. A representation of the adjoint operator L() is given in Mennicken & Möller (2003: Theorem 6.5.1), namely, L()(u; d) = nX i=0 (1)ikiu(i)e ue + LR()d; u 2 Lp0(a; b); d 2 Cn; where ue is the canonical extension of u to R by defining u = 0 on R n [a; b], and where LR  () = nX i=1 (1)i1  W (0)()ei T (i1)a + W (1)()ei T  (i1) b  with the Dirac distributions c at c for c = a and c = b. Theorem 3.5. Assume that the differential operator L defined by (3.3) with the representation (3.4) is Birkhoff regular and that (yi)1i=1 is a sequence of elements in W n p (a; b) which consists of CSEAVs at all eigenvalues of L. Let (vi)1i=1 be a corresponding sequence of CSEAVs (at all eigenvalues of L) such that, under suitable indexing, the biorthogonality relation (2.1) holds. With respect to the decomposition Lp(a; b) Cn  C write vi = (ui; di;1; di;2). Further assume that span fd1;2; : : : ; d;2g = C and span fcW1 bYi : i 2 Ng = C: Then the system (yi)1i=+1 is minimal and complete in Lp(a; b). Proof. Note first that the existence of the sequence (vi)1i=1 in the statement of the theorem is guar- anteed by Proposition 3.4. Putting E = W kp (a; b), F = Lp(a; b) Cn, F1 = Lp(a; b) Cn, F2 = C, and H = Lp(a; b), it follows in the notation of Section 2 that B1y = (y; 0), y 2 E, has a continuous extension B0 onto H given by B0y = (y; 0). By (Mennicken & Möller, 2003: Theorem 8.8.3 & Remark 8.8.4) 148 Acta Wasaensia each function in f 2 Lp(a; b) can be represented by a series P1 j=1 fj , where each fj is a finite linear combination of eigenfunctions and associated functions of L. Therefore the eigenfunctions and associated functions of L form a complete system in Lp(a; b). The operator B2 is represented by B2y = cW1 bY , y 2 E. Thus the operator B2 is represented by the coefficient of  in (LR)() restricted to the last  components of Cn. Then (3.11) shows thatR(B2) consists of linear combina- tions of Dirac distributions and their derivatives, and therefore R(B2) \H 0 = f0g. An application of Theorems 2.3 and 2.6 completes the proof. The condition span fcW1 bYi : i 2 Ng = C is a very technical one and it will not be considered any further in this note in the general case. The case n = 2 will be discussed in the next section. 4 Completeness and minimality for second order differential operators In general, the assumptions of Theorem 3.5 are not easy to verify. Here we consider the case n = 2 with separable boundary conditions. For convenience, and possibly deviating from the arrangement of the rows ofW () in Section 3, it will be assumed that the first boundary condition is at a, whereas the second boundary condition is at b. Therefore W (0)() = w (0) 11 () w (0) 12 () 0 0 ! ; W (1)() = 0 0 w (1) 21 () w (1) 22 () ! ; where neither W (0) nor W (1) is identically zero. If, say, W (0)(0) = 0 for some 0 2 C, then one could factor out  0 from W (0), which would be a rather artificial factor in the boundary matrix. Hence it is reasonable to require that W (0)() 6= 0 and W (1)() 6= 0 for all  2 C. In particular, the boundary condition at a is an initial condition, and therefore each eigenvalue has geometric multiplicity 1. Furthermore, with the notation as in (3.9), W (0) 0 =  0;0 0;1 0 0  C1; W (1) 0 =  0 0 1;0 1;1  C1; for complex numbers i;j such that ( i;0; i;1) 6= (0; 0) for i = 0; 1. Assuming the Sturm-Liouville equation, i.e., k2 = 1, for simplicity and taking k1=22 = i, it follows that C1 =  1 1 i i  ; see (3.6). Therefore, W (0) 0 C1 =  0;0 + i 0;1 0;0 i 0;1 0 0  ; W (1) 0 C1 =  0 0 1;0 + i 1;1 1;0 i 1;1  ; and L = fdiag(1; 0); diag(0; 1)g. Hence B as defined in (3.8) consists of the two matrices 0;0 + i 0;1 0 0 1;0 i 1;1  ;  0 0;0 i 0;1 1;0 + i 1;1 0  : Acta Wasaensia 149 Assuming further for simplicity that all coefficients of wki are real, it follows that all problems in this section are Birkhoff regular. For general k2 and i;j it is obviously easy to determine when the problem is Birkhoff regular. Now the cases  = 0,  = 1, and  = 2 will be considered separately; for notation see Theorem 3.5 and (3.7). It will always be assumed that k2 = 1 and that all wki have real coefficients. 4.1 The case  = 0 This case is well known, it is just included for completeness. Here the assumption on the span is void, and therefore Theorem 4.1. If L is Birkhoff regular and if the boundary conditions are independent of the eigen- value parameter, then the system (yi)1i=1 is minimal and complete in Lp(a; b). 4.2 The case  = 1 Since the geometric multiplicity of each eigenvalue is 1, each CSEAV consist of one CEAV, whose length is the algebraic multiplicity of the corresponding eigenvalue. Hence each yi is therefore an element in such a chain, and we call yi a terminal function if it is the last element of such a CEAV. Note that a terminal function yi is an eigenfunction if the eigenvalue is simple, and an associated function otherwise. Theorem 4.2. Assume that k2 = 1, that k1 = 0, and that L is Birkhoff regular. If y1 is a terminal function or if d1;2 6= 0, then the system (yi)1i=2 is minimal and complete in Lp(a; b). Proof. The minimality has been shown in Möller (2020: Theorem 4). Since  = 1, it therefore suffices to prove that cW1Y 6= 0 whenever y is an eigenfunction. Hence by proof of contradiction, assume that cW1 bY = 0. Then cW1 = eTW1, where  = 1 if the -dependent boundary condition is at a and  = 2 otherwise. Furthermore, W ()bY = 0 for the eigenfunction y implies that also eTW0 bY = 0. But eTW0 and eTW1 are linearly independent by the feasibility assumption, which means that eTW0 bY and eTW1 bY would be linearly independent linear combinations of y and y0 at the corresponding endpoint. Hence both y and y0 would be 0 at that endpoint, which contradicts y 6= 0. 4.3 The case  = 2 Since this note is a continuation of the work in Möller (2020), only the special case y00 + y = 0; (4.1) y0(0) y(0) = 0; (4.2) y(1) y0(1) = 0; (4.3) with ; 2 R n f0g will be considered. 150 Acta Wasaensia Theorem 4.3. Let (yi)1i=1 be a sequence of elements in Wnp (a; b) which consists of CSEAVs at all eigenvalues of (4.1)–(4.3). If y1 and y2 are terminal functions corresponding to sufficiently large real eigenvalues, then the system (yi)1i=3 is minimal and complete in Lp(a; b). Proof. The minimality statement is Möller (2020: Theorem 5). It was also shown there, and it is easy to see, that solutions of (4.1) and (4.2) are multiples of u(x; ) = cos p x+ p  sin p x and that  is an eigenvalue of (4.1)–(4.3) if and only if  satisfies ( 2) cos p + ( + ) p  sin p  = 0: (4.4) It is clear that cW1 bY = y(0) y0(1)  : It has also been shown that u0(1; ) = p  sin p +  cos p  = 2 +  +  cos p ; whereas clearly u(0; ) = . From (4.4) we conclude for eigenvalues  that [( 2)2 + ( + )2] cos2 p  = ( + )2; and therefore (u0(1; ))2 = 2( 2 + )2 ( 2)2 + ( + )2 ; which shows that cW1Yi and cW1Yj are linearly independent for infinitely many pairs of eigenfunc- tions yi and yj . In particular, span fcW1 bYi : i 2 Ng = C2. References Aliyev, Y.N. (2007). Minimality of the system of root functions of Sturm–Liouville problems with decreasing affine boundary conditions. Colloq. Math. 109, 147–162. Aliev, Z.S. & Dun’yamalieva, A.A. (2015). Defect basis property of a system of root functions of a Sturm-Liouville problem with spectral parameter in the boundary conditions. Differ. Equ. 51, 1249– 1266. Aliyev, Z.S. & Guliyeva, S.B. (2018). Spectral properties of a fourth order eigenvalue problem with spectral parameter in the boundary conditions. Filomat 32, 2421–2431. Aliyev, Z.S. & Namazov, F.M. (2017). Spectral properties of a fourth-order eigenvalue problem with spectral parameter in the boundary conditions. Electron. J. Differential Equations 307, 11 pp. Allahverdiev, B.P. (2005). A nonself-adjoint singular Sturm-Liouville problem with a spectral pa- rameter in the boundary condition. Math. Nachr. 278, 743–755. Acta Wasaensia 151 Binding, P.A. & Browne, P.J. (1995). Application of two parameter eigencurves to Sturm-Liouville problems with eigenparameter-dependent boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 125, 1205–1218. Guliyev, N.J. (2019). Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter. J. Math. Phys. 60, 063501, 23 pp. Kerimov, N.B. & Mirzoev, V.S. (2003). On the basis properties of a spectral problem with a spectral parameter in the boundary condition. Sibirsk. Mat. Zh. 44, 1041–1045. Mennicken, R. & Möller, M. (2003). Non-self-adjoint Boundary Eigenvalue Problems. Amsterdam: Elsevier. Möller, M. (2020). Minimality of eigenfunctions and associated functions of ordinary differential operators. Adv. Oper. Theory 5, 1014–1025. Namazov, F.M. (2017). Basis properties of the system of eigenfunctions of a fourth order eigenvalue problem with spectral and physical parameters in the boundary conditions. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 37, 142–149. Schneider, A. (1974). A note on eigenvalue problems with eigenvalue parameter in the boundary conditions. Math. Z. 136, 163–167. Shkalikov, A.A. (1983). Boundary value problems for ordinary differential equations with a pa- rameter in the boundary conditions. Trudy Sem. Petrovsk. 9, 190–229. (In Russian, with English summary.) Shkalikov, A.A. (2019). Basis properties of root functions of ordinary differential operators with spectral parameter in the boundary conditions. Differ. Equ. 55, 631–643. Walter, J. (1973). Regular eigenvalue problems with eigenvalue parameter in the boundary condition. Math. Z. 133, 301–312. School of Mathematics, University of the Witwatersrand, Johannesburg, WITS 2050, South Africa E-mail address: manfred.moller@wits.ac.za Acta Wasaensia 153 PARTIALLY OVERLAPPING EVENT WINDOWS AND TESTING CUMULATIVE ABNORMAL RETURNS IN FINANCIAL EVENT STUDIES Seppo Pynnönen Dedicated to Professor Seppo Hassi on the occasion of his 60th birthday 1 Introduction In financial event studies the interest is to evaluate the effect of an economic event on the value of a firm. For this evaluation data available from financial markets can be successfully used with appropriate statistical testing methodology. The analyses are mainly based on stock or other asset returns. Campbell, Lo & MacKinlay (1997: Chapter 4) is an excellent introduction to financial event studies and related statistical methods. Instead of using returns as such, standardizing them by respective standard deviations homogenizes data and improves testing performance. Because of this improvement, standardized return based tests by Patel (1976) and Boehmer, Musumeci & Poulsen (1991) (BMP) have gained popularity over conventional non-standardized tests in testing event effects on mean security price performance. Har- rington & Shrider (2007) found that in a short-horizon testing of abnormal returns (i.e., systematic deviation from expected behavior), one should always use methods that are robust to cross-sectional variation in the true abnormal returns; for a discussion of true abnormal returns, see (Harrington & Shrider, 2007). They found that BMP is a good candidate for a robust, parametric test in conventional event studies.