Received: 10 July 2018 Revised: 17 February 2019 Accepted: 1 July 2019
DOI: 10.1002/mana.201800300
OR IG INAL PAPER
Generalized boundary triples, I. Some classes of isometric and
unitary boundary pairs and realization problems for subclasses
of Nevanlinna functions
Volodymyr Derkach1,2 Seppo Hassi3 Mark Malamud4
1Department of Mathematics, Vasyl Stus Donetsk National University, Vinnitsya, 21021, Ukraine
2Department of Mathematics, National Pedagogical Dragomanov University, Pirogova 9, Kyiv, 01001, Ukraine
3Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, Vaasa, 65101, Finland
4Peoples’ Friendship University of Russia, 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
Correspondence
SeppoHassi,Department ofMathematics
andStatistics,University of Vaasa,
P.O.Box700, 65101Vaasa, Finland.
Email: seppo.hassi@uwasa.fi
Funding information
VolkswagenFoundation;Vilho,Yrjö andKalle
Väisälä Foundation of the FinnishAcademyof
Science andLetters
Abstract
With a closed symmetric operator 𝐴 in a Hilbert spaceℌ a triple Π =
{,Γ0,Γ1} of
a Hilbert space  and two abstract trace operators Γ0 and Γ1 from 𝐴∗ to  is called
a generalized boundary triple for 𝐴∗ if an abstract analogue of the second Green’s
formula holds. Various classes of generalized boundary triples are introduced and
corresponding Weyl functions 𝑀(⋅) are investigated. The most important ones for
applications are specific classes of boundary triples for which Green’s second identity
admits a certain maximality property which guarantees that the corresponding Weyl
functions are Nevanlinna functions on, i.e.𝑀(⋅) ∈ (), or at least they belong to
the class ̃() of Nevanlinna families on. The boundary condition Γ0𝑓 = 0 deter-
mines a reference operator 𝐴0
(
=ker Γ0
)
. The case where 𝐴0 is selfadjoint implies
a relatively simple analysis, as the joint domain of the trace mappings Γ0 and Γ1
admits a von Neumann type decomposition via 𝐴0 and the defect subspaces of 𝐴.
The case where 𝐴0 is only essentially selfadjoint is more involved, but appears to
be of great importance, for instance, in applications to boundary value problems e.g.
in PDE setting or when modeling differential operators with point interactions. Var-
ious classes of generalized boundary triples will be characterized in purely analytic
terms via theWeyl function𝑀(⋅) and close interconnections between different classes
of boundary triples and the corresponding transformed/renormalized Weyl functions
are investigated. These characterizations involve solving direct and inverse problems
for specific classes of operator functions𝑀(⋅). Most involved ones concern operator
functions𝑀(⋅) ∈ () for which
𝜏𝑀(𝜆)(𝑓, 𝑔) = (2𝑖 Im 𝜆)−1[(𝑀(𝜆)𝑓, 𝑔) − (𝑓,𝑀(𝜆)𝑔)], 𝑓 , 𝑔 ∈ dom𝑀(𝜆),
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original
work is properly cited.
© 2020 The Authors. Mathematische Nachrichten published by Wiley-VCH Verlag GmbH & Co. KGaA
1278 www.mn-journal.org Mathematische Nachrichten. 2020;293:1278–1327.
DERKACH ET AL. 1279
defines a closable nonnegative form on . It turns out that closability of 𝜏𝑀(𝜆)(𝑓, 𝑔)
does not depend on 𝜆 ∈ ℂ± and, moreover, that the closure then is a form domain
invariant holomorphic function on ℂ± while 𝜏𝑀(𝜆)(𝑓, 𝑔) itself need not be domain
invariant. In this study we also derive several additional new results, for instance,
Kreı˘n-type resolvent formulas are extended to the most general setting of unitary and
isometric boundary triples appearing in the present work.
In part II of the present work all the main results are shown to have applications in the
study of ordinary and partial differential operators.
KEYWORD S
boundary triple, boundary value problem, Green’s identities, resolvent, selfadjoint extension, symmetric
operator, trace operator, Weyl family, Weyl function
MSC ( 2 0 1 0 )
Primary: 47A10, 47B25, Secondary: 47A20, 47A48, 47A56, 47B32
1 KEY CONCEPTS AND AN OUTLINE OF THE MAIN RESULTS
1.1 Ordinary boundary triples and Weyl functions
Letℌ be a (complex) Hilbert space, let 𝐴 be a not necessarily densely defined closed symmetric operator inℌ. The adjoint 𝐴∗
of the operator 𝐴 is a linear relation, i.e., a subspace of vectors 𝑔 =
( 𝑔
𝑔′
)
∈ ℌ2 such that
(𝐴𝑓, 𝑔) − (𝑓, 𝑔′) = 0 for all 𝑓 ∈ dom𝐴,
see [4, 16]. In what follows the operator𝐴will be identified with its graph, so that the set () of closed linear operators will be
considered as a subset of ̃() of closed linear relations in. Then𝐴 is symmetric precisely when𝐴 ⊆ 𝐴∗. The defect subspaces
𝔑𝜆 and the deficiency indices of 𝐴 are defined by the equalities 𝔑𝜆 ∶= ker (𝐴∗ − 𝜆), 𝜆 ∈ ℂ± ∶= {𝜆 ∈ ℂ ∶ ±Im 𝜆 > 0 }, and
𝑛±(𝐴) ∶= dim𝔑±𝑖.
The classical J. von Neumann approach to the extension theory of symmetric operators in Hilbert spaces [61] is based on
two fundamental formulas which allow to get a description of all selfadjoint extensions of a symmetric operator by means of
isometric operators from 𝔑𝑖 onto 𝔑−𝑖 (see in this connection the monographs [1, 3, 22]). Another approach to the extension
theory that substantially relied on a concept of abstract Green formula was originated by J.W. Calkin [21]. It turned out to be
more convenient in the study of boundary value problems for ordinary and especially for partial differential equations (ODE and
PDE) (see [19, 20, 32, 33, 36–38, 46, 63, 67]). Some further discussion on Calkin’s paper is given below.
Definition 1.1. A collection Π = {,Γ0,Γ1} consisting of a Hilbert space  and two linear mappings Γ0 and Γ1 from 𝐴∗ to, is said to be an ordinary boundary triple for 𝐴∗ if:
1.1.1 The following abstract Green’s identity holds
(𝑓 ′, 𝑔) − (𝑓, 𝑔′) =
(
Γ1𝑓,Γ0𝑔
)
 −
(
Γ0𝑓,Γ1𝑔
)
 for all 𝑓 =
(
𝑓
𝑓 ′
)
, 𝑔 =
(
𝑔
𝑔′
)
∈ 𝐴∗; (1.1)
1.1.2 The mapping Γ ∶=
(
Γ0
Γ1
)
∶ 𝐴∗ → 2 is surjective.
Note that in the ODE setting formula (1.1) turns into the classical Lagrange identity being a key tool in study of boundary value
problems. The advantage of this approach becomes obvious in applications to boundary value problems for elliptic equations
where the formula (1.1) becomes a second Green’s identity. However, in this case the assumptions of Definition 1.1 are violated
and this circumstance was overcome in the classical papers by M. Višik [67] and G. Grubb [38] (see also [39]). Namely, relying
1280 DERKACH ET AL.
on the Lions–Magenes trace theory ([39, 56]) they regularized the classical Dirichlet and Neumann trace mappings to get a
proper version of Definition 1.1.
The operator Γ in Definition 1.1 is called the reduction operator (in the terminology of [21]). Definition 1.1 immediately
yields a parametrization of the set of all selfadjoint extensions 𝐴 of 𝐴 by means of abstract boundary conditions via
𝐴 = 𝐴Θ ∶=
{
𝑓 ∈ 𝐴∗ ∶ Γ𝑓 ∈ Θ
}
,
where 𝐴Θ ranges over the set of all selfadjoint extensions of 𝐴 when Θ ranges over the set of all selfadjoint relations in 
(subspaces in  ×, see [4]). This correspondence is bijective and in this case Θ ∶= Γ(𝐴). The following two selfadjoint
extensions of 𝐴 are of particular interest:
𝐴0 ∶= ker Γ0 = 𝐴Θ∞ and 𝐴1 ∶= ker Γ1 = 𝐴Θ1;
here Θ∞ = {0} × and Θ1 = 𝕆. These extensions are disjoint, i.e. 𝐴0 ∩ 𝐴1 = 𝐴, and transversal, i.e. they are disjoint and
𝐴0+̂𝐴1 = 𝐴∗. Here the symbol +̂ means the componentwise sum of two linear relations, see (2.1).
In what follows 𝐴0 is considered as a reference extension of 𝐴. Let 𝜌
(
𝐴0
)
be the resolvent set of 𝐴0, and let
?̂?𝜆 ∶=
{
𝑓𝜆 =
(
𝑓𝜆
𝜆𝑓𝜆
)
∶ 𝑓𝜆 ∈ 𝔑𝜆
}
, 𝜆 ∈ 𝜌(𝐴0).
The main analytical tool in the description of spectral properties of selfadjoint extensions of 𝐴 is the abstract Weyl function,
introduced and investigated in [30–32].
Definition 1.2 ([30–32]). The abstract Weyl function and the 𝛾-field of 𝐴, corresponding to an ordinary boundary triple
Π =
{,Γ0,Γ1} are defined by
𝑀(𝜆)Γ0𝑓𝜆 = Γ1𝑓𝜆, 𝛾(𝜆)Γ0𝑓𝜆 = 𝑓𝜆, 𝑓𝜆 ∈ ?̂?𝜆, 𝜆 ∈ 𝜌
(
𝐴0
)
.
Notice that when the symmetric operator 𝐴 is densely defined its adjoint is a single-valued operator and Definitions 1.1
and 1.2 can be used in a simpler form by treating Γ0 and Γ1 as operators from dom𝐴∗ to , see [32, 37, 46]. In what follows
this convention will be tacitly used in most of our examples.
Example 1.3. Let 𝐴 be a minimal symmetric operator in 𝐿2(ℝ+) associated with the Sturm–Liouville differential expression
 ∶= − 𝑑2
𝑑𝑥2
+ 𝑞(𝑥), 𝑞 = 𝑞 ∈ 𝐿1𝑙𝑜𝑐([0,∞)).
Assume the limit-point case at infinity, i.e. assume that 𝑛±(𝐴) = 1. Let 𝑐(⋅, 𝜆) and 𝑠(⋅, 𝜆) be cosine and sine type solutions of the
equation 𝑓 = 𝜆𝑓 subject to the initial conditions
𝑐(0, 𝜆) = 1, 𝑐′(0, 𝜆) = 0; 𝑠(0, 𝜆) = 0, 𝑠′(0, 𝜆) = 1.
The defect subspace 𝔑𝜆 is spanned by the Weyl solution 𝜓(⋅, 𝜆) of the equation 𝑓 = 𝜆𝑓 which is given by
𝜓(𝑥, 𝜆) = 𝑐(𝑥, 𝜆) + 𝑚(𝜆)𝑠(𝑥, 𝜆) ∈ 𝐿2
(
ℝ+
)
.
The function 𝑚(⋅) is called the Titchmarsh–Weyl coefficient of . In this case a boundary triple Π = {ℂ,Γ0,Γ1} can be defined
as Γ0𝑓 = 𝑓 (0), Γ1𝑓 = 𝑓 ′(0). The correspondingWeyl function𝑀(𝜆) coincides with the classical Titchmarsh–Weyl coefficient,
𝑀(𝜆) = 𝑚(𝜆).
In this connection let us mention that the role of the Weyl function 𝑀(𝜆) in the extension theory of symmetric operators
is similar to that of the classical Titchmarsh–Weyl coefficient 𝑚(𝜆) in the spectral theory of Sturm–Liouville operators. For
instance, it is known (see [32, 52]) that if 𝐴 is simple, i.e. 𝐴 does not admit orthogonal decompositions with a selfadjoint
summand, then the Weyl function 𝑀(𝜆) determines the boundary triple Π, in particular, the pair
{
𝐴,𝐴0
}
, uniquely up to
unitary equivalence. Besides, when 𝐴 is simple, the spectrum of 𝐴Θ coincides with the singularities of the operator function
(Θ −𝑀(𝑧))−1; see [32].
DERKACH ET AL. 1281
As was shown in [32, 33] and [58] the Weyl function𝑀(⋅) and the 𝛾-field 𝛾(⋅) both are well defined and holomorphic on the
resolvent set 𝜌
(
𝐴0
)
of the operator 𝐴0. Moreover, the 𝛾-field 𝛾(⋅) and the Weyl function𝑀(⋅) satisfy the identities
𝛾(𝜆) =
[
𝐼 + (𝜆 − 𝜇)
(
𝐴0 − 𝜆
)−1]
𝛾(𝜇), 𝜆, 𝜇 ∈ 𝜌
(
𝐴0
)
, (1.2)
𝑀(𝜆) −𝑀(𝜇)∗ = (𝜆 − ?̄?)𝛾(𝜇)∗𝛾(𝜆), 𝜆, 𝜇 ∈ 𝜌
(
𝐴0
)
. (1.3)
This means that𝑀(⋅) is a 𝑄-function of the operator 𝐴 in the sense of Kreı˘n and Langer [51].
Denote by() the set of bounded linear operators in and by[] the class of Nevanlinna functions, i.e., operator valued
functions 𝐹 (𝜆) with values in (), which are holomorphic on ℂ ⧵ℝ and satisfy the conditions
𝐹 (𝜆) = 𝐹
(
?̄?
)∗
and Im𝐹 (𝜆) ≥ 0 for all 𝜆 ∈ ℂ+, (1.4)
see [44]. It follows from (1.2) and (1.3) that𝑀 belongs to the Nevanlinna class[]. Furthermore, since 𝛾(𝜆) isomorphically
maps onto𝔑𝜆, the relation (1.3) ensures that the imaginary part Im𝑀(𝑧) of𝑀(𝑧) is positively definite, i.e.𝑀(⋅) belongs to
the subclass𝑢[] of uniformly strict Nevanlinna functions:
𝑢[] ∶= {𝐹 (⋅) ∈ [] ∶ 0 ∈ 𝜌(Im𝐹 (𝑖))}.
The converse is also true.
Theorem 1.4 ([33, 52]). The set of Weyl functions corresponding to ordinary boundary triples coincides with the class𝑢[]
of uniformly strict Nevanlinna functions.
1.2 𝑩-generalized and 𝑨𝑩-generalized boundary triples
In BVP’s for Sturm–Liouville operators with an operator potential, for partial differential operators [26], and in point interaction
theory it seems natural to consider more general boundary triples by weakening the surjectivity assumption 1.1.2 in Defini-
tion 1.1. The following notion was introduced in [33] with the name generalized boundary-value space, see also [25], where the
term generalized boundary triplet was used.
Definition 1.5. ([25, 33]) Let 𝐴 be a closed symmetric operator in a Hilbert space ℌ with equal deficiency indices and let 𝐴∗
be a linear relation in ℌ such that 𝐴 ⊂ 𝐴∗ ⊂ 𝐴∗ = 𝐴∗. Then the collection Π =
{,Γ0,Γ1}, where  is a Hilbert space and
Γ =
{
Γ0,Γ1
}
is a single-valued linear mapping from 𝐴∗ into 2, is said to be a 𝐵-generalized boundary triple for 𝐴∗, if:
1.5.1 the abstract Green’s identity (1.1) holds for all 𝑓 =
(
𝑓
𝑓 ′
)
, 𝑔 =
(
𝑔
𝑔′
)
∈ 𝐴∗;
1.5.2 𝐴0 ∶= ker Γ0 is a selfadjoint relation in ℌ;
1.5.3 ran Γ0 = .
The Weyl function𝑀(𝜆) and the 𝛾-field corresponding to a 𝐵-generalized boundary triple are defined by
𝑀(𝜆)Γ0𝑓𝜆 = Γ1𝑓𝜆, 𝛾(𝜆)Γ0𝑓𝜆 = 𝑓𝜆, 𝑓𝜆 ∈ ?̂?𝜆(𝐴∗) ∶= ?̂?𝜆 ∩ 𝐴∗, 𝜆 ∈ 𝜌
(
𝐴0
)
. (1.5)
For every 𝜆 ∈ 𝜌(𝐴) the Weyl function𝑀(𝜆) takes values in () and this justifies the present usage of the term 𝐵-generalized
boundary triple, where “𝐵” stands for aWeyl function whose values are “bounded” operators.