* However, a major problem in statistical tests of returns is that the returns are not normally dis- tributed, see (Fama, 1976). Therefore, not surprisingly, non-parametric rank tests introduced by Corrado (1989; 2011), Corrado & Zivney (1992), Campbell & Wasley (1993), and Kolari & Pyn- nonen (2011), among others, dominate parametric tests both in terms of better size and power, see e.g., (Corrado, 1989; Corrado & Zivney, 1992; Campbell & Wasley, 1993; Kolari & Pynnonen, 2010; 2011; Luoma, 2011). Furthermore, the rank tests of Corrado & Zivney (1992) and Kolari & Pynnonen (2011) that utilize event period re-standardized returns have proven to be robust to event- induced volatility (Kolari & Pynnonen, 2010; 2011), cross-correlation due to event day clusterings (Kolari & Pynnonen, 2010), and autocorrelation (Kolari & Pynnonen, 2011). These are consistent with the view stated in the epilogue of Lehmann (2006: p. v): "Rank tests apply often to relatively simple solutions, such as one-, two-, and s-sample problems, and testing for independence and ran- domness, but for these situations they are often the method of choice." Moreover, the results of rank tests are invariant to monotone transformations of the underlying returns, that is, whether the returns are defined as simple, continuously compounded (log-returns), or gross-returns. The existing rank based tests, however, are not robust to cross-sectional correlation if the event days are partially over- *We define conventional event studies as those focusing only on mean stock price effects. Other types of event studies include (for example) the examination of return variance effects (Beaver, 1968; Patel, 1976), trading volume (Beaver, 1968; Campbell & Wasley, 1996), accounting performance (Barber & Lyon, 1997), and earnings management procedures (Dechow, Sloan & Sweeney, 1995; Kothari, Leone & Wasley, 2005). 154 Acta Wasaensia lapping. That is, when events in calendar time are scattered within an event window more or less randomly rather than clustered on the same calendar day, see (Kolari & Pynnonen, 2010). The cur- rent paper aims to fill this gap in the non-parametric event study testing. Kolari, Pape & Pynnonen (2018) have generalized existing parametric cross-sectional correlation robust testing towards this direction. The rest of the paper is organized as follows. Section 2 reviews some related key literature. Section 3 defines the main concepts and derives some distributional properties of rank statistics. Section 4 introduces the new transformed rank test. Section 5 reports simulation results and Section 6 contains conclusions. 2 Review of related literature Patell and BMP parametric tests apply straightforwardly for testing cumulative abnormal returns (CARs) over multiple day windows. With the correction suggested by Kolari & Pynnonen (2010) these tests are useful also in the case of completely clustered event days, and with the correction suggested by Kolari, Pape & Pynnonen (2018) when the event days are partially clustered. By construction the Corrado (1989) rank test applies for testing single day event returns. Testing for CARs with the same logic implies the need for defining multiple-day returns that match the number of days in the CARs, see Corrado (1989: p. 395) and Campbell & Wasley (1993: footnote 4). In practice this can be carried out by dividing the estimation period and event period into intervals matching the number of days in the CAR. Unfortunately, this procedure is not useful for a number of reasons. Foremost, it does not necessarily lead to a unique testing procedure. Also, the abnormal return model should be re-estimated for each multiple-day CAR definition. Furthermore, for a fixed estimation period, as the number of days accumulated in a CAR increases, the number of multiple- day estimation period observations reduces quickly to an impractically low number and thus, would weaken the abnormal return model estimation, cf. (Kolari & Pynnonen, 2010). Kolari & Pynnonen (2011) solve these issues in their generalized rank test approach. On the other hand, for example, Campbell & Wasley (1993) suggest to use the Corrado (1989) rank test for testing cumulative abnormal returns by simply accumulating the respective ranks to constitute cumulative ranks. This is also the practice adopted in the Eventusr software (Cowan Research L.C., www.eventstudy.com) and is probably, for the time being, the most popular practice in multiple day applications of rank tests. In spite of these attractive properties, the cumulative ranks test does not account for the cross- sectional correlation due to partially overlapping event windows. The correlation biases the standard errors downwards, leading to over-rejection of the null hypothesis of no event effect. This paper proposes an adjustment for the standard errors that corrects the bias. 3 Distributional properties of ranks We begin by fixing some notations and an underlying assumption to facilitate our theoretical discus- sion. Acta Wasaensia 155 Assumption 1. Stock returns rit for firm i are weak white noise continuous random variables and are cross-sectionally independent over non-overlapping calendar days t, or, E [rit] = i for all t; var [rit] =  2 i for all t; cov [rit; riu] = 0 for all t 6= u; rit and rju are independent whenever i 6= j and t 6= u: Note that it is a stylized fact that the variances of the returns are time varying and that there is mild autocorrelation. The time varying volatility problem can be partially captured in terms of generalized autoregressive conditional heteroskedasticity (GARCH) modeling. It is notable that under typical assumptions, GARCH-processes satisfy the weak stationary properties of Assumption 1. Let ARit = ritE [rit] denote the abnormal return of security i on day t, and let day t = 0 indicate the event day. Days from t = T0 + 1 to t = T1 represent the estimation period relative to the event day, and days from t = T1 + 1 to t = T2 represent the event window. The cumulative abnormal return (CAR) from 1 to 2 with T1 < 1  2  T2, is defined as CARi(1; 2) = 2X t=1 ARit: (3.1) The time period from 1 to 2 is called in the following a CAR window or CAR period. Standardized abnormal returns are defined as SARit = ARit S(ARi) ; where S(ARi) = vuut 1 T1 T0 1 T1X t=T0+1 AR2it: Moreover, for the purpose of accounting for the possible event induced volatility, the re-standardized abnormal returns are defined in the manner of Boehmer, Musumeci & Poulsen (1991), see also (Corrado & Zivney, 1992), as SAR0it = ( SARit=SSARt ; T1 < t  T2; SARit; otherwise; where SSARt = vuut 1 n 1 nX i=1 (SARit SARt)2 is the time t cross-sectional standard deviation of the SARits. In the preceding formula n is the number of stocks in the portfolio and SARt = 1n Pn i=1 SARit. Furthermore, let Kit denote the rank number of abnormal returns, where Kit 2 f1; : : : ; Tg, t = T0 + 1; : : : ; T2, T = T2 T0, and i = 1; : : : ; n. In particular, if the available observations on the estimation period vary from one series to another, then it is more convenient to deal with standardized ranks with zero mean and unit variance. For the 156 Acta Wasaensia purpose we use the known results of rank statistics, e.g., Lehmann (2006: Appendix, Section 1), E [Kit] = (T + 1)=2; var [Kit] = (T 2 1)=12; cov [Kis;Kit] = (T + 1)=12; s 6= t; and define, cf. Hagnäs & Pynnonen (2014), Definition 3.1. Standardized ranks are defined as Uit = Kit 12 (T + 1)p (T 2 1)=12 : Thus, E [Uit] = 0, var [Uit] = 1, and cov [Uis; Uit] = 1=(T 1). Next we define cumulative standardized ranks for individual stocks. Definition 3.2. The cumulative standardized ranks of a stock i over the event days window from 1 to 2 are defined as Ui(1; 2) = 2X t=1 Uit; where T1 < 1  2  T2: (3.2) Then immediately i(1; 2) = E [Ui(1; 2)] = 0, and the variance equation var [Ui(1; 2)] = 2X t=t1 var [Uit] + X s6=t cov [Uis; Uit] implies that 2i (1; 2) = var [Ui(1; 2)] = (T ) T 1 ; (3.3) where i = 1; : : : ; n, T1 < 1  2  T2, and  = 2 1 + 1. Rather than investigating individual (cumulative) returns, the practice in event studies is to aggregate individual returns into equally weighted portfolios. Definition 3.3. The average cumulative standardized ranks are defined as the equally weighted portfolio of individual cumulative standardized ranks defined in (3:2), i.e., U(1; 2) = 1 n nX i=1 Ui(1; 2) = 2X t=1 Ut; (3.4) where T1 < 1  2  T2, and Ut = 1 n nX i=1 Uit is the time t average of standardized ranks. The expected value of the average cumulative standardized ranks is the same as that of the cumulative ranks of individual securities, or E  U(1; 2)  = 1 n nX i=1 E [Ui(1; 2)] = 0: Acta Wasaensia 157 If the event days are not clustered, then the cross-sectional correlations of the return series are zero, or at least negligible. Under the assumption of independence, the variance of U(1; 2) is 2 = var  U(1; 2)  = (T ) (T 1)n and by the central limit theorem asymptotically as n!1, Z =  (T 1)n (T )  1 2 U(1; 2)  N(0; 1): (3.5) The situation is more complicated if the event days are partially overlapping in calendar time, which implies cross-correlation. Recalling that the variances of Ui(1; 2) given in equation (3:3) are constants (independent of i), we can write the cross-covariance of Ui(1; 2) and Uj(1; 2) as cov [Ui(1; 2); Uj(1; 2)] = (T ) T 1 ij(1; 2); (3.6) where ij(1; 2) is the cross-sectional correlation of Ui(1; 2) and Uj(1; 2), i; j = 1; : : : ; n. Utilizing this and the variance-of-the-sum formula, the variance of U(1; 2) in (3.4) becomes var  U(1; 2)  = 1 n2 nX i=1 var [Ui(1; 2)] + 1 n2 nX i=1 nX j 6=i cov [Ui(1; 2); Uj(1; 2)] = (T ) (T 1)n (1 + (n 1)n(1; 2)) ; (3.7) where n(1; 2) = 1 n(n 1) nX i=1 nX j=1 j 6=i ij(1; 2) is the average cross-sectional correlation of cumulated ranks. Cross-sectional dependence affects the asymptotic distribution of the statistic in (3.5). However, as discussed in Lehmann (1999: Section 2.8), it is frequently true that the asymptotic normality holds, provided that the average correlation, n(1; 2), tends to zero rapidly enough such that 1 n nX i6=j ij(1; 2) = (n 1)n(1; 2)! as n!1; where is some finite constant. Under this condition the limiting distribution of the Z-statistic in (3.5) becomes N(0; 1 + ). Otherwise, from a practical point of view, the crucial result of (3.7) is that the only unknown pa- rameter to be estimated is the average cross-sectional correlation n(1; 2). Hagnäs & Pynnonen (2014) discuss approaches to implicitly account for this average correlation in cumulated ranks tests when all events share the same calendar day, i.e., the case of complete overlapping (clustering) event periods. These implicit approaches, however, do not work when the event periods are partially over- lapping. Therefore, by utilizing the procedure developed in Kolari, Pape & Pynnonen (2018), this paper proposes a method to estimate explicitly the cross-sectional correlation n(1; 2), and thereby solves the cross-sectional correlation problem also in cases with partially overlapping event periods. 158 Acta Wasaensia 4 Correlation robust test for cumulated ranks Following Kolari, Pape & Pynnonen (2018), let ij , 0  ij   , denote the number of calendar days that stocks i and j have overlapping calendar days within the event windows. By the independence in Assumption 1 the correlation cor [Uiu; Ujv] of the standardized ranks Uiu and Ujv is zero whenever the underlying calendar days of the relative event days, u and v, differ, and can be non-zero when the calendar days are the same. Denoting these non-zero correlations, which are also covariances, by ij , we obtain cov [Ui(1; 2); Uj(1; 2)] = 2X u=1 2X v=1 cor [Uiu; Uju] = ijij : Combining this with (3.6) we obtain ij(1; 2) =  T 1 T   ij  ij : We can assume that the overlapping window lengths ij and the cross-sectional correlations ij are not dependent on each other so that P i6=j ijij = n(n 1) , where  is the average number of overlapping calendar days and  is the average cross-sectional correlation of Ui and Uj .