Example 1.6. LetΩ be a bounded domain inℝ𝑛 with smooth boundary 𝜕Ω. Consider the Laplace operator −Δ in𝐿2(Ω). Let 𝛾𝐷
and 𝛾𝑁 be the Dirichlet and Neumann trace mappings. Moreover, let 𝐴∗ be the pre-maximal operator defined as the restriction
of the maximal Laplace operator 𝐴max to the domain
dom𝐴∗ = 𝐻
3∕2
Δ (Ω) ∶= 𝐻
3∕2(Ω) ∩ dom𝐴max =
{
𝑓 ∈ 𝐻3∕2(Ω) ∶ Δ𝑓 ∈ 𝐿2(Ω)
}
.
It is well known (see e.g. [39, 56]) that the mappings 𝛾𝐷 ∶ 𝐻
3∕2
Δ (Ω)→ 𝐻
1(𝜕Ω) and 𝛾𝑁 ∶ 𝐻
3∕2
Δ (Ω)→ 𝐻
0(𝜕Ω) = 𝐿2(𝜕Ω) are
well defined and surjective.
1282 DERKACH ET AL.
Using the keymapping properties of 𝛾𝐷 and 𝛾𝑁 one can extend the classical Green’s formula to the domain dom𝐴∗. Notice that
the condition 𝛾𝑁𝑓 = 0, 𝑓 ∈ dom𝐴∗, determines the Neumann realization Δ𝑁 of the Laplace operator. Since Δ𝑁 is selfadjoint
and 𝛾𝑁 (dom𝐴∗) = 𝐻0(𝜕Ω), the triple Π =
{
𝐿2(𝜕Ω),Γ0,Γ1
}
with
Γ0 = 𝛾𝑁↾ dom𝐴∗ and Γ1 = 𝛾𝐷↾ dom𝐴∗
is a 𝐵-generalized boundary triple for𝐴∗ with domΓ = dom𝐴∗. Besides, the correspondingWeyl function𝑀(⋅) coincides with
the inverse of the classical Dirichlet-to-Neumann mapΛ(⋅), i.e.𝑀(⋅) = Λ(⋅)−1; see Part II of the present work for further details.
As was shown in [27] for every 𝐵-generalized boundary triple there exists an ordinary boundary triple
{,Γ00,Γ01} and
operators 𝐺,𝐸 = 𝐸∗ ∈ () such that ker 𝐺 = {0} and(
Γ0
Γ1
)
=
(
𝐺−1 0
𝐸𝐺−1 𝐺∗
)(Γ00
Γ01
)
. (1.6)
Weyl functions𝑀 and𝑀0 corresponding to the boundary triples
{,Γ0,Γ1} and {,Γ00,Γ01}, are connected by
𝑀(𝜆) = 𝐺∗𝑀0(𝜆)𝐺 + 𝐸, 𝜆 ∈ 𝜌
(
𝐴0
)
. (1.7)
It should be noted that theWeyl function𝑀(⋅) of a𝐵-generalized boundary triple satisfies the properties (1.2)– (1.4). However,
instead of the property 0 ∈ 𝜌(Im𝑀(𝑖)) one has a weaker condition 0 ∉ 𝜎𝑝(Im𝑀(𝑖)). This motivates the following definition.
Denote by𝑠[] the class of strict Nevanlinna functions
𝑠[] ∶= {𝐹 (⋅) ∈ [] ∶ 0 ∉ 𝜎𝑝(Im𝐹 (𝑖))}.
In fact, it was also shown in [33, Chapter 5] that every 𝑀(⋅) ∈ 𝑠[] can be realized as the Weyl function of a certain
𝐵-generalized boundary triple and hence the following statement holds.
Theorem 1.7 ([33]). The set of Weyl functions corresponding to𝐵-generalized boundary triples coincides with the class𝑠[]
of strict Nevanlinna functions.
This realization result as well as the technique of 𝐵-generalized boundary triples have recently been applied also e.g. to
problems in scattering theory, see [13], in the analysis of discrete and continuous time system theory, and in the boundary
control theory; for some recent achievements, see e.g. [5, 6, 8, 9, 40, 53, 54, 59, 66].
In the present paper we introduce the new class of 𝐴𝐵-generalized boundary triples which is obtained by a weakening of the
surjectivity condition 1.5.3 in Definition 1.5.
Definition 1.8. A collection {,Γ0,Γ1} is said to be an almost 𝐵-generalized boundary triple, or briefly, an 𝐴𝐵-generalized
boundary triple for 𝐴∗, if 𝐴∗ ∶= domΓ is dense in 𝐴∗, the conditions 1.5.1, 1.5.2 are satisfied and
1.8.1 ran Γ0 is dense in .
The Weyl function corresponding to an 𝐴𝐵-generalized boundary triple is again defined by (1.5). One of the main results of
the paper is Theorem 4.4 which states that every 𝐴𝐵-generalized boundary triple can be regularized to produce a 𝐵-generalized
boundary triple in the spirit of (1.6). Another result — Theorem 4.6 gives a characterization of the set of the Weyl functions𝑀
of 𝐴𝐵-generalized boundary triples in the form
𝑀(𝜆) = 𝐸 +𝑀0(𝜆), where 𝑀0 ∈ []
and 𝐸 is a densely defined symmetric operator in, such that ker Im𝑀0(𝜆) ∩ dom𝐸 = {0}.
The class of 𝐴𝐵-generalized boundary triples contains the class of so-called quasi boundary triples, which has been studied
in J. Behrndt and M. Langer [11]. In the definition of a quasi boundary triple Assumption 1.5.3 is replaced by the assumption
that
ran Γ is dense in  ×.
A connection between quasi boundary triples and 𝐴𝐵-generalized boundary triples is given in Corollary 4.9. A joint feature in
𝐴𝐵-generalized boundary triples and quasi boundary triples is that without additional assumptions on themapping Γ =
{
Γ0,Γ1
}
DERKACH ET AL. 1283
these boundary triples are not unitary in the sense of Definition 1.9 presented below. Consequently, their Weyl functions need
not be Nevanlinna functions, i.e., the values𝑀(𝜆) need not be maximal dissipative (accumulative) in ℂ+ (ℂ−); see definitions
in Section 2.1. Special type of isometric boundary triples that will appear in Part II of the present paper are so-called essentially
unitary boundary triples/pairs. As shown therein (cf. [29, Section 7]) quasi boundary triples studied in [11, 12] for elliptic
operators are either special type of unitary boundary triples or they are essentially unitary boundary triples, depending on
the choice of the underlying regularity index of the space used as the domain 𝐴∗ for the boundary triple. For a very recent
contribution and some further development on essentially unitary boundary pairs see also [43].
Different applications of quasi boundary triples in boundary value problems including applications to elliptic theory and trace
formulas can be found e.g. in [11, 14, 15, 40, 62].
1.3 Unitary boundary triples
Ageneral class of boundary triples, to be called here unitary boundary triples, was introduced in [25]. This concept wasmotivated
by the realization problem for the most general class of Nevanlinna functions: realize each Nevanlinna function as the Weyl
function of an appropriate type generalized boundary triple.
To this end denote by() the class of all operator valued holomorphic Nevanlinna functions on ℂ+ (in the resolvent sense)
with values in the set of maximal dissipative (not necessarily bounded) linear operators in . Each 𝑀(⋅) ∈ () is extended
to ℂ− by symmetry with respect to the real line𝑀(𝜆) = 𝑀
(
?̄?
)∗
; see [25, 51]. Analogous to the subclass𝑠[] of Nevanlinna
functions[], the class() contains a subclass𝑠() of strict Nevanlinna functions which satisfy the condition
𝑠() ∶= {𝐹 (⋅) ∈ () ∶ (Im𝐹 (𝑖)ℎ, ℎ) = 0 ⇐⇒ ℎ = 0, ℎ ∈ dom𝐹 (𝑖)}. (1.8)
In order to present the definition of a unitary boundary triple, introduce the fundamental symmetries
𝐽ℌ ∶=
(
0 −𝑖𝐼ℌ
𝑖𝐼ℌ 0
)
, 𝐽 ∶=
(
0 −𝑖𝐼
𝑖𝐼 0
)
, (1.9)
and the associated Kreı˘n spaces
(
ℌ2, 𝐽ℌ
)
and
(2, 𝐽) (see [7, 17]) obtained by endowing the Hilbert spacesℌ2 and2 with
the following indefinite inner products[
𝑓, 𝑔
]
ℌ2 =
(
𝐽ℌ𝑓, 𝑔
)
ℌ2 ,
[
ℎ̂, ?̂?
]
2 =
(
𝐽 ℎ̂, ?̂?
)
2 , 𝑓 , 𝑔 ∈ ℌ2, ℎ̂, ?̂? ∈ 2. (1.10)
This allows to rewrite Green’s identity (1.1) in the form[
𝑓, 𝑔
]
ℌ2 =
[
Γ𝑓,Γ𝑔
]
2 , (1.11)
which means that the mapping Γ is in fact a
(
𝐽ℌ, 𝐽
)
-isometric mapping from the Kreı˘n space
(
ℌ2, 𝐽ℌ
)
to the Kreı˘n space(2, 𝐽). If Γ[∗] denotes the Kreı˘n space adjoint of the operator Γ (see Definition (2.4)), then (1.11) can be simply rewritten as
Γ−1 ⊂ Γ[∗]. The surjectivity of Γ implies that Γ−1 = Γ[∗]. Following Yu. L. Shmuljan [64] a linear operator Γ ∶
(
ℌ2, 𝐽ℌ
)
→(2, 𝐽ℌ) will be called (𝐽ℌ, 𝐽)-unitary, if Γ−1 = Γ[∗].
Definition 1.9 ([25]). A collection {,Γ0,Γ1} is called a unitary (resp. isometric) boundary triple for 𝐴∗, if  is a Hilbert
space and Γ =
(
Γ0
Γ1
)
is a linear operator from ℌ2 to2 such that:
1.9.1 𝐴∗ ∶= domΓ is dense in 𝐴∗ with respect to the topology on ℌ2;
1.9.2 The operator Γ is (𝐽ℌ, 𝐽)-unitary (resp. (𝐽ℌ, 𝐽)-isometric).
The Weyl function 𝑀(𝜆) and the 𝛾-field corresponding to a unitary boundary triple Π are defined again by the same for-
mula (1.5). The transposed boundary triple Π⊤ ∶=
{,Γ1,−Γ0} associated with a unitary boundary triple Π is also a unitary
boundary triple, the corresponding Weyl function takes the form𝑀⊤(𝜆) = −𝑀(𝜆)−1.
The main realization theorem in [25] gives a solution to the inverse problem mentioned above.
Theorem 1.10 ([25]). The class of Weyl functions corresponding to unitary boundary triples coincides with the class 𝑠()
of (in general unbounded) strict Nevanlinna functions.
1284 DERKACH ET AL.
In fact, in [25, Theorem 3.9] a stronger result is stated showing that the class 𝑠() can be replaced by the class () or
even by the class ̃() of Nevanlinna pairs when one allows multi-valued linear mappings Γ in Definition 1.9; see Theorem 3.3
in Section 3.2. Theorem 1.10 plays a key role in the construction of generalized resolvents in the framework of coupling method
that was originally introduced in [24] and developed in its full generality in [26]. It is worth to mention that in [6] it is shown that
a counterpart of the main transform of a unitary boundary triple (with some extra properties) naturally appears in impedance
conservative continuous time input/state/output systems, and, moreover, that the transfer function of such systems is directly
connected with the Weyl function of the unitary boundary triplet. A systematic study of so-called conservative state/signal
systems has been initiated in [5] and, as shown in [6], conservative state/signal systems have a close connection to general
unitary boundary triples in Theorem 1.10; see also Remark 5.7.
Ordinary and 𝐵-generalized boundary triples give examples of unitary boundary triples; see [25], and as noted above the
conditions defining 𝐴𝐵-generalized or quasi boundary triples do not guarantee their unitarity; for a criterion see Corollary 4.7.
Some necessary and sufficient conditions which characterize unitary boundary triples and which differ from the purely analytic
criterion in Theorem 1.10 can be found in [25, Proposition 3.6], [27, Theorem 7.51], some general criteria of geometric nature
have been established in [68, 69], and a further characterization, useful e.g. in applications to elliptic equations, can be found in
Part II of the present paper.
In connection with Definition 1.9 we wish to make some comments on a seminal paper [21] by J. W. Calkin, where a concept
of the reduction operator is introduced and investigated. Although no proper geometric machinery appears in the definition
of Calkin’s reduction operator this notion in the case of a densely defined operator 𝐴 essentially coincides with concept of a
unitary operator between Kreı˘n spaces as in Definition 1.9. An overview on the early work of Calkin and some connections to
later developments can be found in the papers in the monograph [40]; for a further discussion see also Section 3.5.
In Theorem 5.8 we extend Kreı˘n’s resolvent formula to the general setting of unitary boundary triples. Namely, for any proper
extension 𝐴Θ ∈ Ext𝑆 satisfying 𝐴Θ ⊂ domΓ the following Kreı˘n-type formula holds:(
𝐴Θ − 𝜆
)−1 − (𝐴0 − 𝜆)−1 = 𝛾(𝜆)(Θ −𝑀(𝜆))−1𝛾(?̄?)∗, 𝜆 ∈ ℂ ⧵ℝ.
It is emphasized that in this formula 𝐴Θ is not necessarily closed and it is not assumed that𝜆 ∈ 𝜌
(
𝐴Θ
)
, in particular, here the
inverses
(
𝐴Θ − 𝜆
)−1
and
(
Θ −𝑀(𝜆)
)−1
are understood in the sense of relations.
1.4 𝑺-generalized boundary triples
Following [25] we consider a special class of unitary boundary triples singled out by the condition that 𝐴0 ∶= ker Γ0 is a
selfadjoint extension of 𝐴.
Definition 1.11 ([25]). A unitary boundary triple Π = {,Γ0,Γ1} is said to be an 𝑆-generalized boundary triple for 𝐴∗ if the
assumption 1.5.2 holds, i.e. 𝐴0 ∶= ker Γ0 is a selfadjoint extension of 𝐴.
Next following [27, Theorem 7.39] and [25, Theorem 4.13] we present a complete characterization of the Weyl functions
𝑀(⋅) corresponding to 𝑆-generalized boundary triples.
Theorem 1.12. ([25, 27]) Let Π = {,Γ0,Γ1} be a unitary boundary triple for 𝐴∗ and let𝑀(⋅) and 𝛾(⋅) be the corresponding
Weyl function and 𝛾-field, respectively. Then the following statements are equivalent:
(i) 𝐴0 = ker Γ0 is selfadjoint, i.e. Π is an 𝑆-generalized boundary triple;
(ii) 𝐴∗ = 𝐴0 +̂ ?̂?𝜆(𝐴∗) and 𝐴∗ = 𝐴0 +̂ ?̂?𝜇(𝐴∗) for some (equivalently for all) 𝜆 ∈ ℂ+ and 𝜇 ∈ ℂ−;
(iii) ran Γ0 = dom𝑀(𝜆) = dom𝑀(𝜇) for some (equivalently for all) 𝜆 ∈ ℂ+ and 𝜇 ∈ ℂ−;
(iv) 𝛾(𝜆) and 𝛾(𝜇) are bounded and densely defined in  for some (equivalently for all) 𝜆 ∈ ℂ+ and 𝜇 ∈ ℂ−;
(v) dom𝑀(𝜆) = dom𝑀
(
𝜆
)
and Im𝑀(𝜆) is bounded for some (equivalently for all) 𝜆 ∈ ℂ+;
(vi) the Weyl function𝑀(⋅) belongs to𝑠() and it admits a representation
𝑀(𝜆) = 𝐸 +𝑀0(𝜆), 𝑀0(⋅) ∈ [], 𝜆 ∈ ℂ ⧵ℝ, (1.12)
where 𝐸 = 𝐸∗ is a selfadjoint (in general unbounded) operator in .
In Theorem 5.17 this result is extended to the case of 𝑆-generalized boundary pairs {,Γ}, where Γ ∶ 𝐴∗ →  × is
allowed to be multi-valued (see Definitions 3.1 and 5.11).
DERKACH ET AL. 1285
Notice that, for instance, the implications (i)⇒ (ii), (iii) are immediate from the following decomposition of 𝐴∗ ∶= domΓ:
𝐴∗ = 𝐴0 +̂ ?̂?𝜆(𝐴∗), 𝜆 ∈ 𝜌
(
𝐴0
)
. (1.13)
In accordance with (1.12) theWeyl function corresponding to an𝑆-generalized boundary triple is an operator valued Nevanlinna
function with domain invariance property: dom𝑀(𝜆) = dom𝐸 = ran Γ0, 𝜆 ∈ ℂ±. It takes values in the set () of closed (in
general unbounded) operators while the values of the imaginary parts Im𝑀(𝜆) are bounded operators.
As an example wemention that the transposed tripleΠ⊤ =
{
𝐿2(𝜕Ω),Γ1,−Γ0
}
from the PDE Example 1.6 is an𝑆-generalized
boundary triple. The corresponding Weyl function coincides (up to sign change) with the Dirichlet-to-Neumann map Λ(⋅), i.e.
𝑀(⋅)⊤ = −Λ(⋅); further details are given in Part II of the present work.
1.5 𝑬𝑺-generalized boundary triples and form domain invariance
Next we discuss one of the main new objects appearing in the present work.
Definition 1.13. A unitary boundary triple {,Γ0,Γ1} for 𝐴∗ is said to be an essentially selfadjoint generalized boundary
triple, in short, 𝐸𝑆-generalized boundary triple for 𝐴∗, if:
1.13.1 𝐴0 ∶= ker Γ0 is an essentially selfadjoint linear relation in ℌ.
To characterize the class of 𝐸𝑆-generalized boundary triples in terms of the corresponding Weyl functions we associate with
each𝑀(⋅) a family of nonnegative quadratic forms 𝔱𝑀(𝜆) in :
𝔱𝑀(𝜆)[𝑢, 𝑣] ∶=
1
𝜆 − ?̄?
[(𝑀(𝜆)𝑢, 𝑣) − (𝑢,𝑀(𝜆)𝑣)], 𝑢, 𝑣 ∈ dom (𝑀(𝜆)), 𝜆 ∈ ℂ ⧵ℝ. (1.14)
The forms 𝔱𝑀(𝜆) are not necessarily closable. However, it is shown that if 𝔱𝑀(𝜆0) is closable at one point 𝜆0 ∈ ℂ+ (𝜆0 ∈ ℂ−),
then 𝔱𝑀(𝜆) is closable for every 𝜆 ∈ ℂ+ (resp. 𝜆 ∈ ℂ−); for an analytic treatment of this fact see also [28]. In the latter case the
domain of the closure 𝔱𝑀(𝜆) does not depend on 𝜆 ∈ ℂ+ (𝜆 ∈ ℂ−) and therefore theWeyl function𝑀(𝜆) is said to be form domain
invariant in ℂ+ (resp. in ℂ−). In general 𝔱𝑀(𝜆) need not be closable in both half-planes simultaneously; see Proposition 5.26 and
Remark 5.27. On the other hand, if 𝔱𝑀(𝜆) is closable in both half-planes then the form domain does not depend on 𝜆 ∈ ℂ ⧵ℝ;
i.e. form domains coincide also in different half-planes.
In what follows one of the main results established in this connection reads as follows (cf. Theorem 5.24).
Theorem 1.14. Let Π = {,Γ0,Γ1} be a unitary boundary triple for 𝐴∗. Let also 𝑀(⋅) and 𝛾(⋅) be the corresponding Weyl
function and the 𝛾-field, respectively. Then the following statements are equivalent:
(i) Π is an 𝐸𝑆-generalized boundary triple for 𝐴∗;
(ii) 𝛾(𝑖) and 𝛾(−𝑖) are closable;
(iii) 𝛾(𝜆) is closable for every 𝜆 ∈ ℂ+ ∪ ℂ− and dom 𝛾(𝜆) = dom 𝛾(±𝑖), 𝜆 ∈ ℂ+ ∪ ℂ−;
(iv) the forms 𝔱𝑀(𝑖) and 𝔱𝑀(−𝑖) are closable;
(v) the form 𝔱𝑀(𝜆) is closable for every 𝜆 ∈ ℂ+ ∪ ℂ− and dom 𝔱𝑀(𝜆) = dom 𝔱𝑀(±𝑖), 𝜆 ∈ ℂ+ ∪ ℂ−;
(vi) the Weyl function𝑀(⋅) belongs to𝑠() and is form domain invariant in ℂ+ ∪ ℂ−.
The result relies on Theorem 5.5, which contains some important invariance results that unitary boundary triples are shown
to satisfy. If
{,Γ0,Γ1} is an 𝐸𝑆-generalized, but not an 𝑆-generalized, boundary triple for 𝐴∗, then the equality (1.13) fails
to hold and turns out to be an inclusion
𝐴0 +̂ ?̂?𝜆(𝐴∗) ⊊ 𝐴∗ ⊂ 𝐴∗ = 𝐴0 +̂ ?̂?𝜆(𝐴∗), 𝜆 ∈ 𝜌
(
𝐴0
)
.
Indeed, since 𝐴0 is not selfadjoint (while it is essentially selfadjoint), the decomposition 𝐴∗ = 𝐴0 +̂ ?̂?𝜆(𝐴∗) doesn’t hold;
cf. [25, Theorem 4.13]. Then there clearly exists 𝑓 ∈ 𝐴∗ which does not belong to 𝐴0 +̂ ?̂?𝜆(𝐴∗), so that Γ0𝑓 ≠ 0 as well as
Γ0𝑓 ∉ Γ0
(
?̂?𝜆(𝐴∗)
)
= dom𝑀(𝜆). In particular, in this case a strict inclusion dom𝑀(𝜆) ⊊ ran Γ0 holds and, consequently, the
1286 DERKACH ET AL.
Weyl function𝑀(𝜆) can loose the domain invariance property. However, the domain of the closure Γ0 contains the selfadjoint
relation 𝐴0 and admits the decomposition
domΓ0 = 𝐴0 +̂
(
dom
(
Γ0
)
∩ ?̂?𝜆(𝐴∗)
)
, 𝜆 ∈ 𝜌
(
𝐴0
)
.
This implies the equality dom 𝛾(𝜆) = Γ0
(
dom
(
Γ0
)
∩ ?̂?𝜆(𝐴∗)
)
= ran Γ0, which combined with dom 𝔱𝑀(𝜆) = dom 𝛾(𝜆) yields
the form domain invariance property for𝑀 :
dom 𝔱𝑀(𝜆) = ran Γ0.
Passing from the case of an 𝑆-generalized boundary triple to the case of an 𝐸𝑆-generalized boundary triple (which is not
𝑆-generalized) means that 𝐴0 ≠ 𝐴∗0. Then, in particular, conditions (ii) and (iii) in Theorem 1.12 are necessary violated. We
split the situation into two different cases:
Assumption 1.15. 𝑀(𝜆) isn’t domain invariant, i.e. dom𝑀(𝜆1) ≠ dom𝑀(𝜆2) at least for two points 𝜆1, 𝜆2 ∈ ℂ+, 𝜆1 ≠ 𝜆2,
while it is form domain invariant, i.e. dom 𝔱𝑀(𝜆) = dom 𝔱𝑀(±𝑖), 𝜆 ∈ ℂ±.
Assumption 1.16. dom𝑀(𝜆) = dom𝑀(±𝑖), 𝜆 ∈ ℂ±, while dom𝑀(±𝑖) ⫋ ran Γ0.
Both possibilities appear in the spectral theory. An example of a Nevanlinna function satisfying Assumption 1.15 is presented
in Example 5.28. Next we present an example of theWeyl function satisfying Assumption 1.16. Such Nevanlinna functions arise
in the theory of Schrödinger operators with local point interactions.
Example 1.17. Let 𝑋 = {𝑥𝑛}∞1 be a strictly increasing sequence of positive numbers such that lim𝑛→∞ 𝑥𝑛 = ∞. Let 𝑥0 = 0
and denote 𝑑𝑛 ∶= 𝑥𝑛 − 𝑥𝑛−1 > 0, 0 ≤ 𝑑∗ ∶= inf𝑛∈ℕ 𝑑𝑛, and 𝑑∗ ∶= sup𝑛∈ℕ 𝑑𝑛 ≤ ∞.
Let also H𝑛 be a minimal operator associated with the expression −
d2
d𝑥2 in 𝐿
2
0
[
𝑥𝑛−1, 𝑥𝑛
]
. Then H𝑛 is a symmetric operator
with deficiency indices 𝑛±
(
H𝑛
)
= 2 and domain dom
(
H𝑛
)
= 𝑊 2,20
[
𝑥𝑛−1, 𝑥𝑛
]
. Consider in𝐿2
(
ℝ+
)
the direct sum of symmetric
operators H𝑛,
H ∶= Hmin =
∞⨁
𝑛=1
H𝑛, dom
(
Hmin
)
= 𝑊 2,20
(
ℝ+ ⧵𝑋
)
=
∞⨁
𝑛=1
𝑊 2,20
[
𝑥𝑛−1, 𝑥𝑛
]
.
It is easily seen that a boundary triple Π𝑛 =
{
ℂ2,Γ(𝑛)0 ,Γ
(𝑛)
1
}
for H∗𝑛 can be chosen as
Γ(𝑛)0 𝑓 ∶=
(
𝑓 ′
(
𝑥𝑛−1 +
)
𝑓 ′
(
𝑥𝑛 −
) ), Γ(𝑛)1 𝑓 ∶=
(
−𝑓
(
𝑥𝑛−1 +
)
𝑓
(
𝑥𝑛 −
) ), 𝑓 ∈ 𝑊 22 [𝑥𝑛−1, 𝑥𝑛].
The corresponding Weyl function𝑀𝑛 is given by
𝑀𝑛(𝜆) =
−1√
𝜆
⎛⎜⎜⎜⎜⎝
cot
(√
𝜆𝑑𝑛
)
− 1
sin
(√
𝜆𝑑𝑛
)
− 1
sin
(√
𝜆𝑑𝑛
) cot (√𝜆𝑑𝑛)
⎞⎟⎟⎟⎟⎠
.
Clearly, H = Hmin is a closed symmetric operator in 𝐿2(ℝ+). Next we put
 = 𝑙2(ℕ)⊗ ℂ2, Γ =
(
Γ0
Γ1
)
∶=
∞⨁
𝑛=1
(
Γ(𝑛)0
Γ(𝑛)1
)
and note that in accordance with the definition of the direct sum of linear mappings
domΓ ∶=
{
𝑓 = ⊕∞
𝑛=1𝑓𝑛 ∈ dom𝐴
∗ ∶
∑
𝑛∈ℕ
‖‖‖Γ(𝑛)𝑗 𝑓𝑛‖‖‖2𝑛 < ∞, 𝑗 ∈ {0, 1}
}
.
We also put Γ𝑗 ∶= ⊕∞𝑛=1Γ
(𝑛)
𝑗 and note that it is a closure of Γ𝑗 = Γ𝑗↾ domΓ, 𝑗 = 1, 2. It can be seen that the orthogonal sum
Π ∶= ⊕∞
𝑛=1Π𝑛 of the boundary triples Π𝑛 determines an 𝐸𝑆-generalized boundary triple. Moreover, in the case that 𝑑∗ = 0 the
DERKACH ET AL. 1287
Weyl function𝑀(⋅) corresponding to the triple Π = ⊕∞
𝑛=1Π𝑛 satisfies Assumption 1.16, i.e. it is domain invariant, dom𝑀(𝜆) =
dom𝑀(𝑖), 𝜆 ∈ ℂ±, while dom𝑀(𝑖) ⫋ ran Γ0. Hence, by Theorem 1.12, 𝐴0 ≠ 𝐴∗0 and Π = ⊕∞𝑛=1Π𝑛 being 𝐸𝑆-generalized, is
not an 𝑆-generalized boundary triple for H∗. In fact, with 𝑑∗ = 0 the Weyl function𝑀(⋅) as well as its imaginary part Im𝑀(⋅)
take values in the set of unbounded operators. For the details in this example we refer to Part II of the present work, where also
analogous results for moment and Dirac operators with local point interactions are established.
Notice that the minimal operatorH as well as the corresponding tripleΠ forH∗ in Example 1.17 naturally arise when treating
the Hamiltonian H𝑋,𝛼 with 𝛿-interactions in the framework of extension theory. The latter have appeared in various physical
problems as exactly solvable models that describe complicated physical phenomena (see e.g. [2, 3, 34, 48, 49] for details).
Theorem 5.32 offers a renormalization procedure which produces from a form domain invariant Weyl function a domain
invariant Weyl function, whose imaginary part becomes a well-defined bounded operator function on ℂ ⧵ℝ, i.e., the renor-
malized boundary triple is 𝐴𝐵-generalized. Some related results, showing how 𝐵-generalized boundary triples give rise to
𝐸𝑆-generalized boundary triples, are established in Part II of the present work, where these results are applied in the analysis
of regularized trace operators for Laplacians.
Before closing this subsection we wish to mention that other type of examples for 𝐸𝑆-generalized boundary triples are the
Kreı˘n – von Neumann Laplacian and the Zaremba Laplacian for a mixed boundary value problem treated in Part II of the
present work.
1.6 A short description of the contents
For the convenience of the reader in this Introduction we have restricted the exposition of the main definitions and results to
the case of generalized boundary triples, i.e. to boundary triples with a single-valued linear mapping Γ ∶ 𝐴∗ →  × which
admits a decomposition Γ =
{
Γ0,Γ1
}
, where Γ0 and Γ1 give rise to a pair of boundary conditions in (the boundary space) 
typically occurring in boundary value problems in ODE and PDE setting. In the paper itself these results are mostly presented
in a more general setting of boundary pairs, where Γ is allowed to be multi-valued. This generality unifies the presentation in
later Sections and, in fact, often simplifies the description of the particular analytic properties of Weyl functions associated with
different classes of generalized boundary triples and boundary pairs.
In Section 2 we recall basic concepts of linear relations (sums of relations, componentwise sums, defect subspaces, etc.) as
well as unitary and isometric relations in Kreı˘n space. We also introduce the concepts of Nevanlinna functions and families.
In Section 3 we discuss unitary and isometric boundary pairs and triples. We introduce the notions of Weyl functions and
families and discuss their properties. A general version of the main realization result, Theorem 3.3, is presented therein, too. It
completes and improves Theorem 1.10. Besides certain isometric transforms of boundary triples are discussed.
In Section 4 we investigate 𝐴𝐵-generalized boundary pairs and triples. Their main properties can be found in Theorem 4.2
and in various Corollaries appearing in this section. In Theorem 4.4 a connection between 𝐵-generalized and 𝐴𝐵-generalized
boundary triples is established by means of triangular isometric transformations. Connections between 𝐴𝐵-generalized bound-
ary triples and quasi boundary triples are also explained. Moreover, a Kreı˘n type formula for 𝐴𝐵-generalized boundary triples
can be found in Theorem 4.12.
In Section 5 we consider two further subclasses of unitary boundary triples and pairs: 𝑆-generalized and 𝐸𝑆-generalized
boundary triples and pairs. For deriving some of the main results in this connection we have established also some new facts on
the interaction between
(
𝐽ℌ, 𝐽
)
-unitary relations and unitary colligations appearing e.g. in system theory and in the analysis of
Schur functions, see Section 5.1; a background for this connection can be found in [10]. In particular, this connection is applied
to extend Theorem 1.12 to the case of 𝑆-generalized boundary pairs (see Theorem 5.17). In this case representation (1.12) for
the Weyl function remains valid with 𝑀0 ∈ [0] and 0 ⊆  instead of 𝑀0 ∈ 𝑠[]. In Theorem 5.24 the class of Weyl
functions of 𝐸𝑆-generalized boundary pairs is characterized. In Theorem 5.8 it is shown that every unitary boundary triple
admits a Kreı˘n type resolvent formula. Besides, in Theorem 5.32 a connection between 𝐸𝑆-generalized boundary triples and
𝐴𝐵-generalized boundary triples is established via an isometric transform introduced in Lemma 3.12 (see formula (3.23)).