† As a result, we can rewrite (3.7) as var  U(1; 2)  = (T ) (T 1)n (1 + (n 1)); (4.1) where  = (T 1)=((T )) adjusts the average correlation by the fraction of overlapping calendar days within the event window. It is notable that even though the average cross-sectional correlation  in (4.1) is based on n(n1)=2 correlations, it can be computed without estimating individual correlations by utilizing the method introduced by Edgerton & Toops (1928). Instead of the n(n 1)=2 individual correlations, it turns out that we need to compute only n+1 variances, which is a computational problem of order n viz-a- viz of order n2 when averaging the correlations. To illustrate this idea, consider n random variables xj , j = 1; : : : ; n, and define the standardized variables zj = xj=j . Next let z = P j zj=n denote the average of the variables. Then the variance of z is 2z = var [z] = 1 + (n 1) n ; because var [zj ] = 1 and cov [zj ; zk] = cor [zj ; zk] = jk. From this result we obtain  = n2z 1 n 1 : Thus, for large n,   2z . In order to estimate the average cross-sectional correlation, all we need are estimates of n standard deviations of the x-variables and the variance of z. Since in our case the calendar days of different stocks are only partially overlapping, we estimate the variance of the average utilizing the clustering robust estimation technique, see, e.g., (Cameron, †The equation follows by setting P (x x)(y y) =P xy nxy to zero, so thatP xy = nxy. Acta Wasaensia 159 Gelbach & Miller, 2011), suggested in Kolari, Pape & Pynnonen (2018). Following Kolari, Pape & Pynnonen (2018), denote the calendar days of the returns in the combined estimation and event window as t = 1; : : : ; L. In other words, L is the number of clusters which equals the number of separate calendar days on which returns are available in the combined estima- tion and event windows. Let nt denote the number of stocks having returns on calendar day t and define Ut = ntX k=1 Ukt: Then U2t = ntX k=1 U2kt + ntX i6=j UitUjt; which can be rearranged as ntX i6=j UitUjt = U 2 t ntX k=1 U2kt: Summing these up over the calendar days in the combined estimation and event window, we have LX t=1 ntX i6=j UitUjt = LX t=1 U2t LX t=1 ntX k=1 U2kt: (4.2) Taking the average, we get an estimator for the average correlation ^ = 1 M LX t=1 ntX i6=j UitUjt; where M = LX t=1 nt(nt 1) is the number of cross-product terms. It is notable that days for which there is available only one return drop out automatically: if nt = 1 for all t, then ^ = 0. The potentially tedious computation over all cross-products can be materially simplified by utilizing the right-hand side of (4.2) and observing that by rearranging the terms of the rightmost sum it becomes equal to nT , i.e., the number of stocks n multiplied by the combined estimation and event period T . The reason for this is that the sample variances of standardized ranks are all equal to one, see the discussion below Definition 3.1. Therefore, the only component we need is the first sum of squares on the right-hand side. As a result, we can estimate the average correlation efficiently by ^ = N M (s2U 1); where N = PL t=1 nt is the total number of returns, which equals nT if the combined estimation and event windows are of the same length T for all stocks, and s2U is the clustering robust estimator of the variance of standardized ranks in the presence of intra-cluster correlation, i.e., s2U = 1 N LX t=1 U2t : 160 Acta Wasaensia Given the estimator of the average cross-sectional correlation, ^, we can define an appropriate cross- sectional correlation robust test for the null hypothesis of zero cumulative abnormal returns H0 : (1; 2) = E [CAR(1; 2)] = 0 in terms of the cumulated ranks using the z-ratio z = U(1; 2)  p 1 + (n 1)^ ; (4.3) where  is the square root of the variance 2 = (T ) (T 1)n of U(1; 2) for completely non-overlapping event windows in calendar time (i.e., when  = 0 in (4.1)), and  = 2 1 + 1 is the length of the window of cumulated abnormal returns. 5 Simulation results We generate artificial returns utilizing the Fama & French (2015) five-factor model (FF5), (rit rf )t = i + i;mkt(rm rf )t + i;smbSMBt + i;hmlHMLt + i,rmwRMWt + i,cmwCMWt + it; (5.1) where rm rf is the market excess return over the risk-free rate rf , SMB, HML, RMW, and CMW are common market factors proposed by Fama and French. We utilize daily data from January 2, 1990 through October 30, 2020 (7,770 daily returns) to generate 20,000 initial daily return series for this sample period. The regression coefficients for each stock are generated from multivariate normal distribution with mean vector (0; 1; 0:5; 0:5; 0:5; 0:5) and covariance matrix 2i (X 0X)1, in which 2i is the variance of the error term , with i, the standard deviation, generated from the uniform distribution U(1; 3) that corresponds to a range of annual volatilities varying roughly from 10 percent to 48 percent, and X 0X is the cross-product matrix of the Fama-French 5-factor regression model.‡ The 7,770 error terms it for stock i are generated independently from the normal distribution N(0; 2i ). In the simulations we define the abnormal returns with respect to the market model, that is ARit = (ri rf )t ( ^i + ^i(rm rf )t); where ^i and ^i are ordinary least squares (OLS) estimates. Therefore missing common factors introduce cross-sectional correlation between the abnormal returns. The event period is 10 trading days around the event day t = 0 and the estimation period consist of 250 days prior the event periods, i.e., relative days 260; : : : ;11. ‡Factor returns have been downloaded from French’s data library: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. Acta Wasaensia 161 In the experiments we focus on the effect of cross-sectional correlation on the size of the test. Other issues, like event induced volatility, are well reported for example by Kolari & Pynnonen (2010; 2011). Utilizing the base design initiated by Brown & Warner (1985), we generate 1,000 samples of 50 randomly selected stocks (the returns of which are generated by the FF5 model in (5.1)) with four overlapping event days scenarios. In the first case of non-overlapping event days, the returns are cross-sectionally independent. In the second case of completely overlapping events, all firms share the same event day (calendar time), and in the third and fourth scenarios the event days are randomly scattered across 5 and 10 adjacent calendar days, i.e., one and two weeks of trading days, respectively. We report two-tailed rejection rates for the null hypothesis of no event-effect across different event windows of 0, 1, 2, 5, and 10 around the event day, i.e., window lengths  = 1; 3; 5, 11, and 21 days. In addition to the statistic z in (4.3), we also report results for the more traditional rank based test proposed by Campbell & Wasley (1993: p. 85) zcw = P2 t=1 ktp sk ; (5.2) where, with E [Kit] = (T + 1)=2, kt = 1 n X i=1 (Kit E [Kit]) and s2k = 1 T T2X t=T0+1 k2t : Moreover, we also report results for a traditional parametric (cross-sectional correlation non-robust) t-statistics popular in event studies, see, e.g., (Campbell, Lo & MacKinlay, 1997: Chapter 4), t = CAR(1; 2) s.e(CAR) ; (5.3) where CAR(1; 2) is the sample average of CARi(1; 2) defined in (3.1) and s.e(CAR) is the related standard error. Under independence the null distribution of t is asymptotically standard normal. Table 1 summarizes the test statistics and their major features. Table 1. Test statistics and their key features. Robustness due to event correlation caused by Statistic Type volatility complete ovrl partial ovrl z = U(1;2)  p 1+(n1)^ , eq. (4.3) non-parametric yes yes yes zcw = P2 t=1 kt sk , eq. (5.2) non-parametric no yes no t = CAR(1;2) s.e(CAR) , eq. (5.3) parametric yes no no Table 2 reports the simulation results of the two-tailed rejection rates of the null hypothesis of no abnormal return at the 5% nominal rejection rate. The results are clear-cut. Panel A of the table 162 Acta Wasaensia reports the non-overlapping case with zero cross-sectional correlation. All statistics reject close to the nominal rate as would be expected. Panel B reports results of complete overlapping. That is, all events share the same calendar day, and, hence, returns are prone to cross-sectional correlation. The new z and the more traditional cumulative ranks statistic zcw, that both account for cross-sectional correlation, reject reasonably close to the nominal rate up to event windows 5 and exhibit some over-rejection on the longest event window 10, i.e., 21 days. As expected, the parametric, non cross-correlation robust statistic t incrementally over-rejects as the event windows grows longer. Panel C reports partial overlapping with events clustered randomly within 5 trading days (about a week). For the event day testing also the a priori non-robust statistics in this regard are doing fine by rejecting at the nominal rate. However, they start to incrementally over-reject as the event window gets longer. The a priori partial overlapping robust statistic z rejects close to the nominal rate up to the event windows of 5 days and over-rejects to some extend on the longest event windows of 11 and 21 days; albeit far less than the, in this regard, non-robust statistics zcw and t . The results are pretty much similar with the decreased overlapping in Panel D. Thus, we conclude that accounting for cross-sectional correlation is crucial to avoid biased inferences in statistical testing, not only in the case of complete overlapping of event windows but also in the case of partially overlapping cases. For the latter cases this paper has introduced a new test statistics to account for the dependence. Table 2. Rejection rates of the null hypothesis of no event effect at the nominal 5% level when the events are non-overlapping, partially overlapping, and completely overlapping. CAR window length 1 3 5 11 21 Event day (1;+1) (2;+2) (5;+5) (10;+10) Panel A: Non-clustered events z (SCAR) 0:048 0:054 0:049 0:052 0:063 zcw(SCAR) 0:052 0:050 0:051 0:052 0:063 t (CAR) 0:045 0:035 0:049 0:052 0:048 Panel B: Events clustered on the same trading day z (SCAR) 0:059 0:052 0:060 0:064 0:075 zcw(SCAR) 0:059 0:052 0:061 0:064 0:075 t (CAR) 0:087 0:091 0:096 0:085 0:110 Panel C: Events clustered on 5 adjacent trading days z (SCAR) 0:056 0:055 0:058 0:087 0:080 zcw(SCAR) 0:050 0:075 0:093 0:127 0:129 t (CAR) 0:045 0:063 0:077 0:112 0:102 Panel D: Events clustered on 10 adjacent trading days z (SCAR) 0:063 0:051 0:062 0:058 0:080 zcw(SCAR) 0:059 0:062 0:091 0:116 0:133 t (CAR) 0:065 0:057 0:056 0:089 0:105 Acta Wasaensia 163 6 Conclusion This paper proposes a new non-parametric rank based test statistic for testing cumulative abnormal returns in short run event studies. 