2 PRELIMINARY CONCEPTS
2.1 Linear relations in Hilbert spaces
A linear relation 𝑇 from ℌ to ℌ′ is a linear subspace of ℌ ×ℌ′. Systematically a linear operator 𝑇 will be identified with its
graph. It is convenient to write 𝑇 ∶ ℌ→ ℌ′ and interpret the linear relation 𝑇 as a multi-valued linear mapping from ℌ into
1288 DERKACH ET AL.
ℌ′. Ifℌ′ = ℌ one speaks of a linear relation 𝑇 inℌ. Many basic definitions and properties associated with linear relations can
be found in [4, 16, 22].
The following notions appear throughout this paper. For a linear relation 𝑇 ∶ ℌ→ ℌ′ the symbols dom 𝑇 , ker 𝑇 , ran 𝑇 ,
mul 𝑇 and 𝑇 stand for the domain, kernel, range, multi-valued part, and closure, respectively. The inverse 𝑇 −1 is a relation
fromℌ′ toℌ defined by { {𝑓 ′, 𝑓} ∶ {𝑓, 𝑓 ′} ∈ 𝑇 }. The adjoint 𝑇 ∗ is the closed linear relation fromℌ′ toℌ defined by 𝑇 ∗ ={
{ℎ, 𝑘} ∈ ℌ′ ⊕ℌ ∶ (𝑘, 𝑓 )ℌ = (ℎ, 𝑔)ℌ′ , {𝑓, 𝑔} ∈ 𝑇
}
. The sum 𝑇1 + 𝑇2 and the componentwise sum 𝑇1+̂𝑇2 of two linear
relations 𝑇1 and 𝑇2 are defined by
𝑇1 + 𝑇2 =
{(
𝑓
𝑔 + 𝑘
)
∶
(
𝑓
𝑔
)
∈ 𝑇1,
(
𝑓
𝑘
)
∈ 𝑇2
}
,
𝑇1 +̂ 𝑇2 =
{(
𝑓 + ℎ
𝑔 + 𝑘
)
∶
(
𝑓
𝑔
)
∈ 𝑇1,
(
ℎ
𝑘
)
∈ 𝑇2
}
. (2.1)
If the componentwise sum is orthogonal it will be denoted by 𝑇1 ⊕ 𝑇2. If 𝑇 is closed, then the null spaces of 𝑇 − 𝜆, 𝜆 ∈ ℂ,
defined by
𝔑𝜆(𝑇 ) = ker (𝑇 − 𝜆), ?̂?𝜆(𝑇 ) =
{(
𝑓
𝜆𝑓
)
∈ 𝑇 ∶ 𝑓 ∈ 𝔑𝜆(𝑇 )
}
, (2.2)
are also closed. Moreover, 𝜌(𝑇 ) (?̂?(𝑇 )) stands for the set of regular (regular type) points of 𝑇 .
Recall that a linear relation 𝑇 inℌ is called symmetric, dissipative, or accumulative if Im (ℎ′, ℎ) = 0,≥ 0, or≤ 0, respectively,
holds for all {ℎ, ℎ′} ∈ 𝑇 . These properties remain invariant under closures. By polarization it follows that a linear relation 𝑇
in ℌ is symmetric if and only if 𝑇 ⊂ 𝑇 ∗. A linear relation 𝑇 in ℌ is called selfadjoint if 𝑇 = 𝑇 ∗, and it is called essentially
selfadjoint if 𝑇 = 𝑇 ∗. A dissipative (accumulative) linear relation 𝑇 in ℌ is called m-dissipative (m-accumulative) if it has no
proper dissipative (accumulative) extensions.
If the relation 𝑇 is m-dissipative (m-accumulative), then mul 𝑇 = mul 𝑇 ∗ and the orthogonal decomposition
ℌ= (mul 𝑇 )⟂⊕mul 𝑇 induces an orthogonal decomposition of 𝑇 as
𝑇 = gr 𝑇op ⊕ ({0} ×∞), ∞ = mul 𝑇 , gr 𝑇op =
{(
𝑓
𝑔
)
∈ 𝑇 ∶ 𝑔 ∈  ⊖∞
}
,
where 𝑇∞ ∶= {0} ×∞ is a purely multi-valued selfadjoint relation in ∞ and 𝑇op is a densely defined m-dissipative (resp.
m-accumulative) operator in ⊖∞. In particular, if 𝑇 is a selfadjoint relation, then there is such a decomposition, where 𝑇op
is a densely defined selfadjoint operator in  ⊖∞.
A family of linear relations𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ, in a Hilbert space  is called a Nevanlinna family if:
(i) for every 𝜆 ∈ ℂ+(ℂ−) the relation𝑀(𝜆) is m-dissipative (resp. m-accumulative);
(ii) 𝑀(𝜆)∗ = 𝑀
(
?̄?
)
, 𝜆 ∈ ℂ ⧵ℝ;
(iii) for some, and hence for all, 𝜇 ∈ ℂ+(ℂ−) the operator family (𝑀(𝜆) + 𝜇)−1(∈ []) is holomorphic for all 𝜆 ∈ ℂ+(ℂ−).
By the maximality condition, each relation 𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ, is necessarily closed. The class of all Nevanlinna families in
a Hilbert space is denoted by ̃(). If the multi-valued part mul𝑀(𝜆) of 𝑀 ∈ ̃() is nontrivial, then it is independent of
𝜆 ∈ ℂ ⧵ℝ, so that
𝑀(𝜆) = gr𝑀op(𝜆)⊕𝑀∞ ∞ = mul𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ, (2.3)
where𝑀∞ = {0} ×∞ is a purely multi-valued linear relation in ∞ ∶= mul𝑀(𝜆) and𝑀op(⋅) ∈ ( ⊖∞), cf. [51, 52,
55]. Identifying operators in  with their graphs one can consider classes
𝑢[] ⊂ 𝑠[] ⊂ 𝑠() ⊂ ()
introduced in Section 1 as subclasses of ̃(). In addition, a Nevanlinna family𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ, which admits a holomorphic
extrapolation to the negative real line (−∞, 0) (in the resolvent sense as in item (iii) of the above definition) and whose values
𝑀(𝑥) are nonnegative (nonpositive) selfadjoint relations for all 𝑥 < 0 is called a Stieltjes family (an inverse Stieltjes family,
respectively).
DERKACH ET AL. 1289
2.2 Unitary and isometric relations in Kreı˘n spaces
Let ℌ and  be Hilbert spaces and let (ℌ2, 𝐽ℌ) and (2, 𝐽) be Kreı˘n spaces with fundamental symmetries 𝐽ℌ, 𝐽 and
indefinite inner products [⋅, ⋅]ℌ, [⋅, ⋅] defined in (1.9) and (1.10), respectively. If Γ is a linear relation from the Kreı˘n space(
ℌ2, 𝐽ℌ
)
to the Kreı˘n space
(2, 𝐽), then the adjoint linear relation Γ[∗] is defined by
Γ[∗] =
{(
?̂?
𝑔
)
∈
(2
ℌ2
)
∶
[
𝑓, 𝑔
]
ℌ2 =
[
ℎ̂, ?̂?
]
2 for all
(
𝑓
ℎ̂
)
∈ Γ
}
. (2.4)
Definition 2.1. ([64]) A linear relation Γ from the Kreı˘n space (ℌ2, 𝐽ℌ) to the Kreı˘n space (2, 𝐽) is said to be (𝐽ℌ, 𝐽)-
isometric if Γ−1 ⊂ Γ[∗] and
(
𝐽ℌ, 𝐽
)
-unitary, if Γ−1 = Γ[∗].
The following two statements are due to Yu. L. Shmul’jan [64]; see also [25].
Proposition 2.2. Let Γ be a (𝐽ℌ, 𝐽)-unitary relation from the Kreı˘n space (ℌ2, 𝐽ℌ) to the Kreı˘n space (2, 𝐽). Then:
(i) domΓ is closed if and only if ran Γ is closed;
(ii) the following equalities hold: ker Γ = (domΓ)[⟂], mul Γ = (ran Γ)[⟂].
A
(
𝐽ℌ, 𝐽
)
-unitary relation Γ ∶
(
ℌ2, 𝐽ℌ
)
→
(2, 𝐽) may be multi-valued, nondensely defined, and unbounded. It is the
graph of an operator if and only if its range is dense in2. In this case it need not be densely defined or bounded; and even if it
is bounded it need not be densely defined.
3 UNITARY AND ISOMETRIC BOUNDARY PAIRS AND THEIR WEYL
FAMILIES
3.1 Definitions and basic properties
Let 𝐴 be a closed symmetric linear relation in the Hilbert space ℌ. It is not assumed that the defect numbers of 𝐴 are equal or
finite. Following [25, 27] a unitary/isometric boundary pair for 𝐴∗ is defined as follows.
Definition 3.1. Let 𝐴 be a closed symmetric linear relation in a Hilbert space ℌ, let  be an auxiliary Hilbert space and let
Γ be a linear relation from the Kreı˘n space
(
ℌ2, 𝐽ℌ
)
to the Kreı˘n space
(2, 𝐽). Then {,Γ} is called a unitary/isometric
boundary pair for 𝐴∗, if:
3.1.1 𝐴∗ ∶= domΓ is dense in 𝐴∗ with respect to the topology on ℌ2;
3.1.2 the linear relation Γ is (𝐽ℌ, 𝐽)-unitary/isometric.
In particular, it follows from this definition that for all vectors
{
𝑓, ℎ̂
}
,
{
𝑔, ?̂?
}
∈ Γ of the form (1.10) the abstract Green’s
identity (cf. Definition 1.1) holds
(𝑓 ′, 𝑔)ℌ − (𝑓, 𝑔′)ℌ = (ℎ′, 𝑘) − (ℎ, 𝑘′) . (3.1)
Let {,Γ} be a unitary boundary pair for 𝐴∗ and let 𝐴∗ = domΓ. According to [25, Proposition 2.12] the domain 𝐴∗ of Γ
is a linear relation in ℌ, such that
𝐴 ⊂ 𝐴∗ ⊂ 𝐴
∗, 𝐴∗ = 𝐴∗.
The eigenspaces 𝔑𝜆(𝐴∗) and ?̂?𝜆(𝐴∗) of 𝐴∗ are defined as in (2.2),
𝔑𝜆(𝐴∗) = ker (𝐴∗ − 𝜆), ?̂?𝜆(𝐴∗) =
{(
𝑓𝜆
𝜆𝑓𝜆
)
∈ 𝐴∗ ∶ 𝑓𝜆 ∈ 𝔑𝜆(𝐴∗)
}
.
1290 DERKACH ET AL.
Definition 3.2. The Weyl family 𝑀 of 𝐴 corresponding to the unitary or isometric boundary pair {,Γ} is defined by
𝑀(𝜆) ∶=Γ
(
?̂?𝜆(𝐴∗)
)
, i.e.,
𝑀(𝜆) ∶=
{
ℎ̂ ∈ 2 ∶ {𝑓𝜆, ℎ̂} ∈ Γ for some 𝑓𝜆 = ( 𝑓𝜆𝜆𝑓𝜆
)
∈ ℌ2
}
(𝜆 ∈ ℂ ⧵ℝ).
In the case where𝑀 is single-valued it is called theWeyl function of𝐴 corresponding to {,Γ}. The 𝛾-field of𝐴 corresponding
to the unitary/isometric boundary pair {,Γ} is defined by
𝛾(𝜆) ∶=
{{
ℎ, 𝑓𝜆
}
∈  ×ℌ ∶
{(
𝑓𝜆
𝜆𝑓𝜆
)
,
(
ℎ
ℎ′
)}
∈ Γ for some ℎ′ ∈ 
}
,
where 𝜆 ∈ ℂ ⧵ℝ. Moreover, ?̂?(𝜆) stands for
?̂?(𝜆) ∶=
{{
ℎ, 𝑓𝜆
}
∈  ×ℌ2 ∶
{
𝑓𝜆,
(
ℎ
ℎ′
)}
∈ Γ for some ℎ′ ∈ 
}
. (3.2)
With 𝛾(𝜆) the relation Γ↾ ?̂?𝜆(𝐴∗) can be rewritten as follows
Γ↾ ?̂?𝜆(𝐴∗) ∶=
{{(
𝛾(𝜆)ℎ
𝜆𝛾(𝜆)ℎ
)
,
(
ℎ
ℎ′
)}
∶
(
ℎ
ℎ′
)
∈ 𝑀(𝜆)
}
, 𝜆 ∈ ℂ ⧵ℝ. (3.3)
Associate with Γ the following two linear relations which are not necessarily closed:
Γ0 =
{{
𝑓, ℎ
}
∶
{
𝑓, ℎ̂
}
∈ Γ, ℎ̂ =
(
ℎ
ℎ′
)}
, Γ1 =
{{
𝑓, ℎ′
}
∶
{
𝑓, ℎ̂
}
∈ Γ, ℎ̂ =
(
ℎ
ℎ′
)}
. (3.4)
The 𝛾-field 𝛾(⋅) associated with {,Γ} is the first component of the mapping ?̂?(𝜆) in (3.2). Observe, that
?̂?(𝜆) ∶=
(
Γ0↾ ?̂?𝜆(𝐴∗)
)−1
, 𝜆 ∈ ℂ ⧵ℝ,
is a linear mapping from Γ0
(
?̂?𝜆(𝐴∗)
)
= dom𝑀(𝜆) onto ?̂?𝜆(𝐴∗): it is single-valued in view of (3.1); cf. (3.7), (3.8). Conse-
quently, the 𝛾-field is a single-valued mapping from dom𝑀(𝜆) onto𝔑𝜆(𝐴∗) and it satisfies 𝛾(𝜆)Γ0𝑓𝜆 = 𝑓𝜆 for all 𝑓𝜆 ∈ ?̂?𝜆(𝐴∗).
If Γ is single-valued then these component mappings decompose Γ, Γ = Γ0 × Γ1, and the triple
{,Γ0,Γ1} will be called a
unitary/isometric boundary triple for 𝐴∗. In this case the Weyl function corresponding to the unitary/isometric boundary triple{,Γ0,Γ1} can be also defined via
𝑀(𝜆)Γ0𝑓𝜆 = Γ1𝑓𝜆, 𝑓𝜆 ∈ ?̂?𝜆(𝐴∗). (3.5)
When 𝐴 admits real regular type points it is useful to extend Definition 3.2 of the Weyl family to the points on the real line by
setting𝑀(𝑥) ∶= Γ
(
?̂?𝑥(𝐴∗)
)
or, more precisely,
𝑀(𝑥) ∶=
{
ℎ̂ ∈ 2 ∶ {𝑓𝑥, ℎ̂} ∈ Γ for some 𝑓𝑥 = ( 𝑓𝑥𝑥𝑓𝑥
)
∈ ℌ2, 𝑥 ∈ ℝ
}
.
3.2 Unitary boundary pairs and unitary boundary triples
The following theorem shows that the set of all Weyl families of unitary boundary pairs coincides with ̃() (see [25, Theo-
rem 3.9]). Recall that a unitary boundary pair {,Γ} for 𝐴∗ is said to be minimal, if
ℌ = ℌmin ∶= span {𝔑𝜆(𝐴∗) ∶ 𝜆 ∈ ℂ+ ∪ ℂ− }.
Theorem 3.3. Let {,Γ} be a unitary boundary pair for 𝐴∗. Then the corresponding Weyl family 𝑀 belongs to the class of
Nevanlinna families ̃().
Conversely, if𝑀 belongs to the class ̃(), then there exists a unique (up to a unitary equivalence) minimal unitary boundary
pair {,Γ} whose Weyl function coincides with𝑀 .
DERKACH ET AL. 1291
Notice that Theorem 1.10 contains a general analytic criterion for an isometric boundary triple to be unitary; theWeyl function
should be a Nevanlinna function, cf. Theorem 1.10.
Corollary 3.4. The class of Weyl functions corresponding to unitary boundary triples coincides with the class 𝑠() of (in
general unbounded) strict Nevanlinna functions.
Proof. The statement is immediate when combining Theorem 3.3 with Proposition 4.5 from [25]. □
As a consequence of (3.1) and (3.3) the following identity holds (cf. (2.3))
(𝜆 − ?̄?)(𝛾(𝜆)ℎ, 𝛾(𝜇)𝑘)ℌ =
(
𝑀op(𝜆)ℎ, 𝑘
)
 −
(
ℎ,𝑀op(𝜇)𝑘
)
 , (3.6)
where ℎ ∈ dom𝑀(𝜆) and 𝑘 ∈ dom𝑀(𝜇), 𝜆, 𝜇 ∈ ℂ ⧵ℝ.
As was already mentioned in Section 1 every operator valued function 𝑀 from 𝑢[] (𝑠[]) can be realized as a Weyl
function of some ordinary boundary triple (𝐵-generalized boundary triple, respectively).
The multi-valued analog for the notion of 𝐵-generalized boundary triple was introduced in [25, Section 5.3], a formal defini-
tion reads as follows.
Definition 3.5. Let 𝐴 be a symmetric operator (or relation) in the Hilbert space ℌ and let  be another Hilbert space. Then a
linear relation Γ ∶ 𝐴∗ →  ⊕ with dense domain in 𝐴∗ is said to be a 𝐵-generalized boundary pair for 𝐴∗, if the following
three conditions are satisfied:
3.5.1 the abstract Green’s identity (3.1) holds;
3.5.2 ran Γ0 = ;
3.5.3 𝐴0 = ker Γ0 is selfadjoint,
where Γ0 stands for the first component of Γ; see (3.4).
As was shown in [25, Proposition 5.9] every Weyl function of a 𝐵-generalized boundary pair belongs to the class[] and,
conversely, every operator valued function𝑀 ∈ [] can be realized as the Weyl function of a 𝐵-generalized boundary pair.
3.3 Isometric boundary pairs and isometric boundary triples
Let Γ be a
(
𝐽ℌ, 𝐽
)
-isometric relation from the Kreı˘n space
(
ℌ2, 𝐽ℌ
)
to the Kreı˘n space
(2, 𝐽). In view of (1.9)–(1.11) this
just means that the abstract Green’s identity (3.1) holds. It follows from (3.1) that
ker Γ ⊂ (domΓ)[⟂], mul Γ ⊂ (ran Γ)[⟂],
compare Proposition 2.2. Let Γ0 and Γ1 be the linear relations determined by (3.4). The kernels 𝐴0 ∶= ker Γ0 and 𝐴1 ∶= ker Γ1
need not be closed, but they are symmetric extensions of ker Γ which are contained in the domain 𝐴∗ = domΓ of Γ; cf. [25,
Proposition 2.13]. If 𝐴∗ = domΓ is dense in 𝐴∗ then the pair {,Γ} is viewed as an isometric boundary pair for 𝐴∗; cf.
Definition 3.1. In general𝐴 ∶= (𝐴∗)∗ = (domΓ)[⟂] is an extension of ker Γwhich need not belong to domΓ; for some sufficient
conditions for the equality 𝐴 = ker Γ, see [26, Section 2.3] and [27, Section 7.8].
With
{
𝑓𝜆, ℎ̂
}
,
{
𝑔𝜇, ?̂?
}
∈ Γ, 𝜆, 𝜇 ∈ ℂ, Green’s identity (3.1) gives, cf. (3.6),
(ℎ′, 𝑘) − (ℎ, 𝑘′) = (𝜆 − ?̄?)
(
𝑓𝜆, 𝑔𝜇
)
ℌ. (3.7)
In particular, with 𝜇 = 𝜆 (3.7) implies that Im (ℎ′, ℎ) = Im 𝜆‖𝑓𝜆‖2. Hence, for all 𝜆 ∈ ℂ ⧵ℝ
ker
(
Γ ↾ ?̂?𝜆(𝐴∗)
)
= {0} and ker
(
Γ𝑗 ↾ ?̂?𝜆(𝐴∗)
)
= {0} (𝑗 = 0, 1). (3.8)
Moreover, with 𝜇 = ?̄? (3.7) implies that
𝑀
(
?̄?
)
⊆ 𝑀(𝜆)∗, 𝜆 ∈ ℂ ⧵ℝ. (3.9)
Here equality does not hold if Γ is not unitary. However, with the Weyl family the multi-valued part of Γ can be described
explicitly; see [27, Lemma 7.57], cf. also [25, Lemma 4.1].
1292 DERKACH ET AL.
Lemma 3.6. Let {,Γ} be an isometric boundary pair with the Weyl family 𝑀 . Then the following equalities hold for all
𝜆 ∈ ℂ ⧵ℝ:
(i) 𝑀(𝜆) ∩𝑀(𝜆)∗ = mul Γ;
(ii) ker𝑀(𝜆) × {0} = mul Γ ∩ ( × {0});
(iii) {0} × mul𝑀(𝜆) = mul Γ ∩ ({0} ×);
(iv) ker(𝑀(𝜆) −𝑀(𝜆)∗) = mul Γ0;
(v) ker
(
𝑀(𝜆)−1 −𝑀(𝜆)−∗
)
= mul Γ1.
If Γ itself is single-valued, then the Weyl family𝑀 is an operator valued function, i.e. mul𝑀(𝜆) = 0, belonging to the class
𝑠(), see [25, Proposition 4.5]. Moreover, ker Im (𝑀(𝜆)) = {0} and ker Im (𝑀(𝜆)−1) = {0}, in particular, ker𝑀(𝜆) = 0.
Recall that when Γ is single-valued 𝑀(𝜆) can equivalently be defined by the equality (3.5). Hence, if ℎ ∈  is given
and ℎ∈Γ0
(
?̂?𝜆(𝐴∗)
)
, then 𝛾(𝜆)ℎ solves a boundary eigenvalue problem, i.e., 𝛾(𝜆)ℎ ∈ ker(𝐴∗ − 𝜆) and Γ0?̂?(𝜆)ℎ = ℎ, while
Γ1?̂?(𝜆)ℎ = 𝑀(𝜆)ℎ. Also for an operator valued function𝑀(⋅) the identity (3.7) can be rewritten in the form
(𝜆 − ?̄?)(𝛾(𝜆)ℎ, 𝛾(𝜇)𝑘)ℌ = (𝑀(𝜆)ℎ, 𝑘) − (ℎ,𝑀(𝜇)𝑘) , (3.10)
where ℎ ∈ dom𝑀(𝜆) and 𝑘 ∈ dom𝑀(𝜇), 𝜆, 𝜇 ∈ ℂ ⧵ℝ. This is an analog of (3.6) for an isometric boundary triple.
Let Γ be an isometric relation and let 𝐴0 = ker Γ0. Then 𝐴0 is a symmetric, not necessarily closed, relation and one can write
for every 𝜆 ∈ ℂ ⧵ℝ,
𝐴0 =
{( (
𝐴0 − 𝜆
)−1
ℎ
ℎ + 𝜆
(
𝐴0 − 𝜆
)−1
ℎ
)
∶ ℎ ∈ ran
(
𝐴0 − 𝜆
)}
.
The linear mapping
𝐻(𝜆) ∶ ℎ→
{( (
𝐴0 − 𝜆
)−1
ℎ
ℎ + 𝜆
(
𝐴0 − 𝜆
)−1
ℎ
)}
(3.11)
from ran
(
𝐴0 − 𝜆
)
onto 𝐴0 is clearly bounded with bounded inverse.
Lemma 3.7. Let {,Γ} be an isometric boundary pair and let 𝐴0 = ker Γ0. Then the following assertions hold:
(i) Γ1𝐻(𝜆) is closable for one (equivalently for all) 𝜆 ∈ ℂ ⧵ℝ if and only if Γ1↾𝐴0 is closable;
(ii) Γ1𝐻(𝜆) is closed for one (equivalently for all) 𝜆 ∈ ℂ ⧵ℝ if and only if Γ1↾𝐴0 is closed;
(iii) Γ1𝐻(𝜆) is a bounded operator for one (equivalently for all) 𝜆 ∈ ℂ ⧵ℝ if and only if Γ1↾𝐴0 is a bounded operator;
(iv) domΓ1𝐻(𝜆) is dense in ℌ for some (equivalently for all) 𝜆, ?̄? ∈ ℂ ⧵ℝ if and only if 𝐴0 is essentially selfadjoint;
(v) domΓ1𝐻(𝜆) = ℌ for some (equivalently for all) 𝜆, ?̄? ∈ ℂ ⧵ℝ if and only if 𝐴0 is selfadjoint;
(vi) ran Γ1𝐻(𝜆) = Γ1
(
𝐴0
) [
= ran
(
Γ1↾𝐴0
)]
for all 𝜆 ∈ ℂ ⧵ℝ.
Proof. By definition 𝐴0 = ker Γ0 ⊂ domΓ1, so that domΓ1𝐻(𝜆) = ran (𝐴0 − 𝜆), 𝜆 ∈ ℂ ⧵ℝ. Since𝐻(𝜆) ∶ ran
(
𝐴0 − 𝜆
)
→𝐴0
is bounded with bounded inverse, all the statements are easily obtained by means of the equality Γ1↾𝐴0 =(
Γ1𝐻(𝜆)
)
𝐻(𝜆)−1. □
Similar facts can be stated for the restriction Γ0↾𝐴1, where 𝐴1 = ker Γ1.
The inclusion (3.13) in the next proposition was stated for a single-valued Γ with dense range in [27, Proposition 7.59]; here
a direct proof for this inclusion is given in the general case.
Lemma 3.8. Let {,Γ} be an isometric boundary pair, let 𝛾(𝜆) be its 𝛾-field, and let𝐻(𝜆) be as defined in (3.11). Then
Γ𝐻(𝜆) ⊂
(
0
𝛾
(
?̄?
)∗) +̂ ({0} × mul Γ ), 𝜆 ∈ ℂ ⧵ℝ, (3.12)
DERKACH ET AL. 1293
where the adjoint 𝛾
(
?̄?
)∗ of 𝛾(?̄?) is in general a linear relation. In particular,
Γ1𝐻(𝜆) ⊂ 𝛾
(
?̄?
)∗ +̂ ({0} × mul Γ1), 𝜆 ∈ ℂ ⧵ℝ, (3.13)
and if, in addition, mul Γ1 = {0}, then
Γ1𝐻(𝜆) ⊂ 𝛾
(
?̄?
)∗
, 𝜆 ∈ ℂ ⧵ℝ. (3.14)
Furthermore, the following statements hold:
(i) if 𝛾
(
?̄?
)
is densely defined for some ?̄? ∈ ℂ ⧵ℝ, then 𝛾
(
?̄?
)∗ is a closed operator and if, in addition, mul Γ1 = {0}, then
Γ1𝐻(𝜆) is a closable operator;
(ii) if 𝐴0 = ker Γ0 is essentially selfadjoint, then 𝛾
(
?̄?
)
is closable for all 𝜆 ∈ ℂ ⧵ℝ;
(iii) if 𝐴0 = ker Γ0 is selfadjoint, then dom 𝛾
(
?̄?
)∗ = ℌ and 𝛾(?̄?) is a bounded operator for all 𝜆 ∈ ℂ ⧵ℝ.
Proof. Let ℎ ∈ dom 𝛾
(
?̄?
)
= dom𝑀
(
?̄?
)
and 𝑘𝜆 ∈ ran
(
𝐴0 − 𝜆
)
. Then
{
?̂?
(
?̄?
)
ℎ, {ℎ, ℎ′}
}
∈ Γ and, since 𝐻(𝜆)𝑘𝜆 ∈ 𝐴0 =
ker Γ0, one has the inclusion
{
𝐻
(
𝜆
)
𝑘𝜆, {0, 𝑘′′}
}
∈ Γ for some 𝑘′′ ∈ . On the other hand, {𝑘𝜆, 𝑘′} ∈ Γ1𝐻(𝜆) means that{
𝑘𝜆, {𝑘, 𝑘′}
}
∈ Γ𝐻(𝜆) for some 𝑘 ∈  which combined with {𝐻(𝜆)𝑘𝜆, {0, 𝑘′′}} ∈ Γ implies that {{0, 0}, {𝑘, 𝑘′ − 𝑘′′}} ∈ Γ.
Now applying Green’s identity (3.1) shows that(
?̄?𝛾
(
?̄?
)
ℎ,
(
𝐴0 − 𝜆
)−1
𝑘𝜆
)
−
(
𝛾
(
?̄?
)
ℎ,
(
𝐼 + 𝜆
(
𝐴0 − 𝜆
)−1)
𝑘𝜆
)
= 0 −
(
ℎ, 𝑘′′
)
 .
This identity can be rewritten equivalently in the form(
𝛾
(
?̄?
)
ℎ, 𝑘𝜆
)
= (ℎ, 𝑘′′)
for all ℎ ∈ dom 𝛾
(
?̄?
)
and 𝑘𝜆 ∈ ran
(
𝐴0 − 𝜆
)
. This proves that
{
𝑘𝜆, 𝑘
′′} ∈ 𝛾(?̄?)∗. Hence, if{𝑘𝜆,(𝑘𝑘′
)}
∈ Γ𝐻(𝜆) then
{
𝑘𝜆,
(
𝑘
𝑘′
)}
=
{
𝑘𝜆,
(
0
𝑘′′
)}
+
{
0,
(
𝑘
𝑘′ − 𝑘′′
)}
(3.15)
with
{
𝑘𝜆, 𝑘
′′} ∈ 𝛾(?̄?)∗ and {𝑘, 𝑘′ − 𝑘′′} ∈ mul Γ from which the formulas (3.12) and (3.13) follow. If mul Γ1 = {0}, then
{𝑘, 𝑘′ − 𝑘′′} ∈ mul Γ implies that 𝑘′ = 𝑘′′ and therefore the above argument shows that
{
𝑘𝜆, 𝑘
′} ∈ 𝛾(?̄?)∗ for all {𝑘𝜆, 𝑘′} ∈
Γ1𝐻(𝜆); i.e. (3.14) is satisfied.
It remains to prove the statements (i)–(iii).
(i) If 𝛾
(
?̄?
)
is densely defined then clearly 𝛾
(
?̄?
)∗
is a closed operator and if Γ1 is single-valued then (3.14) shows that Γ1𝐻(𝜆)
is closable as a restriction of 𝛾
(
?̄?
)∗
.
(ii) By Lemma 3.7 𝐴0 is essentially selfadjoint if and only if Γ1𝐻(𝜆) is densely defined, in which case also 𝛾
(
?̄?
)∗
is densely
defined, so that 𝛾
(
?̄?
)
is closable.
(iii) If 𝐴0 is selfadjoint, then domΓ1𝐻(𝜆) = ℌ and, therefore, also dom 𝛾(?̄?)∗ = ℌ. In addition 𝛾
(
?̄?
)
is closable, thus clos 𝛾
(
?̄?
)
and 𝛾(?̄?
)
are bounded operators.
□
Proposition 3.9. Let 𝐴 be a closed symmetric relation in the Hilbert space ℌ and let {,Γ} be an isometric boundary pair,
let 𝑀(⋅) and 𝛾(⋅) be the corresponding Weyl function and the 𝛾-field and, in addition, assume that 𝐴0 = ker Γ0 is selfadjoint.
Then:
(i) 𝐴∗ ∶= domΓ admits the decomposition 𝐴∗ = 𝐴0 +̂ ?̂?𝜆(𝐴∗) and ?̂?𝜆(𝐴∗) is dense in ?̂?𝜆(𝐴∗) for all 𝜆 ∈ ℂ ⧵ℝ;
(ii) with a fixed 𝜆 ∈ ℂ ⧵ℝ the graph of Γ admits the following representation:
Γ = Γ𝐴0 +̂
{{(
𝛾(𝜆)ℎ
𝜆𝛾(𝜆)ℎ
)
,
(
ℎ
ℎ′
)}
∶
(
ℎ
ℎ′
)
∈ 𝑀(𝜆)
}
;
1294 DERKACH ET AL.
(iii) if Γ̃ ∶
(
ℌ2, 𝐽ℌ
)
→
(2, 𝐽) is an isometric extension of Γ with the Weyl function 𝑀 and the 𝛾-field ?̃?(⋅) such that
𝐴∗ ∶= dom Γ̃⊂𝐴∗, then with a fixed 𝜆 ∈ ℂ ⧵ℝ the following equivalence holds:
Γ̃ = Γ ⇔ 𝑀(𝜆) = 𝑀(𝜆).