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Department of Mathematics and Statistics, University of Vaasa, 65101 Vaasa, Finland E-mail address: sjp@uwasa.fi Acta Wasaensia 165 ON THE KREI˘N-VON NEUMANN AND FRIEDRICHS EXTENSION OF POSITIVE OPERATORS Zoltán Sebestyén and Zsigmond Tarcsay Dedicated to Seppo Hassi on the occasion of his 60th birthday 1 Introduction In his profound paper (von Neumann, 1931), J. von Neumann introduced the concept of the adjoint of a densely defined possibly unbounded operator J : K ! H between two Hilbert spaces as the operator J : H ! K, having the domain dom J = fg 2 H : 9k0 2 K such that (Jk j g) = (k j k0) 8k 2 dom Jg; by setting Jg := k0; g 2 dom J: Although the adjoint operator behaves nicer than the original one (because it is always closed), it is not necessarily densely defined. An essential question arises therefore: when is the domain dom J a dense subspace ofH? Von Neumann himself gave an elegant answer to that question. Namely, he proved that J is densely defined if and only if J is a closable operator. Moreover, in that case the second adjoint J of J exists and it is equal to the closure J of J : J = J: At the same time, JJ and JJ are positive self-adjoint operators in the Hilbert spaces H and K, respectively. Note also that we have dom (JJ)1=2 = dom J and dom (JJ)1=2 = dom J on the domains, and ran (JJ)1=2 = ran J and ran (JJ)1=2 = ran J on the ranges. Here, for a given positive self-adjoint operator A, A1=2 denotes the unique positive self-adjoint square root of A; see, e.g., (Sebestyén & Tarcsay, 2017). However, if J is not closed, then JJ and JJ are not self-adjoint operators in general. In fact, it is not even clear whether those operators are densely defined, and therefore it is also a non- trivial question whether they have any positive self-adjoint extensions at all. From classical works by Friedrichs, Kreı˘n, and von Neumann, we know that a densely defined positive and symmetric operator may be extended to a positive self-adjoint operator, see, e.g., (Friedrichs, 1934; Kreı˘n, 1947; von Neumann, 1931). In that case, there exist two distinguished self-adjoint extensions AN and AF of any positive symmetric operator A, such that AN  AF ; 166 Acta Wasaensia and every positive self-adjoint extension eA of A is between AN and AF : AN  eA  AF . The smallest extension AN of A is called the Kreı˘n-von Neumann extension, while the largest extension AF of A is called the Friedrichs extension. The problem of the existence of positive self-adjoint extensions has its relevance even in the non- densely defined case. Although the Friedrichs extension exists only for a densely defined operator, the smallest extension always exists if there exists any extension, see, e.g., (Sebestyén & Stochel, 1991) and also (Sebestyén & Stochel, 2007; Hassi, 2004). In the present paper we revise the main result Theorem 1 of Sebestyén & Stochel (1991) and give some new characterizations for a not necessarily densely defined positive symmetric operator to admit positive self-adjoint extensions. More specifically, in Section 2 we collect some new properties for an operator to be closable. Based on this new characterization of closability, we establish in Section 3 the correct version of the "duality theorem" stated in Jorgensen, Pearse & Tian (2018: Theorem 5). In Section 4 we give a short proof of the fact that the "modulus square" operator T T of any densely defined operator T always has a positive self-adjoint extension, cf. (Sebestyén & Tarcsay, 2012: Theorem 2.1). At the same time, we shall see that this is not the case with TT ; that operator might be even non-closable. However, we are going to establish necessary and sufficient conditions for the extendibility of TT . In particular, our construction of the Kreı˘n-von Neumann extension in Section 4 will be used to exhibit a counterexample to (Jorgensen, Pearse & Tian, 2018: Theorem 5). Finally, in Section 5 we treat the problem of the existence of the Friedrichs extension of a densely defined positive symmetric operator. In particular, we discuss there the case when the Friedrichs extension of the operator T T is identical with T T . 2 Closable operators Let J be a densely defined operator between the real or complex Hilbert spaces K andH. Note that J is closable if for each sequence (gn)n2N  dom J , such that gn ! 0 and Jgn ! h, it follows that h = 0. On the other hand, a profound theorem by von Neumann tells us that J is closable if and only if J is densely defined, that is, (dom J)? = f0g: In the following theorem, we give an extension of von Neumann’s result and collect some new characteristic properties for an operator J to be closable. For further characterizations of closability and closedness, see, e.g., (Popovici & Sebestyén, 2014; Sebestyén & Tarcsay, 2016; 2019; 2020). Theorem 2.1. Let J be a densely defined operator between the real or complex Hilbert spaces K andH. Then the following properties are equivalent: (i) J is closable; (ii) (dom J)? = f0g; (iii) (dom J)? \ (ran J)?? = f0g; (iv) (dom J)?  ran (I + JJ): Acta Wasaensia 167 Proof. (i)) (ii) Consider a vector h 2 (dom J)?, then (0; h) 2 fJk; kg : k 2 dom J = G(J): Since J is closable, this implies h = 0. (ii)) (iii) This implication is trivial. (iii) ) (i) Consider a sequence (gn)n2N  dom J such that gn ! 0, and Jgn ! h. Then h 2 ran J = (ran J)??. On the other hand, for every f 2 dom J (f jh) = lim n!1(f j Jgn) = limn!1(J f j gn) = 0; which means that h 2 (dom J)?. Consequently, h = 0 by (ii) and therefore J is closable. (ii)) (iv) This implication is clear. (iv)) (ii) We are going to show that dom J is dense in H. To this aim, take g 2 (dom J)?. By (iv), there exists h 2 dom JJ such that g = h+ JJh. In particular, h 2 dom J and one has 0 = (g jh) = (h+ JJh jh) = (h jh) + (JJh jh) = khk2 + kJhk2; so that h = 0. This implies that g = 0 and therefore (iv) implies (ii). 3 Duality theorems Let H1 and H2 be Hilbert spaces with a common vector subspace D. In Jorgensen, Pearse & Tian (2018: Theorem 5) a necessary and sufficient condition is stated for the existence of a positive and self-adjoint operator  onH1 with the duality property (' j )1 = (' j )2; '; 2 D; cf. also (Jorgensen & Pearse, 2016: Theorem 4.1). Unfortunately, there is a simple but serious error in their proof and the statement itself is not true in that form either (a counterexample will be exhibited in Example 4.2 below). In Theorem 3.3 we are going to establish the correct form of that statement. Its proof depends on the following lemma. Lemma 3.1. LetH andK be Hilbert spaces and let J : K ! H be a densely defined linear operator between them. Then the following three statement are equivalent: (i) ran J  dom J; (ii) J is closable and dom J  dom JJ; (iii) there exists a positive self-adjoint operator A in K such that dom J  dom A and (Ag j k) = (Jg j Jk); g; k 2 dom J: (3.1) 168 Acta Wasaensia Proof. (i)) (ii) Since ran J  dom J, it follows that (ran J)??  (dom J)?? = H (dom J)?; and, consequently, (dom J)? \ (ran J)?? = f0g: Applying Theorem 2.1 we see that J is closable. On the other hand, ran J  dom J implies that dom J = dom JJ  dom JJ. (ii)) (iii) If J is closable, then A := JJ is a positive self-adjoint operator inH, and by (ii) one has dom J  dom A. On the other hand, (Ag j k) = (JJg j k) = (Jg j Jk) = (Jg j Jk); for every g; k 2 dom J . (iii) ) (i) Suppose that A is a positive operator with dom J  dom A which satisfies (3.1). Let k 2 dom J be arbitrary, then for every g 2 dom J (Jg j Jk) = (Ag j k) = (g jAk); which implies Jk 2 dom J. Therefore, ran J  dom J: Remark 3.2. Let J be a closed operator. Then the inclusion dom J  dom JJ (3.2) is only possible if J is continuous and everywhere defined onH1, see, e.g., (Tarcsay, 2012: Lemma 2.1). This suggests that Lemma 3.1 is only relevant if J is a closable but not a closed operator. The erroneous observation in the proof of (Jorgensen, Pearse & Tian, 2018: Theorem 5) is that (3.2) holds true provided that both J and J are densely defined. This makes it necessary to provide the following correct version of (Jorgensen, Pearse & Tian, 2018: Theorem 5), which can also be considered as a noncommutative version of the Lebesgue-Radon-Nikodym decomposition theorem. Theorem 3.3. Let H1 and H2 be real or complex Hilbert spaces which contain a common linear manifold D as a vector space. Suppose that D is dense inH1 and set D := h 2 H2 : 9Ch  0 such that j(' jh)2j  Chk'k1 8' 2 D : Then the following two conditions are equivalent: (i) D  D inH2; (ii) there exists a positive self-adjoint operator  inH2 such that D  dom  inH1 and (' j )1 = (' j )2; '; 2 D: (3.3) Proof. Let J be the operator from D  H1 to H2 defined by the identification J' := ', ' 2 D. Then J is a densely defined operator such that its adjoint J has domain D: dom J = D. The desired equivalence follows now from Lemma 3.1. Acta Wasaensia 169 4 Von Neumann’s problem on positive self-adjoint extendibility Given a positive symmetric operator A in a real or complex Hilbert space K, the question arises whether there exists a positive self-adjoint extension of A. If the operator in question is densely defined, then we know from classical papers by Friedrichs, Kreı˘n, and von Neumann that the operator has a positive self-adjoint extension, see (Friedrichs, 1934; Kreı˘n, 1947; von Neumann, 1931); cf. also (Ando & Nishio, 1970; Arlinskiı˘ et al., 2001; Prokaj & Sebestyén, 1996a;b; Schmüdgen, 2012). However, uniqueness of the extension occurs only in the very special case when the operator in question is essentially self-adjoint. In all other cases, the set of positive self-adjoint extensions is an operator interval [AN ; AF ], whereAN is the smallest (the so-called Kreı˘n-von Neumann) extension, while AF is the largest (the so-called Friedrichs) extension of A: Recall that the partial ordering among the set of positive self-adjoint operators is given by A  B () (I +B)1  (I +A)1: Equivalently, by means of the square roots, one has A  B if and only if dom B1=2  dom A1=2 and kA1=2kk2  kB1=2kk2; 8k 2 dom B1=2: The problem of the existence of positive self-adjoint extensions has its relevance even in the non- densely defined case, and was treated in detail by Sebestyén & Stochel (1991), see also (Sebestyén & Stochel, 2007; Hassi, 2004). In the next result we revise (Sebestyén & Stochel, 1991: Theorem 1) on the existence of the Kreı˘n- von Neumann extension of a positive and symmetric operator A. In this case it is convenient to introduce the linear space D(A) by D(A) := fk 2 K : sup fj(Ag j k)j : g 2 dom A; (Ag j g)  1g < +1g: (4.1) Theorem 4.1. Let A be a positive and symmetric operator on a real or complex Hilbert space K. Then the following statements are equivalent: (i) D(A) as in (4.1) is dense in K; (ii) for every sequence (gn)n2N  dom A and k 2 K such that (Agn j gn)K ! 0 and Agn ! k; it follows that k = 0; (iii) there exist a Hilbert space E and a densely defined linear operator V : K ! E such that dom A  dom V , (V (dom A))? = f0g; and hV g; V hiE = (Ag jh)K; g 2 dom A; h 2 dom V ; (4.2) (iv) there exists a positive self-adjoint extension of A. If any, and hence all, assertions of (i)-(iv) are satisfied, then there exists the smallest positive exten- sion AN of A. 170 Acta Wasaensia Proof. (i)) (ii) Assume that (Ag j g) = 0 for some g 2 dom A. Then sup j(Ag; k)j < 1 for all k 2 D(A), which implies (Ag; k) = 0. Since D(A)  K is dense by (i), it follows that Ag = 0. This means that hAg;AhiE := (Ag jh)K; g; h 2 dom A; (4.3) defines an inner product on ran A. Denote by E the completion of that space and consider the natural inclusion operator JA : E  ran A! K, JA(Ag) := Ag 2 K; g 2 dom A: (4.4) Clearly, ran A forms a dense linear manifold in E by definition, so that JA is densely defined. On the other hand, one has dom JA = D(A); (4.5) thanks to the identities (JA(Ag) jh)K = (Ag jh)K; g 2 dom A; h 2 K; and hAg;AgiE = (Ag j g)K; g 2 dom A: From (4.5) and (i) we see that JA is densely defined and therefore JA is closable. From this it follows that A fulfills (ii). (ii) ) (iii) Note that the condition in (ii) implies that (4.3) defines an inner product. With the notation as in the proof of the implication (i) ) (ii), (ii) expresses that the canonical inclusion operator JA : E ! K is closable. Its adjoint JA : K ! E is therefore a densely defined operator such that hJAg;AhiE = (g j JA(Ah))K = (g jAh)K = hAg;AhiE ; g; h 2 dom A; whence we conclude that JAg = Ag 2 E ; g 2 dom A: (4.6) As a conseqence, JA provides a factorization for A in the sense of (iii): hJAg; JAhiE = hAg; JAhiE = (Ag jh)K; g 2 dom A; h 2 D(A): (4.7) Moreover, by (4.6) we see that JA(dom A) = fAg : g 2 dom Ag; where the right-hand side is dense in H by definition. Hence, V := JA satisfies all requirements of (iii). (iii)) (iv) Let V : K ! E be a densely defined closable operator satisfying the properties in (iii). By (4.2) we conclude that V g 2 dom V  for every g 2 dom A and that V V g = Ag; g 2 dom A: (4.8) This means that dom V  includes the dense set V (dom A), and therefore