Proof.
(i) By von Neumann’s formula 𝐴∗ = 𝐴0 +̂ ?̂?𝜆(𝐴∗). Since 𝐴∗ ∶= domΓ is dense in 𝐴∗ and 𝐴0 ⊂ 𝐴∗, it follows that
𝐴∗ =𝐴0 +̂ ?̂?𝜆(𝐴∗) and that ?̂?𝜆(𝐴∗) is dense in ?̂?𝜆(𝐴∗) for every 𝜆 ∈ ℂ ⧵ℝ.
(ii) In view of (i) for every
{
𝑓, ?̂?
}
∈ Γ there exist unique elements 𝑓0 ∈ 𝐴0 and 𝑓𝜆 ∈ ?̂?𝜆(𝐴∗), 𝜆 ∈ ℂ ⧵ℝ, such that
𝑓 = 𝑓0 + 𝑓𝜆. Moreover, if
{
𝑓𝜆, ℎ̂
}
∈ Γ then ℎ̂ = {ℎ, ℎ′} ∈ 𝑀(𝜆) and one can write (uniquely) 𝑓𝜆 = ?̂?(𝜆)ℎ; see (3.3). The
stated representation for Γ is now clear.
(iii) It follows from Γ ⊂ Γ̃ that 𝐴0 ⊂ ker Γ̃0. Since ker Γ̃0 is symmetric and 𝐴0 is selfadjoint, the equality 𝐴0 = ker Γ̃0 holds.
Now recall that two linear relations with𝐻1 ⊂ 𝐻2 are equal precisely when the equalities dom𝐻1 = dom𝐻2 andmul𝐻1 =
mul𝐻2 hold; see [4]. By Lemma 3.6 (i) mul Γ = 𝑀(𝜆) ∩𝑀(𝜆)∗. Therefore, 𝑀(𝜆) = 𝑀(𝜆) implies that mul Γ̃ = mul Γ.
Moreover, we have dom𝑀(𝜆) = dom𝑀(𝜆) and, since ̂̃𝛾(𝜆)maps dom𝑀(𝜆) onto ?̂?𝜆
(
𝐴∗
)
and ?̂?(𝜆)maps dom𝑀(𝜆) onto
?̂?𝜆(𝐴∗), we conclude from (i) that dom Γ̃ = domΓ. Therefore, 𝑀(𝜆) = 𝑀(𝜆) implies Γ̃ = Γ. The reverse implication is
clear. □
The Weyl function of an isometric or a unitary boundary pair takes values which need not be invertible, and in general can be
unbounded, possibly multi-valued, operators. In what follows Weyl functions𝑀(𝜆), whose domain (or form domain) does not
dependent on 𝜆 ∈ ℂ ⧵ℝ are of special interest. Here a characterization for domain invariant Weyl families will be established.
We start with the next lemma concerning the domain inclusion dom𝑀(𝜆) ⊂ dom𝑀(𝜇).
Lemma 3.10. Let {,Γ} be an isometric boundary pair with 𝐴∗ = domΓ, let𝑀(⋅) and 𝛾(⋅) be the corresponding Weyl family
and the 𝛾-field, and let 𝐴0 = ker Γ0. Then for each fixed 𝜆, 𝜇 ∈ ℂ ⧵ℝ with 𝜆 ≠ 𝜇 the inclusion
dom𝑀(𝜇) ⊂ dom𝑀(𝜆) (3.16)
is equivalent to the inclusion
ran 𝛾(𝜇) ⊂ ran
(
𝐴0 − 𝜆
)
. (3.17)
If one of these conditions is satisfied, then the 𝛾-field 𝛾(⋅) satisfies the identity
𝛾(𝜆)ℎ =
[
𝐼 + (𝜆 − 𝜇)
(
𝐴0 − 𝜆
)−1]
𝛾(𝜇)ℎ, ℎ ∈ dom 𝛾(𝜇). (3.18)
Proof. By Definition 3.2 dom𝑀(𝜆) = dom 𝛾(𝜆) = Γ0
(
?̂?𝜆(𝐴∗)
)
and, moreover, ran 𝛾(𝜆) = 𝔑𝜆(𝐴∗), 𝜆 ∈ ℂ ⧵ℝ. Now assume
that (3.16) holds and let ℎ ∈ dom𝑀(𝜇) ⊂ dom𝑀(𝜆). It follows from (3.2) that there exist ℎ′, ℎ′′ ∈  such that{(
𝛾(𝜆)ℎ
𝜆𝛾(𝜆)ℎ
)
,
(
ℎ
ℎ′
)}
∈ Γ↾ ?̂?𝜆(𝑇 ) ⊂ Γ,
{(
𝛾(𝜇)ℎ
𝜇𝛾(𝜇)ℎ
)
,
(
ℎ
ℎ′′
)}
∈ Γ↾ ?̂?𝜇(𝑇 ) ⊂ Γ.
This implies {(
(𝛾(𝜆) − 𝛾(𝜇))ℎ
(𝜆𝛾(𝜆) − 𝜇𝛾(𝜇))ℎ
)
,
(
0
ℎ′ − ℎ′′
)}
∈ Γ,
and hence (
(𝛾(𝜆) − 𝛾(𝜇))ℎ
(𝜆𝛾(𝜆) − 𝜇𝛾(𝜇))ℎ
)
∈ 𝐴0 and
(
(𝛾(𝜆) − 𝛾(𝜇))ℎ
(𝜆 − 𝜇)𝛾(𝜇)ℎ
)
∈ 𝐴0 − 𝜆. (3.19)
Therefore, 𝛾(𝜇)ℎ ∈ ran
(
𝐴0 − 𝜆
)
for every ℎ ∈ dom𝑀(𝜇) and thus (3.17) follows.
DERKACH ET AL. 1295
Conversely, assume that (3.17) holds and let ℎ ∈ dom𝑀(𝜇) = dom 𝛾(𝜇). This implies that{(
𝛾(𝜇)ℎ
𝜇𝛾(𝜇)ℎ
)
,
(
ℎ
ℎ′
)}
∈ Γ (3.20)
for some ℎ′ ∈ . Moreover, since 𝛾(𝜇)ℎ ∈ ran(𝐴0 − 𝜆), there exists an element 𝑘 ∈ ℌ such that {𝑘, 𝛾(𝜇)ℎ + 𝜆𝑘} ∈ 𝐴0 =
ker Γ0. Consequently, there exists 𝜑 ∈  such that{(
(𝜆 − 𝜇)𝑘
(𝜆 − 𝜇)𝛾(𝜇)ℎ + 𝜆(𝜆 − 𝜇)𝑘
)
,
(
0
𝜑
)}
∈ Γ. (3.21)
It follows from (3.20) and (3.21) that {(
𝛾(𝜇)ℎ + (𝜆 − 𝜇)𝑘
𝜆(𝛾(𝜇)ℎ + (𝜆 − 𝜇)𝑘)
)
,
(
ℎ
ℎ′ + 𝜑
)}
∈ Γ.
Therefore, ℎ ∈ Γ0
(
?̂?𝜆(𝐴∗)
)
= dom𝑀(𝜆). This proves the inclusion (3.16).
Finally, observe that the assumption (3.16) implies (3.19). Since 𝐴0 is symmetric,
(
𝐴0 − 𝜆
)−1
is a bounded operator on
ran
(
𝐴0 − 𝜆
)
and, thus, (3.19) leads to (3.18). □
The next result characterizes domain invariance of the Weyl family corresponding to an arbitrary isometric boundary pair
{,Γ}. In the special case of a unitary boundary pair {,Γ} items (i) and (iii) contain [25, Proposition 4.11, Corollary 4.12].
Proposition 3.11. Let the assumptions and notations be as in Lemma 3.10. Then:
(i) dom𝑀(𝜆) is independent from 𝜆 ∈ ℂ+ (resp. from 𝜆 ∈ ℂ−) if and only if
𝔑𝜇(𝐴∗) ⊂ ran
(
𝐴0 − 𝜆
)
for all 𝜆, 𝜇 ∈ ℂ+ (resp. for all 𝜆, 𝜇 ∈ ℂ−), 𝜆 ≠ 𝜇,
in this case the 𝛾-field 𝛾(⋅) satisfies
𝛾(𝜆) =
[
𝐼 + (𝜆 − 𝜇)
(
𝐴0 − 𝜆
)−1]
𝛾(𝜇), 𝜆, 𝜇 ∈ ℂ+(ℂ−);
(ii) if 𝐴0 is selfadjoint, then dom𝑀(𝜆) does not dependent on 𝜆 ∈ ℂ ⧵ℝ;
(iii) if dom𝑀(𝜆) does not dependent on 𝜆 ∈ ℂ ⧵ℝ, then 𝐴0 is essentially selfadjoint.
Proof. The assertions (i) and (ii) follow directly from Lemma 3.10.
To see (iii) one can use the same argument that is presented in [25, Corollary 4.12]. □
3.4 Some transforms of boundary triples
In this subsection a specific transform of isometric boundary triples is treated. In what follows such transforms are used repeat-
edly and, in fact, they appear also in concrete boundary value problems in ODE and PDE setting. To formulate a general result
in the abstract setting consider in the Kreı˘n space
(2, 𝐽) the transformation operator 𝑉 whose action is determined by the
triangular operator
𝑉 =
(
𝐺−1 0
𝐸𝐺−1 𝐺∗
)
, 𝐸 ⊂ 𝐸∗, dom𝐸 = dom𝐺 = ran𝐺 = , ker 𝐺 = {0}. (3.22)
By assumptions on 𝐺 one has ker 𝐺∗ = mul𝐺∗ = {0}, so that the adjoint 𝐺∗ is an injective operator in . To keep a wider
generality, 𝐺 is not assumed to be a closed operator, while in applications that will often be the case. In particular, it is possible
that𝐺∗ is not densely defined and also its range need not be dense. Since𝐸 is a densely defined symmetric operator, it is closable
and its closure 𝐸 ⊂ 𝐸∗ is also symmetric. With the assumptions on 𝑉 in (3.22) a direct calculation shows that(
𝐽𝑉 𝑓, 𝑉 𝑔
)
2 =
(
𝐽𝑓, 𝑔
)
2 , 𝑓 , 𝑔 ∈ dom𝑉 .
Hence, 𝑉 is an isometric operator in theKreı˘n space
(2, 𝐽). Moreover,𝑉 is injective. These observations lead to the following
(unbounded) extension of [26, Proposition 3.18].
1296 DERKACH ET AL.
Lemma 3.12. Let {,Γ0,Γ1} be an isometric boundary triple for 𝐴∗ such that ker Γ = 𝐴, let 𝛾(𝜆) and 𝑀(𝜆) be the corre-
sponding 𝛾-field and the Weyl function, and let 𝑉 be as defined in (3.22). Then 𝑉 is isometric in the Kreı˘n space
(2, 𝐽) and
moreover:
(i) the transform Γ̃ = 𝑉 ◦Γ (
Γ̃0𝑓
Γ̃1𝑓
)
=
(
𝐺−1Γ0𝑓
𝐸𝐺−1Γ0𝑓 + 𝐺∗Γ1𝑓
)
, 𝑓 ∈ domΓ, (3.23)
defines an isometric boundary triple with domain 𝐴∗ ∶= dom Γ̃ and kernel ker Γ̃ = 𝐴;
(ii) the 𝛾-field and the Weyl function of Γ̃ are in general unbounded nondensely defined operators given by
?̃?(𝜆)𝑘 = 𝛾(𝜆)𝐺𝑘, 𝑀(𝜆)𝑘 = 𝐸𝑘 + 𝐺∗𝑀(𝜆)𝐺𝑘, 𝑘 ∈ dom𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ.
Proof.
(i) By the assumptions in (3.22) 𝑉 is an isometric operator in the Kreı˘n space
(2, 𝐽) and since Γ is an isometric operator
from
(
ℌ2, 𝐽ℌ
)
to
(2, 𝐽) the composition operator 𝑉 ◦Γ is also an isometric operator from (ℌ2, 𝐽ℌ) to (2, 𝐽). Since
𝑉 is injective, one has ker Γ̃ = ker Γ = 𝐴. In general 𝑉 is not everywhere defined, so that 𝐴∗ is typically a proper linear
subset of 𝐴∗ = domΓ which is not necessarily dense in 𝐴∗.
(ii) By Definition 3.2 the Weyl function𝑀(𝜆) of Γ̃ is given by𝑀(𝜆) = 𝑉 ◦𝑀(𝜆) or, more explicitly, by
𝑀(𝜆) =
{(
𝐺−1ℎ
𝐸𝐺−1ℎ + 𝐺∗𝑀(𝜆)ℎ
)
∶ ℎ ∈ dom𝐸𝐺−1 ∩ dom𝐺∗𝑀(𝜆)
}
=
{(
𝑘
𝐸𝑘 + 𝐺∗𝑀(𝜆)𝐺𝑘
)
∶ ℎ = 𝐺𝑘 ∈ dom𝐺
∗𝑀(𝜆),
𝑘 ∈ dom𝐺 ∩ dom𝐸
}
= 𝐸 + 𝐺∗𝑀(𝜆)𝐺.
Similarly,
(
𝐺−1Γ0↾ ?̂?𝜆
(
𝐴∗
))−1
=
(
Γ0↾ ?̂?𝜆
(
𝐴∗
))−1
𝐺 implies that ?̃?(𝜆) = 𝛾(𝜆)𝐺 with dom ?̃?(𝜆) = dom𝑀(𝜆).
□
Example 3.13.
(i) If 𝐺 = 𝐼 then the condition Γ̃1𝑓 = 0 reads as Γ1𝑓 + 𝐸 Γ0𝑓 = 0. In applications such conditions are called Robin type
boundary conditions. This corresponds to the transposed boundary triple
{,Γ1 + 𝐸 Γ0,−Γ0} which is also isometric and
has −(𝑀(𝜆) + 𝐸)−1 as its Weyl function.
(ii) As indicated𝐺 need not be closable. An extreme situation appears when𝐺 is a singular operator; cf. [47]. By definition this
means that dom𝐺 ⊂ ker 𝐺 or, equivalently, that ran𝐺 ⊂ mul𝐺. Thus, in this case dom𝐺∗ = ran𝐺∗ = {0}. If, for instance,
Γ is an ordinary boundary triple for 𝐴∗ then 𝐴0 = ker Γ0 and 𝐴1 = ker Γ1 are selfadjoint. It is easy to check that
𝐴∗ =
{
𝑓 ∈ 𝐴∗ ∶ Γ1𝑓 = 0
}
= ker Γ1 = 𝐴1, ker Γ̃0 = 𝐴0 ∩ 𝐴1 = 𝐴.
Moreover, ran Γ̃ = 𝐸↾ dom𝐺 is a symmetric operator in and dom𝑀(𝜆) = dom ?̃?(𝜆) is trivial.
3.5 Some additional remarks
Despite of the fact that the paper [21] has been quoted byM.G. Kreı˘n [50] and a discussion on [21] appears in themonograph [23]
the actual results of Calkin on reduction operators remained widely unknown among experts in extension theory. Apparently
this was caused by the fact that the paper [21] was ahead of time – it was using the new language of binary linear relations with
hidden ideas on geometry of indefinite inner product spaces, concepts which were not well developed at that time. The concept
of a bounded reduction operator investigated therein (see [21, Chapter IV]) essentially covers the notion of an ordinary boundary
triple in Definition 1.1 as well as the notion of𝐷-boundary triple introduced in [60] for symmetric operators with unequal defect
DERKACH ET AL. 1297
numbers. An overview on the early work of Calkin and more detailed description on its connections to boundary triples and
unitary boundary pairs (boundary relations) can be found in the monograph [40]. In fact, [40] contains a collection of articles
reflecting various recent activities in different fields of applications with related realization results for Weyl functions, including
analysis of differential operators, continuous time state/signal systems and boundary control theory with interconnection analysis
of port-Hamiltonian systems involving Dirac and Tellegen structures etc.
4 𝑨𝑩-GENERALIZED BOUNDARY PAIRS AND BOUNDARY TRIPLES
In this section we present a new generalization of the class of 𝐵-generalized boundary triples from [33] (cf. Definition 1.5).
Definition 4.1. Let 𝐴 be a symmetric operator (or relation) in the Hilbert space ℌ and let  be another Hilbert space. Then
a linear relation Γ ∶ 𝐴∗ →  ⊕ with domain dense in 𝐴∗ is said to be an almost 𝐵-generalized boundary pair, in short,
𝐴𝐵-generalized boundary pair for 𝐴∗, if the following three conditions are satisfied:
4.1.1 the abstract Green’s identity (3.1) holds;
4.1.2 ran Γ0 is dense in ;
4.1.3 𝐴0 = ker Γ0 is selfadjoint.
A single-valued 𝐴𝐵-generalized boundary pair is also said to be an almost 𝐵-generalized boundary triple, shortly, an
𝐴𝐵-generalized boundary triple for 𝐴∗.
If Γ is an 𝐴𝐵-generalized boundary pair for 𝐴∗, then the same is true for its closure. Indeed, since Γ is an extension of
Γ, it is clear that domΓ is dense in 𝐴∗ and ran
(
Γ
)
0 is dense in . By Assumption 4.1.1 Γ is isometric (in the Kreı˘n space
sense), i.e. Γ−1 ⊂ Γ[∗]. Thus, clearly Γ
−1
⊂ Γ[∗] = Γ
[∗]
. Hence, the closure satisfies Green’s identity (3.1) and this implies that
the corresponding kernels ker
(
Γ
)
0 ⊃ ker Γ0 = 𝐴0 and ker
(
Γ
)
1 ⊃ kerΓ1 = 𝐴1 are symmetric. Therefore, ker
(
Γ
)
0 = 𝐴0 must
be selfadjoint.
4.1 Characteristic properties of 𝑨𝑩-generalized boundary pairs
The next theorem describes the central properties of an 𝐴𝐵-generalized boundary pair.
Theorem 4.2. Let 𝐴 be a closed symmetric relation in ℌ, let {,Γ} be an 𝐴𝐵-generalized boundary pair for 𝐴∗, and let Γ0
and Γ1 be the corresponding component mappings from domΓ into. Moreover, let 𝛾(⋅) and𝑀(⋅) be the corresponding 𝛾-field
and the Weyl function, 𝜆 ∈ ℂ ⧵ℝ. Then:
(i) ker Γ = 𝐴;
(ii) 𝐴∗ ∶= domΓ admits the decomposition 𝐴∗ = 𝐴0 +̂ ?̂?𝜆(𝐴∗) and ?̂?𝜆(𝐴∗) is dense in ?̂?𝜆(𝐴∗);
(iii) the 𝛾-field 𝛾(𝜆) is a densely defined bounded operator from ran Γ0 onto ?̂?𝜆(𝐴∗). It is domain invariant and
dom 𝛾(𝜆) = ran Γ0, ker 𝛾(𝜆) = mul Γ0;
(iv) the adjoint 𝛾(𝜆)∗ is a bounded everywhere defined operator and, moreover, equalities hold in (3.12), (3.13),
Γ𝐻(𝜆) =
(
0
𝛾
(
?̄?
)∗) +̂ ({0} × mul Γ), Γ1𝐻(𝜆) = 𝛾(?̄?)∗ +̂ ({0} × mul Γ1); (4.1)
(v) the closure of the 𝛾-field 𝛾(𝜆) is a bounded operator from  into ?̂?𝜆(𝐴∗) satisfying the identity
𝛾(𝜆) =
[
𝐼 + (𝜆 − 𝜇)
(
𝐴0 − 𝜆
)−1]
𝛾(𝜇), 𝜆, 𝜇 ∈ ℂ ⧵ℝ;
(vi) the Weyl function 𝑀(⋅) is a densely defined operator function which is domain invariant, dom𝑀(𝜆) = ran Γ0,
𝑀(𝜆) ⊂ 𝑀
(
?̄?
)∗, and the imaginary part Im𝑀(𝜆) = (𝑀(𝜆) −𝑀(𝜆)∗)∕2𝑖 is bounded with dom Im𝑀(𝜆) = ran Γ0 and
ker Im𝑀(𝜆) = mul Γ0. Furthermore,𝑀(𝜆) admits the representation
𝑀(𝜆) = 𝐸 +𝑀0(𝜆), 𝜆 ∈ ℂ ⧵ℝ, (4.2)
1298 DERKACH ET AL.
where 𝐸 = Re𝑀(𝜇) is a symmetric densely defined operator in  and 𝑀0(⋅) is the restriction of a Nevanlinna function
𝑀0(⋅) ∈ [] onto the domain dom𝐸.
Proof. (i) It is clear from Green’s identity that ker Γ ⊂ (domΓ)∗ = (𝐴∗)∗ = 𝐴; cf. [27, Lemma 7.3]. To prove the reverse inclu-
sion, the property that 𝛾(𝜆), 𝜆 ∈ ℂ ⧵ℝ, is densely defined will be used (and this is independently proved in (iii) below). Assump-
tion 4.1.3 implies that 𝐴 = (𝐴∗)∗ ⊂ 𝐴∗0 = 𝐴0 = ker Γ0 ⊂ domΓ. On the other hand, if 𝑘𝜆 ∈ ran (𝐴 − 𝜆) then by Lemma 3.8{
𝑘𝜆, 𝑘
′′} ∈ 𝛾(?̄?)∗ for some 𝑘′′ and thus
(𝑘′′, ℎ) =
(
𝑘𝜆, 𝛾
(
?̄?
)
ℎ
)
= 0 for all ℎ ∈ dom 𝛾
(
?̄?
)
.
Assumption 4.1.2 combined with dom 𝛾
(
?̄?
)
= ran Γ0 (see proof of (iii) below) shows that 𝛾
(
?̄?
)
is densely defined and, hence,
𝛾
(
?̄?
)∗
is an operator and 𝑘′′ = 𝛾
(
?̄?
)∗
𝑘𝜆 = 0. Now apply the formula (3.15) in the proof of Lemma 3.8 to 𝑘𝜆 ∈ ran(𝐴 − 𝜆):
therein
{
𝑘𝜆, {𝑘, 𝑘′}
}
∈ Γ𝐻(𝜆) and 𝑘′′ = 0 so that (3.15) reads as{
𝑘𝜆,
(
𝑘
𝑘′
)}
=
{
𝑘𝜆,
(
0
0
)}
+
{
0,
(
𝑘
𝑘′
)}
,
(
𝑘
𝑘′
)
∈ mul Γ.
Hence,𝐻(𝜆)𝑘𝜆 ∈ ker Γ and 𝐴 = 𝐻(𝜆)(ran (𝐴 − 𝜆)) ⊂ ker Γ. Therefore, ker Γ = 𝐴.
(ii) This holds by Proposition 3.9 (i).
(iii) & (iv) The decomposition of 𝐴∗ in (ii) combined with 𝐴0 = ker Γ0 implies that
Γ0(𝐴∗) = Γ0
(
?̂?𝜆(𝐴∗)
)
= dom𝑀(𝜆) = dom 𝛾(𝜆), 𝜆 ∈ ℂ ⧵ℝ.
Hence, dom𝑀(𝜆) = dom 𝛾(𝜆) = ran Γ0 does not depend on 𝜆 ∈ ℂ ⧵ℝ. Now Assumption 4.1.2 shows that 𝛾(𝜆) and 𝑀(𝜆)
are densely defined for all 𝜆 ∈ ℂ ⧵ℝ. Moreover, according to Lemma 3.8 (iii) 𝛾(𝜆) is a bounded operator and the equality
dom 𝛾(𝜆)∗ = ℌ holds for all 𝜆 ∈ ℂ ⧵ℝ. Since 𝛾(𝜆) is densely defined in, the adjoint 𝛾(𝜆)∗ is a bounded everywhere defined
operator from ℌ into Γ1
(
𝐴0
)
. Since 𝑀
(
?̄?
)
⊂ 𝑀(𝜆)∗, see (3.9), the adjoint 𝑀(𝜆)∗ and the closure of 𝑀
(
?̄?
)
are also densely
defined operators. In view of (3.10) one has
(𝜆 − ?̄?)(𝛾(𝜆)ℎ, 𝛾(𝜇)𝑘)ℌ =
((
𝑀(𝜆) −𝑀(𝜇)∗
)
ℎ, 𝑘
)
 , 𝜆, 𝜇 ∈ ℂ ⧵ℝ, (4.3)
for all ℎ, 𝑘 ∈ dom 𝛾(𝜆) = ran Γ0. In particular, 2𝑖Im 𝜆‖𝛾(𝜆)ℎ‖2ℌ = ((𝑀(𝜆) −𝑀(𝜆)∗)ℎ, ℎ) holds for all ℎ ∈ dom 𝛾(𝜆) =
dom𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ. By Lemma 3.6 (4.3) implies that
ker 𝛾(𝜆) = ker
(
𝑀(𝜆) −𝑀(𝜆)∗
)
= mul Γ0, 𝜆 ∈ ℂ ⧵ℝ.
It remains to prove (4.1). Observe, that domΓ1𝐻(𝜆) = dom 𝛾
(
?̄?
)∗ = ℌ and clearly the multi-valued parts on both sides of the
inclusion in (3.12), (3.13) are equal. Hence, the inclusions (3.12), (3.13) must prevail actually as equalities (by the criterion
from [4]).
(v) Since dom𝑀(𝜆) = dom 𝛾(𝜆) = ran Γ0 does not depend on 𝜆 ∈ ℂ ⧵ℝ, the following equality holds(
𝐼 + (𝜆 − 𝜇)
(
𝐴0 − 𝜆
)−1)
𝛾(𝜇)ℎ = 𝛾(𝜆)ℎ for all 𝜆, 𝜇 ∈ ℂ ⧵ℝ, ℎ ∈ ran Γ0 (4.4)
by Proposition 3.11. According to (iii) 𝛾(𝜆) is bounded and densely defined, so that its closure 𝛾(𝜆) is bounded and defined
everywhere on . The formula in (iv) is obtained by taking closures in (4.4).
(vi) It suffices to prove the representation (4.2) for 𝑀(𝜆), since all the other assertions were already shown above when
proving (iii) & (iv). It follows from (4.3) and (4.4) that
(𝑀(𝜆)ℎ, 𝑘) =
(
𝑀(𝜇)∗ℎ, 𝑘
)
+ (𝜆 − ?̄?)
((
𝐼 + (𝜆 − 𝜇)
(
𝐴0 − 𝜆
)−1)
𝛾(𝜇)ℎ, 𝛾(𝜇)𝑘
)
= (Re𝑀(𝜇)ℎ, 𝑘) +
((
(𝜆 − Re𝜇) + (𝜆 − 𝜇)(𝜆 − ?̄?)
(
𝐴0 − 𝜆
)−1)
𝛾(𝜇)ℎ, 𝛾(𝜇)𝑘
)
,
ℎ, 𝑘 ∈ dom 𝛾(𝜆) = dom𝑀(𝜇) = ran Γ0, 𝜆, 𝜇 ∈ ℂ ⧵ℝ. Here 2Re𝑀(𝜇) = 𝑀(𝜇) +𝑀(𝜇)∗ and hence 2(Re𝑀(𝜇))∗ ⊃ 𝑀(𝜇)∗ +
𝑀(𝜇) ⊃ 2Re𝑀(𝜇), so that𝐸 ∶= Re𝑀(𝜇) is a symmetric operator with dom𝐸 = dom𝑀(𝜇) = ran Γ0. On the other hand, since
𝛾(𝜆) and its adjoint 𝛾(𝜆)∗ are bounded everywhere defined operators, it follows that the closure of
𝑀0(𝜆) ∶= 𝛾(𝜇)∗
(
(𝜆 − Re𝜇) + (𝜆 − 𝜇)(𝜆 − ?̄?)
(
𝐴0 − 𝜆
)−1)
𝛾(𝜇)
DERKACH ET AL. 1299
defines a holomorphic operator valued Nevanlinna function in the class [], such that 𝑀(𝜆) = 𝐸 +𝑀0(𝜆). This completes
the proof. □
For an 𝐴𝐵-generalized boundary pair it is possible to describe the graph of Γ, (ran Γ)[⟂], and the closure of ran Γ explicitly.
Corollary 4.3. Let Γ be an 𝐴𝐵-generalized boundary pair for 𝐴∗ and let 𝛾(⋅) and 𝑀(⋅) = 𝐸 +𝑀0(⋅) be the corresponding
𝛾-field and Weyl function as in Theorem 4.2 with 𝐸 = Re𝑀(𝜇) for some fixed 𝜇 ∈ ℂ ⧵ℝ. Then:
(i) with a fixed 𝜆 ∈ ℂ ⧵ℝ the graph of Γ admits the following representation:
Γ =
{{
𝐻(𝜆)𝑘𝜆,
(
0
𝛾
(
?̄?
)∗
𝑘𝜆
)}
+
{(
𝛾(𝜆)ℎ
𝜆𝛾(𝜆)ℎ
)
,
(
ℎ
𝑀(𝜆)ℎ
)}
∶ 𝑘𝜆 ∈ ran
(
𝐴0 − 𝜆
)
ℎ ∈ dom𝑀(𝜆)
}
;
(ii) the range of Γ satisfies
(ran Γ)[⟂] = 𝐸∗ ↾ ker 𝛾(𝜆) and ran Γ =
(
𝐸∗ ↾ ker 𝛾(𝜆)
)∗
,
and here ker 𝛾(𝜆) = ker
(
Im𝑀(𝜆)
)
= ker
(
𝑀0(𝜆)
)
does not depend on 𝜆 ∈ ℂ ⧵ℝ. In particular, ran Γ is dense in if and
only if dom𝐸∗ ∩ ker 𝛾(𝜆) = {0} for some or, equivalently, for every 𝜆 ∈ ℂ ⧵ℝ.
(iii) Γ is a single-valued mapping if and only if mul Γ0 = {0} or, equivalently, if and only if ker Im𝑀(𝜆) (= ker 𝛾(𝜆)) = 0,
𝜆 ∈ ℂ ⧵ℝ.
Proof.
(i) Using the representation of Γ𝐻(𝜆) in (4.1), the inclusion mul Γ ⊂ 𝑀(𝜆) in Lemma 3.6, and the fact that by Theorem 4.2
𝑀(𝜆) is an operator, one concludes that the representation of Γ given in Proposition 3.9 (ii) can be rewritten in the form
as stated in (i).
(ii) The description in (i) shows that
ran Γ = Γ
(
𝐴0
)
+̂ 𝑀(𝜆) =
(
0
ran 𝛾
(
?̄?
)∗) +̂ 𝑀(𝜆), (4.5)
for all 𝜆 ∈ ℂ ⧵ℝ. Therefore, (ran Γ)[⟂] =
(
{0} × ran 𝛾
(
?̄?
)∗)[⟂] ∩𝑀(𝜆)∗. Hence ?̂? = {𝑘, 𝑘′} ∈ (ran Γ)[⟂] if and only
if ?̂?∈𝑀(𝜆)∗ and 𝑘 ∈
(
ran 𝛾
(
?̄?
)∗)⟂ = ker 𝛾(?̄?). Since 𝛾(𝜆) and Im𝑀(𝜆) are bounded and 𝐸 = Re𝑀(𝜇), one has
Re𝑀0(𝜇) = 0 and ker 𝛾(𝜇) = ker
(
Im𝑀0(𝜇)
)
= ker
(
𝑀0(𝜇)
)
. This kernel does not depend on 𝜇 ∈ ℂ ⧵ℝ due to
𝑀0(⋅) ∈[]; cf. Theorem 4.2 (v). This proves that
(ran Γ)[⟂] = 𝑀(𝜆)∗↾ ker 𝛾(𝜆) =
(
𝐸∗ +𝑀0(𝜆)∗
)
↾ ker 𝛾(𝜆) = 𝐸∗↾ ker 𝛾(𝜆).