V is closable. Moreover, by (4.8) we see that A  V V  V V , i.e., the positive self-adjoint operator V V  extends A. Acta Wasaensia 171 (iv) ) (i): Let B be a positive self-adjoint extension of A. Then for every k 2 dom B1=2 and g 2 dom A with (Ag j g)  1, we obtain that j(Ag j k)j = j(Bg j k)j = j(B1=2g jB1=2k)j  kB1=2gkkB1=2kk = (Ag j g)1=2kB1=2kk  kB1=2kk; whence k 2 D(A). This implies that dom B1=2  D(A); (4.9) where the former subspace is dense in K. Hence, D(A) is dense in K, i.e., (i) holds. Finally, let any, and hence all, assertions of (i)-(iv) be satisfied. First note that the operator JA defined in (4.4) is closable by (i). Hence, from (4.6) and (4.7) it follows that AN := J  A J  A (4.10) is a positive self-adjoint extension of A. Furthermore we have D(A) = dom JA = dom (JA JA)1=2 (4.11) and the density of ran A inH implies for every k 2 D(A) that k(JA JA)1=2kk2K = kJAkk2E = sup jhAg; JAkiE j2 : g 2 dom A; hAg;AgiE  1 = sup j(JA(Ag) j k)Kj2 : g 2 dom A; (Ag j g)K  1 = sup j(Ag j k)Kj2 : g 2 dom A; (Ag j g)K  1 : Next we show thatAN as in (4.10) is the smallest self-adjoint extension ofA. Let thereforeB be any positive self-adjoint extension of A. Since the positive self-adjoint operator B has no proper self- adjoint extension, applying the above construction forB, we infer thatB = JB J  B . By the inclusion (4.9) we have dom B1=2  dom A1=2N , see (4.10) and (4.11), and from the above calculation we obtain that, for every k 2 dom B1=2, kA1=2N kk2 = k(JA JA)1=2kk2 = sup j(Ag j k)j2 : g 2 dom A; (Ag j g)  1  supj(Bg j k)j2 : g 2 dom B; (Bg j g)  1 = k(JB JB)1=2kk2K = kB1=2kk2K: Hence AN  B, as it is stated. As was mentioned in the previous section, the statement of (Jorgensen, Pearse & Tian, 2018: Theo- rem 5) is not correct, as with the notation used in Theorem 3.2, they assert that the existence of the positive self-adjoint operator  satisfying (3.3) is equivalent to D being dense inH2. Based on the preceding theorem and its proof, it will be shown by a counterexample that their assertion is not true in general. Example 4.2. Consider an unbounded positive self-adjoint operator A in a Hilbert space K and set D := ran A: 172 Acta Wasaensia Denote by E the "energy space" associated with A and by J the corresponding inclusion operator J : E  ran A! K as in the proof of Theorem 4.1. Then D is a common vector subspace of E and K such that D  E is dense. Furthermore, D :=fk 2 K : 9Ck  0 such that j(' j k)2Kj  Chk'k2E 8' 2 Dg =fk 2 K : 9Ck  0 such that j(Ah j k)2Hj  Ck(Ah jh)K 8h 2 dom Ag; from which we conclude that D = D(A) = dom J; so D  K is dense. Suppose that the conclusion of (Jorgensen, Pearse & Tian, 2018: Theorem 5) is true, then by that theorem the density of D in K implies that there exists a positive self-adjoint operator  : E ! E , D  dom , such that h';'iE = (' j')K; ' 2 D: From this we conclude that (J(Ah) jAk)K = (Ah jAk)K = hAh;(Ak)iE ; h 2 dom A; which in turn means that Ak 2 dom J and J(Ak) = (Ak). As a consequence we see that ran A  dom J, and since dom A  dom J holds true as well, we obtain that K = dom A+ ran A  dom J: So J is an everywhere defined bounded operator on K, and therefore so is A = JJ. This is in contradiction to the assumption that A is an unbounded operator. Thanks to a classical result of J. von Neumann (von Neumann, 1931) we know that T T and TT  are positive self-adjoint operators whenever T is densely defined and closed. In Sebestyén & Tarcsay (2014) we proved the converse of that statement: if both T T and TT  are self-adjoint then T is necessarily closed, see also (Gesztesy & Schmüdgen, 2019) and (Sandovici, 2018) for the case of linear relations. This means that if T is not closed (or not even closable), then either T T or TT  (or even both) fail to be self-adjoint. In fact, TT  might even be non-closable; however, surprisingly, T T behaves well. Namely, it was proved in Sebestyén & Tarcsay (2012: Theorem 2.1) that T T always has a positive self-adjoint extension. We provide a short proof of that result. Theorem 4.3. Let T : K ! H be a densely defined linear operator between the real or complex Hilbert spaces K andH. Then T T has a positive self-adjoint extension. Proof. Consider the positive symmetric operator A := T T . We are going to show that dom T  D(A): Indeed, for g 2 dom A and k 2 dom T , we have j(Ag j k)j2 = j(T Tg j k)j2 = j(Tg jTk)j2  (Tg jTg)(Tk jTk) = (T Tg j g)(Tk jTk) = (Ag j g)(Tk jTk): Hence D(A) is dense in K. Thus A has by Theorem 4.1 a positive self-adjoint extension. Acta Wasaensia 173 In the next result we deal with the positive extendibility of TT . Theorem 4.4. Let T : K ! H be a densely defined operator between the real or complex Hilbert spaces K andH. Then the following two statements are equivalent: (i) TT  has a positive self-adjoint extension; (ii) T jdomT \ ranT is a closable operator. Proof. The positive symmetric operator A := TT  has a positive self-adjoint extension if and only if it satisfies condition (ii) of Theorem 4.1. That is, according to that result, T T has a positive self-adjoint extension if and only if for every sequence (hn)  dom TT  and every vector f 2 H the conditions (TT hn jhn) = kT hnk2 ! 0 and TT hn ! f; imply that f = 0. Evidently, this is equivalent to the closability of the restriction of T to the set ran T  \ dom T . In the following example, we show that TT  may have a bounded positive self-adjoint extension in some cases even if T is not even closable. Example 4.5. Let K be a separable Hilbert space and consider two orthonormal bases in it fen;m : n;m 2 Ng and ffn : n 2 Ng: Let us define the operator T on the vectors en;m by setting Ten;m := mfn; n;m 2 N; and then extend it by linearity to dom T := span fen;m : n;m 2 Ng. It follows from this definition that dom T  = f0g. In order to see this, observe that for z 2 dom T  and n 2 N we have (z j fn) = 1 m (z jTen;m) = 1 m (T z j en;m); for any m 2 N. Letting m!1 gives that (z j fn) = 0 and, hence, z = 0. Consequently, T is non- closable (in fact, T is a maximal singular operator), but A = 0 is a (bounded) positive self-adjoint extension of TT . The previous example demonstrated that TT  can behave nicely even though T is singular. However, as the following example illustrates, there exists an operator T such that TT  is non-closable. Example 4.6. Consider a maximal singular operator T in a Hilbert spaceK, that is, an operator such that dom T  = f0g (take e.g. the operator T from Example 4.5). Consider the following operator J : K  dom T ! KK; Jg := fg; Tgg: Then it is easy to verify that dom J = K  dom T  = K  f0g, and Jfk; 0g = k. In particular, dom JJ = dom T  f0g, and JJfg; 0g = fg; Tgg; g 2 dom T: 174 Acta Wasaensia Furthermore, we claim that J is not closable. Indeed, take any nonzero k 2 K, then there exists a sequence (gn)n2N in dom T such that gn ! 0 and Tgn ! k. Then JJfgn; 0g = fgn; T gng ! f0; kg; which means that JJ may not be closable. Theorem 4.7. Let T : K ! H be a densely defined closable linear operator, such that dom T  ran T : (4.12) Then T T  agrees with the Kreı˘n-von Neumann extension of TT . Proof. Denote by E the completion of ran TT  under the inner product hTT h; TT fi := (TT h j f) = (T h jT f); h; f 2 dom TT : By the construction of the proof of Theorem 4.1, the Kreı˘n-von Neumann extension of TT  is of the form JJ, where J is the natural inclusion operator from E  ran TT  intoH: J(TT h) := TT h; h 2 dom TT : Note that by (4.12) we have the identity dom T = fT g : g 2 dom TT g. Consequently, dom (JJ)1=2 = dom J = D(TT ) = n h 2 H : sup n j(TT f jh)j : f 2 dom TT ; kT fk2  1 o < +1 o = dom T  = dom (T T )1=2: At the same time we have that k(JJ)1=2hk2 = kJhk2E = sup jhTT f; Jhi2j : f 2 dom TT ; kT fk2  1 = sup j(T f jT h)j2 : f 2 dom TT ; kT fk2  1 = kT hk2; for every h 2 dom T . We have therefore proved that T T   JJ, and since JJ is the smallest positive self-adjoint extension of TT , we obtain that T T  = JJ. 5 The Friedrichs extension A densely defined positive symmetric operator A on a real or complex Hilbert space K always has a positive self-adjoint extension. Indeed, the generalized Schwarz inequality j(Ag jh)j2  (Ag j g)(Ah jh); g; h 2 dom A implies that dom A  D(A) and, therefore, A admits a positive self-adjoint extension according to Theorem 4.1. In particular, by the same theorem, the Kreı˘n-von Neumann extension AN of A Acta Wasaensia 175 exists. In that case it is known that the so-called Friedrichs extension, that is, the largest positive self-adjoint extension, exists as well. Using the procedure described in Theorem 4.1, we prove the existence of the Friedrichs extension. Our method is very similar to that of Prokaj & Sebestyén (1996a), but somewhat simpler. Theorem 5.1. LetA be a densely defined positive symmetric operator in the real or complex Hilbert space K. Then there exists the largest positive self-adjoint extension AF of A. Proof. Let us recall the construction of the proof of Theorem 4.1 and consider the energy Hilbert space E and the inclusion operator JA : E  ran A! K defined by JA(Ag) := Ag; g 2 dom A: By (4.5) we have dom JA = D(A)  dom A, and therefore we may consider the restriction QA of JA to dom A, i.e., QA := J  AjdomA: By (4.6), QAg = Ag 2 E ; g 2 dom A: On the other hand, from QA  JA we get JA  QA and QA  JA, whence it follows that AF := Q  AQ  A is a positive self-adjoint extension of A. We claim that AF is the largest among the set of all positive self-adjoint extensions of A. Indeed, let B  A be any positive self-adjoint extension of A. Repeating the above process we apparently have B = QBQ  B . Then dom (QAQ  A ) 1=2 = dom QA = dom QA = fk 2 K : 9(kn)n2N  dom A; kn ! k; (A(kn km) j kn km)! 0g; and, accordingly, dom (QBQ  B ) 1=2 = fk 2 K : 9(kn)n2N  dom B; kn ! k; (B(kn km) j kn km)! 0g  fk 2 K : 9(kn)n2N  dom A; kn ! k; (A(kn km) j kn km)! 0g = dom (QAQ  A ) 1=2: Finally, for k 2 dom (QAQA )1=2  dom (QBQB )1=2, take (kn)n2N  dom A such that kn ! k and (A(kn km) j kn km)! 0; then QAkn ! QA k in E , and hence k(AF )1=2kk2 = (QAQA )1=2kk2 = kQA kk2E = lim n!1 kQAknk 2 E = lim n!1(Akn j kn): Moreover, since B  A, kB1=2kk2 = k(QBQB )1=2kk2 = lim n!1(Bkn j kn) = limn!1(Akn j kn): As a consequence we see that AF  B, as desired. Theorem 5.2. Let T : K ! H be a densely defined linear operator satisfying ran T  dom T : (5.1) 176 Acta Wasaensia Then T is closable and the Friedrichs extension of the positive symmetric operator T T is equal to T T : (T T )F = T T : (5.2) Proof. Condition (5.1) guarantees, according to Lemma 3.1, that T is closable. Hence, T  exists and T T  is a positive self-adjoint extension of T T , thanks to von Neumann, see (Schmüdgen, 2012: Proposition 3.18). Our duty is therefore to establish identity (5.2). To this end we need only to prove the domain inclusion dom (T T )1=2F  dom (T T )1=2; (5.3) because we know that dom (T T )1=2F  dom (T T )1=2 and that k(T T )1=2F kk2 = k(T T )1=2kk2; k 2 dom (T T )1=2F ; see the proof of Theorem 5.1. First we note that dom (T T )1=2 = dom T  = fk 2 K : 9(kn)n2N  dom T; kn ! k; Tkn Tkm ! 0g: Recalling the proof of Theorem 5.1, let us denote by E the "energy space" associated with T T , that is, the completion of ran T T endowed with the inner product hT Tk; T Tfi := (Tk jTf); k; f 2 dom T T: Consider the operator Q : K ! E given by dom Q = dom T T = dom T , Q(T Tk) := T Tk 2 E ; k 2 dom T; then we have (T T )F = QQ, again according to the proof of Theorem 5.1. Consequently, the domain dom (T T )1=2F can be described as follows: dom (T T )1=2F = dom (Q Q)1=2 = dom Q = fk 2 K : 9(kn)n2N  dom T; kn ! k; kT Tkn T Tkmk2E ! 0g = fk 2 K : 9(kn)n2N  dom T; kn ! k; kTkn Tkmk2K ! 0g = dom T  = dom (T T )1=2: This proves identity (5.3) and therefore (T T )F = T T , as is claimed. Acknowledgement: The authors are extremely grateful to Henk de Snoo and Rudi Wietsma for carefully reading the manuscript and for their invaluable suggestions. Acknowledgement: The corresponding author Zs. Tarcsay was supported by DAAD-TEMPUS Cooperation Project "Harmonic Analysis and Extremal Problems" (grant no. 308015), by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and by the ÚNKP–20-5-ELTE- 185 New National Excellence Program of the Ministry for Innovation and Technology. "Application Domain Specific Highly Reliable IT Solutions" project has been implemented with the support pro- vided from the National Research, Development and Innovation Fund of Hungary, financed under the Thematic Excellence Programme TKP2020-NKA-06 (National Challenges Subprogramme) funding scheme. Acta Wasaensia 177 References Ando, T. & Nishio, K. (1970). Positive selfadjoint extensions of positive symmetric operators. To- hoku Math. J. 22, 65–75. Arlinskiı˘, Y.M., Hassi, S., Sebestyén, Z. & de Snoo, H.S.V. (2001). On the class of extremal exten- sions of a nonnegative operator. Oper. Theory Adv. Appl. 127, 41–81. Friedrichs, K. (1934). 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Sebestyén, Z. & Tarcsay, Zs. (2014). A reversed von Neumann theorem. Acta Sci. Math. (Szeged) 80, 659–664. 178 Acta Wasaensia Sebestyén, Z. & Tarcsay, Zs. (2016). Adjoint of sums and products of operators in Hilbert spaces. Acta Sci. Math. (Szeged) 82, 175–191. Sebestyén, Z. & Tarcsay, Zs. (2017). On square root of positive selfadjoint operators. Period. Math. Hungar. 75, 268–272. Sebestyén, Z. & Tarcsay, Zs. (2019). On the adjoint of Hilbert space operators. Linear and Multilin- ear Algebra 67, 625–645. Sebestyén, Z. & Tarcsay, Zs. (2020). Range-kernel characterizations of operators which are adjoint of each other. Adv. Oper. Theory 5, 1026–1038. Tarcsay, Zs. (2012). Operator extensions with closed range. Acta Math. Hungar. 135, 325–341. Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, Pázmány Péter sétány 1/c, Budapest H-1117, Hungary, and Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, Budapest H-1053, Hungary E-mail address: tarcsay@cs.elte.hu Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, Pázmány Péter sétány 1/c, Budapest H-1117, Hungary E-mail address: sebesty@cs.elte.hu Acta Wasaensia 179 THE CHARACTERIZATION OF BROWNIAN MOTION AS AN ISOTROPIC I.I.D.-COMPONENT LÉVY PROCESS Tommi Sottinen This work is dedicated to Professor Seppo Hassi for his 60th birthday 1 Introduction The Brownian motion is arguably the most important stochastic process there is. It has a long history in particle physics dating back to at least the Roman poet and philosopher Lucretius and his scientific poem De rerum natura ca. 50 BC; for a recent English translation, see Slavitt (2008). Since that time the Brownian motion has proven to be central in such diverse fields as physics, economics, quantitative finance, and evolutionary biology, just to mention a few. The process is christened "Brownian motion" to honor the pioneering work of the Scottish botanist Robert Brown in his work in 1827 on pollen movement in water; although it is obvious that Brown was not the first one to observe the Brownian motion. The first mathematical study of the Brown- ian motion is apparently by the Danish astronomer Thorvald Nicolai Thiele in 1880, see Lauritzen (1981) for a discussion of Thiele’s work. The Brownian motion is also sometimes called the "Wiener process" in honor of Norbert Wiener for his pioneering contributions to the mathematical study of the process. To honor Wiener, we use the symbol W for the Brownian motion. From a mathematical and modeling point of view, the Brownian motion is extremely convenient. It belongs to the intersection of many mathematical models: it is Gaussian, it is Markovian, it is a Lévy process, it is a martingale, and it is a self-similar process. The Brownian motion has many characterizations. It is, for example, the scaling limit of ran- dom walks, the independent-increment stationary-increment Gaussian process, the 12 -self-similar stationary-increment Gaussian process, the Markov process with Laplacian as its generator, the con- tinuous Lévy process, or the continuous local martingale with the (raw) bracket [W ]t = t. Of all the characterizations of the Brownian motion, let us just give the shortest one here. Definition 1.1 (The Brownian motion as a Gaussian process). A d-dimensional centered stochastic process W = (W 1; : : : ;W d) with W0 = 0 is the Brownian motion if it is Gaussian with variance- covariance matrix given by E[W itW is ] = (t ^ s)ij , where t ^ s = min(t; s) and ij =  1; if i = j; 0; if i 6= j: Definition 1.1 is of course concise and opaque. We will have better definitions later. Nevertheless, let us show that Definition 1.1 is not vacuous, i.e., such a process does indeed exist. We do this by using the Karhunen-Loève type expansion by which the Brownian motion is constructed as the isonormal Gaussian process on the separable Hilbert space L2([0; T ]). 180 Acta Wasaensia Theorem 1.2 (Brownian motion series construction). Let jk, j = 1; : : : ; d, k 2 N, be independent and identically distributed standard Gaussian random variables. For every t 2 [0; T ] set W jt = 1X k=1 Z t 0 ek(s) ds  j k; j = 1; : : : ; d; (1.1) where (ek)k0 is your favorite orthonormal basis of L2([0; T ]). Then the series in (1.1) converges in L2 and the process W = (W 1; : : : ;W d) is the Brownian motion on the time interval [0; T ]. To see that (1.1) defines the Brownian motion (in the sense of Definition 1.1), one simply calculates the covariance. In this paper we provide a new characterization, or a definition if you like, for the d-dimensional Brownian motion for d  2 as the isotropic (i.e., rotationally invariant) Lévy process with inde- pendent and identically distributed (i.i.d.) components. Our proof is short and simple, but not elementary. Moreover, in proving the rise of Gaussianity we do not use, at least not directly, the central limit theorem. The rest of the paper is organized as follows. In Section 2 we give some basic results of Lévy pro- cesses for the convenience of those readers who are not familiar with Lévy processes, and put our result in context. In particular, we recall the Lévy–Khintchine representation theorem for Lévy pro- cesses, and show by using it that the Brownian motion can be characterized as being the continuous Lévy process. In Section 3 we state and prove our new characterization: Theorem 3.1. In Section 4 we discuss our new characterization and its implications. In particular, Remark 4.4 gives an open problem on generalizing our result to the Markovian setting, and its connection to a qualitative char- acterization of the Laplace operator. Finally, in Remarks 4.5 and 4.6 we discuss the implications of our result for the modeling point of view. 2 Brownian motion and Lévy processes The following is a common textbook definition of the Brownian motion, see, e.g., the recent book on Brownian motion by Mörters & Peres (2010). Definition 2.1 (The Brownian motion, a textbook definition). A d-dimensional centered stochastic process W = (W 1; : : : ;W d) with W0 = 0 is the Brownian motion if: (i) for all times 0  t1      tn the random vectors Wtn Wtn1 ;Wtn1 Wtn2 ; : : : ;Wt2 Wt1 are independent; i.e., the process W has independent increments; (ii) for every t  0 and h  0, the distribution of the increment Wt+h Wt does not depend on t; i.e., the process W has stationary increments; (iii) the process (Wt)t0 has almost surely continuous paths; (iv) for every t  0 and h  0 the incrementWt+hWt is multivariate normally distributed with mean zero and variance-covariance matrix h Id, where Id is the d d identity matrix. Acta Wasaensia 181 Remark 2.2 (Standard Brownian motion and relaxed definition). Property (iv) of Definition 2.1 states that, in addition to being Gaussian, the processW also has i.i.d.-componentsW i, i = 1; : : : ; d, and E[(W i1)2] = 1. Sometimes one only insists that the process W is Gaussian. Under the assump- tions (i)–(iii) of Definition 2.1 this would mean that Wt+h Wt is a centered Gaussian vector with variance-covariance matrix h, where  is the variance-covariance matrix of W1. With this more relaxed definition, one usually says that if  = Id, then W is the standard Brownian motion. The connection between the standard Brownian motion and the relaxed definition of the Brownian mo- tion is simple. Indeed, if W is the standard Brownian motion, and we decompose  as KKT, then KW is a Brownian motion in the relaxed sense. The following is a common textbook definition for Lévy processes, see, e.g, Bertoin (1996) or Sato (2013). Definition 2.3 (Lévy process, textbook definition). A stochastic process L = (L1; : : : ; Ld) with L0 = 0 is a Lévy process if (i) it has independent increments; (ii) it has stationary increments; (iii) it is stochastically continuous, i.e., for every t  0 and " > 0 lim h!0 P [jLt+h Ltj > "] = 0: Remark 2.4 (Lévy process, càdlàg version). Sometimes one adds the following property to the definition of Lévy processes (iv) the paths (Lt)t0 are right-continuous with left limits, i.e., they are càdlàg (continue à droite, limite à gauche). However, it can be shown that under the assumptions (i)–(iii) of Definition 2.3 a Lévy process admits a version with càdlàg paths. Thus, the continuity-type assumption (iv) is not really necessary. Finally, regarding the mild continuity assumption (iii) in Definition 2.3, it should be noted that due to assumption (ii), the stationarity of the increments, it is actually equivalent to assuming that for every " > 0 lim h!0 P [jLhj > "] = 0: So, the assumption (iii) is mild, indeed. Thus, Lévy processes are processes with stationary independent increments satisfying a mild con- tinuity assumption. The Brownian motion is a continuous Lévy process that is also Gaussian. Ac- tually, the Gaussianity of the Brownian motion follows from the Lévy property and the continuity. Indeed, property (iv) of Definition 2.1 can be replaced by the following much weaker property (iv) the process W = (W 1; : : : ;W d) has i.d.d.-components with E[(W i1)2] = 1. The fact that Gaussianity follows from continuity is usually not appreciated in the common textbook definitions, such as Definition 2.1 above. That fact follows from the following Lévy–Khintchine 182 Acta Wasaensia representation theorem for Lévy processes. For that we recall the typical notation hx; yi = dX j=1 xjyj for the Euclidean inner product on Rd, kxk = p hx; xi for the Euclidean norm in Rd, and 1A(x) =  1; if x 2 A; 0; if x 62 A; for the indicator of the set A. Finally let Bd be the closed unit ball in Rd. Theorem 2.5 (Levy–Khintchine representation theorem). A stochastic process L = (L1; : : : ; Ld) is a Lévy process if and only if its characteristic function is of the form E h eih;Lti i = et (); where the characteristic exponent is of the form () = ihm; i+ 1 2 h;i+ Z Rd h 1 eih;xi + ih; xi1Bd(x) i (dx): Here m 2 Rd is the drift parameter, the symmetric non-negative definite matrix  2 Rdd is the diffusion parameter, and , the so-called Lévy measure, is a measure on Rd satisfying (f0g) = 0 and Z Rd kxk2 ^ 1 (dx) <1: The triplet (m;; ) is called the Lévy triplet of the process L. Now, to see that the Gaussianity of the Brownian motion follows from the Lévy–Khintchine repre- sentation, just note that 1st for continuous Lévy processes one must have   0; 2nd then, for centered Lévy processes one must have m  0; 3rd and finally, for i.i.d.-component Lévy processes one must have  =  Id, and since one has E[(W i1)2] = 1, it follows that  = 1. Thus () = 12kk2, and the Gaussianity follows from this. Remark 2.6. If we did not assume independence (and identical distribution) of the components in the reasoning above, we would still obtain from the continuity that () = 1 2 h;i; which would still imply Gaussianity. Thus the (relaxed sense) Brownian motion is characterized as being the continuous Lévy process. Acta Wasaensia 183 3 A new characterization Theorem 3.1. Let d  2. A Lévy process W = (W 1; : : : ;W d) is (a multiple of) the standard Brow- nian motion if and only if it is centered and isotropic with independent and identically distributed components. Proof. The only if part is clear. For the if part, let (m;; ) be the Lévy triplet of W . Since W has independent components,  is concentrated on the coordinate axes. Since W is isotropic,  is also isotropic. Consequently,   0. Since W is centered, m  0. Finally, since W has i.i.d.- components,  =  Id. Thus (m;; ) = (0;  Id; 0), proving the claim. 4 Discussion Remark 4.1 (The importance of having d  2). In Theorem 3.1 it is important that d  2. Indeed, for d = 1 symmetric processes are isotropic and there are many discontinuous symmetric Lévy processes for d = 1. Indeed, one can construct such Lévy processes by using the Lévy triplet (0; 0; ), where  is any symmetric measure on R satisfying (f0g) = 0 and Z R  x2 ^ 1 (dx) <1: Remark 4.2 (Gaussianity without the central limit theorem). As stated in the introduction, we did not (directly) invoke the central limit theorem in proving the rise of Gaussianity in our new char- acterization of the Brownian motion in Theorem 3.1. Instead, we used the Lévy–Khintchine repre- sentation of Theorem 2.5. Now, the classical way of proving the Lévy–Khintchine representation theorem does involve the central limit theorem and the so-called infinitely divisible distributions that are closely related to central limit-type theorems. There are, however, ways of proving the Lévy–Khintchine representation without resorting to the central limit theorem. For example, Jacod & Shiryaev (2003: Chapter 4) contains a nice derivation of the Lévy–Khinchine formula that only uses stochastic analysis and "compensator calculus". Remark 4.3 (The history of the rise of Gaussianity). It seems that the rise of Gaussianity through independence of the components and the rotational invariance has a long history dating back at least to the works of Herschel and Maxwell in 1850’s, see Jaynes (2003: Section 7.2). The key ingredients in the Herschel–Maxwell argument are (looking at a fixed time point and assuming continuous distribution) that the components are i.i.d. and that the process is isotropic. Hereby the first property implies that the distribution takes the form p(x) dx = dY j=1 f(xj) dxj (4.1) in Cartesian coordinates, and the second property implies that the distribution takes the form p(x) dx = g(r)r drd (4.2) in polar (or hyper-spherical) coordinates. 184 Acta Wasaensia Equating (4.1) and (4.2) one is given the functional equation dY j=1 f(xj) = g(kxk): (4.3) The solution of the functional equation (4.3) is of the form f(x) = c1e c2kxk2 ; which is the Gaussian density. Einstein (1905) also used similar arguments in his investigation of the Brownian motion in con- nection to the existence of atoms and molecules. Of course, neither the Herschel–Maxwell nor the Einstein derivation can be used in our setting of Theorem 3.1, since the distribution of a Lévy process at any fixed time is not a priori continuous. Remark 4.4 (Open problem: from Lévy to Markov). Lévy processes in general, and Brownian motion in particular, are Markovian processes that admit generators. Indeed, recall that the generator, if it exists, of a Markov process X = (X1; : : : ; Xd) is the linear operator given as Af(x) = lim t!0 1 t [Ex [f(Xt)] f(x)] : Here Ex means that the process X is started from x, or, in the case of Lévy processes, we are considering the translated process X + x. The generator of a Lévy process with triplet (m;; ) is Af(x) = hm;rf(x)i+ 1 2 hr;rf(x)i (4.4) + Z Rd [f(x+ y) f(x) hy;rf(x)i1Bd(y)] (dy): Thus the generator of the Brownian motion is the Laplacian up to a factor 1=2. This provides also an alternative proof for Theorem 3.1. Indeed, one only has to show that (a multiple) of the Laplacian is the only linear operator of the form (4.4) with d  2 that satisfies (i) Ax = Ay for any rotation y = Rx (isotropy); (ii) Ax = Pd j=1 Axj (i.i.d.-components). It would be interesting to know in which class of operators the properties (i)–(ii) above characterize the (multiple of the) Laplacian. For example, could it be possible to extend Theorem 3.1 from Lévy processes to (a larger class of) Markov processes? Remark 4.5 (Fractional Laplacian). Recently there has been much interest in models involving the fractional Laplacian () =2, which is a non-local pseudo-differential operator given by the Cauchy principal value integral () =2f(x) = 2 ( =2 + 1=2) 1=2( =2) Z Rd f(x) f(y) kx yk1+ dy: From the probabilistic point of view the fractional Laplacian can be understood as the generator of an isotropic -stable Lévy process. With = 2 the fractional Laplacian is just the Laplacian Acta Wasaensia 185 (actually, with our probabilistic parametrization it is 12). Now, -stable processes X are natural models in the sense that they are Lévy processes satisfying the scaling property (cXc t)t0 d = (Xt)t0; where d= means equality in the sense of finite-dimensional distributions. Thus, assuming -stability of some given random Lévy d-variate time-series X , we are faced with two natural (and maybe contradicting) assumptions: ( 1) The time-series X is rotationally invariant. ( 2) The time-series X has i.i.d.-components. If = 2, then the time-series is generated by a Brownian motion, and the assumptions ( 1) and ( 2) are the same. If, however, 6= 2 (and then necessarily 2 (0; 2)), the assumptions ( 1) and ( 2) are mutually exclusive. Assumption ( 1) corresponds to the Lévy process with generator () =2x that is rotationally invariant, while assumption ( 2) corresponds to the Lévy process with generator A x = dX j=1 ( =2xj ) that acts coordinatewise. Remark 4.6 (Modeling implications). Remark 4.5 above illustrates the message of our new charac- terization, Theorem 3.1, for the modeling point of view. In the context of d-variate Lévy processes for d  2 the two natural assumptions • rotational invariance; • independent components, are mutually exclusive unless your model is the Brownian motion. References Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge: Cam- bridge University Press. Einstein, A. (1905). Über die von der molekularkinetischen Theorie der Wärme geforderte Bewe- gung von in ruhenden Flüßigkeiten suspendierten Teilchen. Annalen der Physik 322, 549–560. Jacod, J. & Shiryaev, A.N. (2003). Limit Theorems for Stochastic Processes. Second edition. Grundlehren der Mathematischen Wissenschaften, vol. 288. Berlin: Springer-Verlag. Jaynes, E.T. (2003). Probability Theory: The Logic of Science. Cambridge: Cambridge University Press. Lauritzen, S. (1981). Time series analysis in 1880: A discussion of contributions made by T.N. Thiele. Internat. Statist. Rev. 49, 319–331. 186 Acta Wasaensia Mörters, P. & Peres, Y. (2010). Brownian Motion. Cambridge Series in Statistical and Probabilis- tic Mathematics, vol. 30. Cambridge: Cambridge University Press. (With an appendix by Oded Schramm and Wendelin Werner.) Sato, K.-I. (2013). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Ad- vanced Mathematics, vol. 68. Cambridge: Cambridge University Press. (Translated from the 1990 Japanese original, revised edition of the 1999 English translation). Slavitt, D. (2008). De Rerum Natura (The Nature of Things): A Poetic Translation. Berkeley: Uni- versity of California Press. Department of Mathematics and Statistics, University of Vaasa, 65101 Vaasa, Finland E-mail address: tommi.sottinen@iki.fi Acta Wasaensia 187 THE ROLE OF MATHEMATICS AND STATISTICS IN THE UNIVERSITY OF VAASA; THE FIRST FIVE DECADES Ilkka Virtanen In dedication to Professor Seppo Hassi on the occasion of his 60th birthday The tragedy of the world is that those who are imaginative have but slight experience, and those who are experienced have feeble imaginations. Fools act on imagination without experience. Pedants act on experience without imagination. The task of the university is to weld together imagination and experience. Alfred North Whitehead A Cambridge mathematician 1 From a business school towards a university with a clearly defined academic profile 1.1 The beginning The University of Vaasa is one of the four “new universities” founded in Finland in 1966. Practical activities of the new units started two to five years after the foundation decision was made by the state authorities. Three of the new universities were founded in Eastern Finland whereas the Univer- sity of Vaasa came to be placed on the west coast of Finland. All these new universities had different academic profiles. Vaasa got a private business school which, however, from the beginning – the first students started their studies in 1968 – got the major part of its financing from the state. After ten years the business school became a state university. Thus the University of Vaasa started as a business school, but it was clear from the beginning that the final target was to develop it into a mul- tidisciplinary university. The University of Lappeenranta was a technical university, the University of Kuopio had disciplines in medicine, biosciences, and social sciences and, finally, the University of Joensuu was based on an earlier teacher training college having also natural sciences, humanities, and social sciences in its discipline palette. The Eastern Finland universities were state universities from the beginning. 188 Acta Wasaensia The post-war baby boom generation was coming to the university in the 1960s and 1970s, and competition for resources between universities was great. So it turned out that the development of the business school towards the multidisciplinary target would be a hard and long-lasting exercise. Step by step the goal was, however, reached, at least to a decent extent. Today the University of Vaasa is – on the Finnish scale – a small or medium-sized business-oriented and multidisciplinary science university. The strategic focus areas of the university are management and change, energy and sustainable development, as well as financial and economical decision- making. The university has about 5000 students and about 300 of them are international degree students. The total number of employees is about 510 of whom about 100 are international. The university has been organized into schools. There are four schools for research and teaching: the School of Management, the School of Accounting and Finance, the School of Marketing and Communication, and the School of Technology and Innovations. The unit of mathematical sciences belongs to the School of Technology and Innovations, although it is responsible for all the teaching of mathematics and mathematics-based quantitative methods in the whole university. The teachers and researchers of this unit are in close cooperation with employees of the other schools. 1.2 Strategy and values of the university today In its strategy for 2030 the university defines itself to be an internationally competitive, productive and specialized research university with a strong focus on impactful basic scientific research. The core competence of the university consists of high-level expertise in business, technology, manage- ment, and communications. The university is focused on responsible business. Its fundamental purpose is to cultivate new know- ledge and nurture civilization as a core value of our society. This is why the focus is on global challenges and opportunities. They provide the university with the core source of motivation for its education and research. The university uses its work as a means to advance positive and sustainable development for individuals, communities, and the world at large. Based on the strategy for 2030, the university has defined its vision, mission and values as follows. • Vision: The University of Vaasa is regarded internationally as a successful and impactful research university. • Mission: The university carries out impactful research and educates experts that address the needs of society today, and in the future. The university advances competitiveness, innovation and sustainable development in business, technology, and society. • Values: The values of the university are Courage, Community, and Responsibility. Acta Wasaensia 189 2 The role of mathematics and statistics in the university’s palette of sciences 2.1 No own degree programme in master level – a recognized role, however The role of mathematics and statistics in the University of Vaasa is not typical for universities. Ac- cording to the strategy applied in the university today and during the past decades, these disciplines have never been offered as the main subjects of bachelor and master level degrees. This has not meant, however, that the disciplines were considered as less important for studies and research than the disciplines with their own degree programmes. On the contrary, the methodological disciplines have for example always been represented by professor chairs. It is noteworthy that even the first chair established in the university – a business school in the beginning – and also having got its first office-holder was the combined chair in economic mathematics and statistics. The holder of the chair in economic mathematics and statistics was also the first rector of the business school. He started his work already one year earlier than the school was opened and received its first students. That is, he had a central role in planning the first year’s curriculum for the school. According to the experiential opinion of the author of this article, the result was that the quantitative sciences got a stronger role in the curriculum for the business students than was the case in the other Finnish business schools at that time. Nowadays the situation has changed. Other business schools have also became aware that modern economic and business education as well as research needs a good comprehension of the quantitative methods as well as the skill to use them. A large number of students graduated as doctors in business economics in the University of Vaasa who, having been recruited as professors and other academic employees to other business schools, have had a marked effect in this process. 2.2 Mathematical sciences – a close and united academic unit At the beginning the teaching staff consisted of one professor and one lecturer, both posts being combined posts for business mathematics and statistics. As the number of students increased the unit obtained another professorship and an additional lectureship. This made it possible to focus the duties of all the posts only on business mathematics or statistics, respectively. An assistantship was also an important addition to the staff. In the beginning of the 1990s the university started to enlarge its branch of activities towards techno- logy. This could progress step by step only. First, technical elements were included in the curriculum of two master programs in economics and business administration and an unofficial label “industrial economist” was given to the graduates. The main subject in these programmes was either industrial management or information technology. In the next step the university started to educate civil engi- neers in co-operation with the Helsinki Technical University. The University of Vaasa recruited the students who carried out half of their studies in Vaasa and graduated from Helsinki Technical Uni- versity. In the beginning of the 2000s the University of Vaasa was allowed to educate civil engineers completely as its own activity. 190 Acta Wasaensia It was clear that the enlargement of the branch of activities into technology presumed additional resources also for the unit of mathematical sciences. The total number of students increased and the amount, type, and level of mathematics needed in technical studies is different from what is needed in economics, business administration, and in the social sciences. The requirements were resolved by providing the unit with a professorship and a lectureship in mathematics. Today the permanent staff in the unit (department) consists of three professors, three senior lecturers with doctor degrees, one university teacher, and a varying number (3–5) of doctoral students and post-doctoral researchers. Although administratively the unit is a part of the School of Technology and Innovations, the main sphere of responsibility of business mathematics and statistics has been supporting the School of Accounting and Finance and other business oriented schools by teaching quantitative methods both for education and research purposes. The unit of mathematical sciences is small. Therefore it is important that the mathematically oriented disciplines form also administratively an integrated own unit. Together the unit’s academic subjects are stronger, they can flexibly use common teaching resources, and they create and maintain a high- level and international academic atmosphere in research. This administrative solution guarantees that the department can form a close and united academic society. One example of the manifestation of this coherence is that the unit takes care also of maintaining close contacts with its staff in retirement. Holders of the professor offices in the department have been mainly recruited from other universities. Statistics is an exception. The last two professor appointments in statistics have been candidates who have done their doctoral studies and qualified for professorship when working at the University of Vaasa. All the senior lecturers of the unit have received their doctor degrees while working at the university. As a result, the staff members represents a high quality group of experts in their own fields, but, at the same time, they possess understanding of and positive attitude towards the needs of quantitative methods appearing among the students and researchers of other disciplines in the university. 2.3 The unit’s supporting role in undergraduate education, its own intensive postgraduate education, and high-quality international research As has been mentioned earlier, the unit of mathematical sciences doesn’t have and has never had any own master-level education programme in the curriculum of the university. In undergraduate studies the role of mathematical sciences is to contribute as a strong and high-standard supporting discipline offering basic university level knowledge and relevant advanced tools in mathematical and statistical modelling for the students of the other education programmes of the university, especially for students in business and technology. The demand of skills in the use of modern quantitative methods and models is outstandingly high in the postgraduate level of the studies. Besides offering methodological courses the professors of the unit participate as tutors in other subject’s doctoral seminars. Postgraduate studies in mathematics and statistics have been possible in the university from the be- ginning. Their role has become more and more important during the years. The recruiting of students is challenging due to the non-existing own master level education. The achievements are, however, good and the department is for example a pioneer in the university in recruiting international students for doctoral studies. Today, a majority of the unit’s postgraduate students is of foreign origin. Acta Wasaensia 191 As the mathematical sciences are represented on the professor level in the university, scientific re- search is, of course, strongly on the department’s agenda. The research groups in which the de- partment’s researchers participate represent the top quality in the university. The researchers have created and maintain one of the university’s research programmes with the theme of mathematical modelling. The projects in the programme are both discipline oriented, especially in mathematics, and more application oriented in statistics and business mathematics. The research groups work actively in co-operation with other researchers inside the university and with researchers in other universities, both in Finland and abroad. Visiting foreign scholars are commonly seen working in the department and the staff members work frequently abroad. The professors and other staff members participate actively in the general management of the Uni- versity. Two professors in economic mathematics have served as rectors of the University, several professors as faculty deans and as heads of multidisciplinary departments. Professors and other staff members are members of several managerial working groups. 3 Discussion The chosen policy for the mathematical sciences in the University of Vaasa – i.e., to operate as an academic education and research unit without any own undergraduate education programme – is challenging. But the work done during the five first decades has shown that the chosen option has been successful. The main measures for reaching success are: • Careful attention has been paid when recruiting new members to the staff. Besides compe- tence, the new members must also have an understanding of and a positive attitude towards the other disciplines in the university with which they are expected to co-operate both in education and research. • Active co-operation with other disciplines inside the university has been a necessity for being able to give relevant and up-to-date methodological support to both students and researchers of other disciplines. • To reach and maintain a high level competence in the area of everyone’s own expertise an active communication and co-operation with the representatives of one’s own discipline in other universities in Finland and abroad have been strongly on the agenda. • The absence of own undergraduate education has not meant absence of postgraduate edu- cation. On the contrary, active doctoral programmes have guaranteed continuity in research and have helped in recruiting to the unit new employees who possess a relevant orientation towards the operating principle of the unit. Of course, external recruiting has also been im- portant for guaranteeing high levels of competence. • Special attention has been paid for activating international co-operation in research and post- graduate education. A small university like the University of Vaasa must carefully focus its activities on areas about which it has successful experience from the past, which are still relevant today and are also or are expected to become crucial in the future, and which form a coherent entity. In education and research, also the needs of the region and the whole society as well as the region’s possibilities to 192 Acta Wasaensia offer co-operation and support must have a strong role in the agenda. The strategy and values of the university meet these requirements well. Similarly, the unit of mathematical sciences as a small unit inside the university must have a clear and specific high-standard academic culture in its operations. Conclusions from the scrutiny above show that the unit has been successful in creating this culture. Department of Mathematics and Statistics, University of Vaasa, 65200 Vaasa, Finland E-mail address: ilkka.virtanen@uva.fi