As to the closure observe that
ran Γ =
(
(ran Γ)[⟂]
)[⟂] = ((ran Γ)∗)∗ = (𝐸∗↾ ker 𝛾(𝜆))∗.
Thus, ran Γ =  × if and only if 𝐸∗↾ ker 𝛾(𝜆) = {0, 0} or, equivalently, dom𝐸∗ ∩ ker 𝛾(𝜆) = {0}, since 𝐸∗ together
with 𝐸 (⊂ 𝐸∗) is a densely defined operator in.
(iii) In view of (i) this follows from mul Γ0 = ker Im𝑀(𝜆) = ker 𝛾(𝜆); see Lemma 3.6. □
Corollary 4.3 shows that for an 𝐴𝐵-generalized boundary pair the inclusionmul Γ ⊂ (ran Γ)[⟂] is in general strict. In particu-
lar, the range of Γ for a single-valued𝐴𝐵-generalized boundary pair, i.e., an𝐴𝐵-generalized boundary triple, need not be dense
in  ×. Notice that an 𝐴𝐵-generalized boundary pair with the surjectivity condition ran Γ0 =  is called a 𝐵-generalized
boundary pair for 𝐴∗; see Definition 3.5. The next result gives a connection between 𝐴𝐵-generalized boundary pairs and
𝐵-generalized boundary pairs.
1300 DERKACH ET AL.
Theorem 4.4. Let {,Γ} be a 𝐵-generalized boundary pair for 𝐴∗, and let𝑀(⋅) and 𝛾(⋅) be the corresponding Weyl function
and 𝛾-field. Let also 𝐸 be a symmetric densely defined operator in  and let Γ = {Γ0,Γ1} where Γ𝑖 = 𝜋𝑖Γ, 𝑖 = 0, 1, be the
corresponding components of Γ as in (3.4). Then the transform(
Γ̃0
Γ̃1
)
=
(
𝐼 0
𝐸 𝐼
)(
Γ0
Γ1
)
(4.6)
defines an 𝐴𝐵-generalized boundary pair for 𝐴∗. The corresponding Weyl function𝑀(⋅) and ?̃?(⋅)-field are connected by
𝑀(𝜆) = 𝐸 +𝑀(𝜆), ?̃?(𝜆) = 𝛾(𝜆)↾ dom𝐸, 𝜆 ∈ ℂ ⧵ℝ.
Furthermore, Γ̃ ∶=
{
Γ̃0, Γ̃1
}
in (4.6) is closed if and only if 𝐸 is a closed symmetric operator in, in particular, the closure of
Γ̃ is given by (4.6) with 𝐸 replaced by its closure 𝐸.
Conversely, if
{, Γ̃} is an 𝐴𝐵-generalized boundary pair for 𝐴∗ then there exists a 𝐵-generalized boundary pair {,Γ}
for 𝐴∗ and a densely defined symmetric operator 𝐸 in  such that Γ̃ is given by (4.6).
Proof. (⇒) By Lemma 3.12 the block triangular transformation 𝑉 in (4.6) acting on  × is an isometric operator. Con-
sequently, Γ̃ = 𝑉 ◦Γ is isometric. It is clear from (4.6) that 𝐴0 ∶= ker Γ0 ⊂ ker Γ̃0, which by symmetry of ker Γ̃0 implies that
ker Γ̃0 = 𝐴0. Clearly ran Γ̃0 is dense in , since ran Γ0 =  and 𝐸 is densely defined. Thus Γ̃ admits all the properties in
Definition 4.1. Since in addition ker Γ̃ = ker Γ, it follows from Theorem 4.2 (i) that 𝐴∗ = dom Γ̃ is dense in 𝐴∗. Therefore,{, Γ̃} is an 𝐴𝐵-generalized boundary pair for 𝐴∗. The connections between the Weyl functions and 𝛾-fields are clear from the
definitions; cf. Lemma 3.12.
To treat the closedness properties of Γ̃ consider the representation of Γ̃ in Corollary 4.3. Let 𝜆 ∈ ℂ ⧵ℝ be fixed and
assume that the sequence
{
𝑓𝑛, ?̂?𝑛
}
∈ Γ̃ converges to
{
𝑓, ?̂?
}
. Then 𝑓𝑛 = 𝐻(𝜆)𝑘𝑛,𝜆 + ?̂?(𝜆)ℎ𝑛 with unique 𝑘𝑛,𝜆 ∈ ran
(
𝐴0 − 𝜆
)
and
ℎ𝑛 ∈ dom𝑀(𝜆) = dom𝐸 and, since the angle between the graphs of 𝐴0 and ?̂?𝜆
(
𝐴∗
)
is positive, it follows that
𝑘𝑛,𝜆 → 𝑘𝜆 ∈ ran
(
𝐴0 − 𝜆
)
. Moreover, the representation of
{
𝑓𝑛, ?̂?𝑛
}
∈ Γ̃ in Corollary 4.3 shows that ℎ𝑛 → ℎ ∈ . Accord-
ing to Theorem 4.2 𝛾(𝜆) and 𝛾
(
?̄?
)∗
are bounded operators and, since 𝑀(𝜆) = 𝐸 +𝑀(𝜆), where 𝑀(𝜆) is bounded (see
[26, Proposition 3.16]), it follows from Corollary 4.3 that{
𝐻(𝜆)𝑘𝑛,𝜆,
(
0
𝛾
(
?̄?
)∗
𝑘𝑛,𝜆
)}
+
{(
𝛾(𝜆)ℎ𝑛
𝜆𝛾(𝜆)ℎ𝑛
)
,
(
ℎ𝑛
𝐸ℎ𝑛 +𝑀(𝜆)ℎ𝑛
)}
∈ Γ̃
converges to {
𝐻(𝜆)𝑘𝜆,
(
0
𝛾
(
?̄?
)∗
𝑘𝜆
)}
+
{(
𝛾(𝜆)ℎ
𝜆𝛾(𝜆)ℎ
)
,
(
ℎ
ℎ′′ +𝑀(𝜆)ℎ
)}
∈ clos Γ̃, (4.7)
where {ℎ, ℎ′′} ∈ 𝐸. It is also clear that the limit element in (4.7) belongs to Γ̃ if and only if lim𝑛→∞{ℎ𝑛, 𝐸ℎ𝑛} = {ℎ, ℎ′′} ∈ 𝐸.
Therefore, Γ̃ is closed if and only if 𝐸 is closed and, moreover, the closure of Γ̃, which is also an 𝐴𝐵-generalized boundary pair
for 𝐴∗ (as stated after Definition 4.1), is given by (4.6) with 𝐸 replaced by its closure 𝐸.
(⇐) Let
{, Γ̃} be an 𝐴𝐵-generalized boundary pair. According to Theorem 4.2 the corresponding Weyl function 𝑀 is of
the form𝑀 = 𝐸 +𝑀 , where𝑀 ∈ [] and 𝐸 (= Re𝑀(𝜇)) is a symmetric densely defined operator in.
To construct Γ̃ directly from an associated 𝐵-generalized boundary pair, define(
Γ̂0
Γ̂1
)
∶=
(
𝐼 0
−𝐸 𝐼
)(
Γ̃0
Γ̃1
)
. (4.8)
Since𝑀(𝜆) = Γ̃
(
?̂?𝜆
(
𝐴∗
))
⊂ ran Γ̃, where 𝐴∗ = dom Γ̃, and dom𝑀(𝜆) = dom𝐸, it follows that the graph of𝑀(𝜆) belongs to
the domain of the block operator
(
𝐼 0
−𝐸 𝐼
)
, i.e., ?̂?𝜆
(
𝐴∗
)
⊂ dom Γ̂ for all 𝜆 ∈ ℂ ⧵ℝ. Moreover,
Γ̂
(
?̂?𝜆
(
𝐴∗
))
= −𝐸 +𝑀(𝜆) = 𝑀(𝜆) ↾ dom𝐸 ⊂ ran Γ̂
for all 𝜆 ∈ ℂ ⧵ℝ. Since 𝑀 ∈ [] this implies that ran Γ̂0 is dense in . Clearly, ker Γ̂0 = ker Γ̃0 = 𝐴0 and since
𝐴∗ =𝐴0 +̂ ?̂?𝜆
(
𝐴∗
)
one concludes that 𝐴∗ = dom Γ̂ = dom Γ̃ is dense in 𝐴∗. Thus, Γ̂ is also an 𝐴𝐵-generalized boundary
DERKACH ET AL. 1301
pair for 𝐴∗ and, consequently, also its closure is an 𝐴𝐵-generalized boundary pair for 𝐴∗, too. Denote the closure of Γ̂ by Γ(0).
Then the corresponding Weyl function𝑀 (0)(⋅) is an extension of𝑀 and its closure is equal to𝑀 . Since Γ(0) is closed, it must
be unitary by [27, Theorem 7.51] (cf. [25, Proposition 3.6]). In particular, 𝑀 (0)(⋅) is also closed, i.e., 𝑀 (0)(⋅) = 𝑀 ∈ [].
Thus, ran Γ(0)0 = dom𝑀
(0)(⋅) =  and hence Γ(0) is a 𝐵-generalized boundary pair for 𝐴∗; see Definition 3.5. Finally, in view
of (4.8) one has
(
Γ̃0
Γ̃1
)
=
(
𝐼 0
𝐸 𝐼
)(
Γ̂0
Γ̂1
)
⊂
(
𝐼 0
𝐸 𝐼
)
Γ(0) =∶ Γ̃(0).
Here equality Γ̃ = Γ̃(0) holds by Proposition 3.9 (iii), since𝑀 (0)(⋅) = 𝐸 +𝑀(⋅) = 𝑀(⋅). □
The proof of Theorem 4.4 contains also the following result.
Corollary 4.5. If {, Γ̃} is an 𝐴𝐵-generalized boundary pair for 𝐴∗ with the Weyl function𝑀(⋅) and 𝐸 = Re𝑀(𝜇) for some
𝜇 ∈ 𝜌(𝑀), then the closure of Γ =
(
𝐼 0
−𝐸 𝐼
)
Γ̃ defines a 𝐵-generalized boundary pair for 𝐴∗ with the bounded Weyl function
𝑀(⋅) = clos(𝑀(⋅) − 𝐸).
Theorems 4.2 and 4.4 imply the following characterization for theWeyl functions corresponding to𝐴𝐵-generalized boundary
pairs.
Corollary 4.6. The class of 𝐴𝐵-generalized boundary pairs coincides with the class of isometric boundary pairs whose Weyl
function is of the form
𝑀(𝜆) = 𝐸 +𝑀0(𝜆), 𝜆 ∈ ℂ ⧵ℝ, (4.9)
with 𝐸 a symmetric densely defined operator in  and 𝑀0(⋅) ∈ []. In particular, every function 𝑀 of the form (4.9) is a
Weyl function of some 𝐴𝐵-generalized boundary pair.
Proof. By Theorem 4.2 the Weyl function𝑀 of an 𝐴𝐵-generalized boundary pair
{, Γ̃} is of the form (4.9), where 𝐸 ⊂ 𝐸∗
is densely defined and𝑀0(⋅) ∈ [].
Conversely, if 𝑀 is given by (4.9) with 𝑀0(⋅) ∈ [], then by [26, Proposition 3.16] 𝑀0(⋅) is the Weyl function of a
𝐵-generalized boundary pair {,Γ} for 𝐴∗. Now according to the first part of Theorem 4.4 the transform Γ̃ of Γ defined
in (4.6) is an 𝐴𝐵-generalized boundary pair for 𝐴∗ such that the corresponding Weyl function is equal to (4.9). □
By Definition 3.5 every 𝐵-generalized boundary pair is also an 𝐴𝐵-generalized boundary pair. Hence, the notions of an
𝐴𝐵-generalized boundary triple and 𝐴𝐵-generalized boundary pair generalize the earlier notions of “a generalized boundary
value space” as introduced in [33, Definition 6.1] and “a boundary relation with the Weyl function in[]” as defined in [25,
Proposition 5.9]. It is emphasized that 𝐵-generalized boundary pairs are not only isometric: they are also unitary in the Kreı˘n
space sense, see Definition 3.1. The characteristic properties of the classes of 𝐵-generalized boundary triples and pairs can be
found in Theorem 1.7, see also [25, Propositions 5.7, 5.9] and [26, Proposition 3.16]. Some further characterizations connected
with 𝐴𝐵-generalized boundary pairs are given in the next corollary.
Corollary 4.7. Let {, Γ̃} be an 𝐴𝐵-generalized boundary pair for 𝐴∗ as in Theorem 4.4 and let 𝐸 be a symmetric densely
defined operator in as in (4.6). Then:
(i)
{, Γ̃} is a unitary boundary pair (boundary relation) for 𝐴∗ if and only if the operator 𝐸 is selfadjoint;
(ii)
{, Γ̃} has an extension to a unitary boundary pair for 𝐴∗ if and only if the operator 𝐸 has equal defect numbers and in
this case the formula (4.6) defines a unitary extension of Γ̃ when 𝐸 is replaced by some selfadjoint extension 𝐸0 of 𝐸;
(iii)
{, Γ̃} is a 𝐵-generalized boundary pair for 𝐴∗ if and only if the operator 𝐸 is bounded and everywhere defined (hence
selfadjoint);
(iv)
{, Γ̃} is an ordinary boundary triple for 𝐴∗ if and only if ran Γ =  ⊕, or equivalently, if and only if ran Γ is closed,
𝐸 is bounded, and ker Im𝑀(𝜆) = 0, 𝜆 ∈ ℂ ⧵ℝ.
1302 DERKACH ET AL.
Proof.
(i) By Theorem 4.4 Γ̃ is closed if and only if 𝐸 is closed. Moreover, 𝐸 = 𝐸∗ if and only if𝑀 is a Nevanlinna function. Now
the statement follows from [25, Proposition 3.6] (or [27, Theorem 7.51]).
(ii) This is clear from part (i) and Theorem 4.4.
(iii) This follows from Theorem 4.2 (v) by the equalities ran Γ̃0 = dom𝑀 = dom𝐸 (= ).
(iv) The first equivalence is contained in [25, Proposition 5.3]. To prove the second criterion, we apply Corollary 4.3, in par-
ticular, the representation of ran Γ in (4.5):
ran Γ = Γ
(
𝐴0
)
+̂ 𝑀(𝜆) =
(
{0} × ran 𝛾
(
?̄?
)∗) +̂ 𝑀(𝜆). (4.10)
Clearly, 𝐸 is bounded precisely when𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ, is bounded. In this case the angle between the last two subspaces
in (4.10) is positive and then ran Γ is closed if and only if ran 𝛾
(
?̄?
)∗
and 𝑀(𝜆) both are closed. By Theorem 4.2𝛾(𝜆) is
bounded and dom 𝛾(𝜆) = dom𝑀(𝜆) = , when 𝑀(𝜆) is closed and bounded. Then 𝛾(𝜆) is closed and (ran 𝛾(?̄?)∗)⟂ =
ker 𝛾(𝜆) = ker Im𝑀(𝜆). Therefore, the conditions ran Γ is closed, 𝐸 is bounded, and ker Im𝑀(𝜆) = 0 imply that ran Γ is
also dense in × and, thus, Γ is surjective. The converse is clear.
□
The class of 𝐴𝐵-generalized boundary triples contains the class of so-called quasi boundary triples, which has been studied
in J. Behrndt and M. Langer [11].
Definition 4.8 ([11]). Let 𝐴 be a densely defined symmetric operator in ℌ. A triple Π = {,Γ0,Γ1} is said to be a quasi
boundary triple for 𝐴∗, if 𝐴∗ ∶= domΓ is dense in 𝐴∗ and the following conditions are satisfied:
4.8.1 Green’s identity (1.1) holds for all 𝑓, 𝑔 ∈ 𝐴∗;
4.8.2 𝐴0 = ker Γ0 is a selfadjoint operator in ℌ;
4.8.3 the range of Γ is dense in  ×.
For isometric boundary pairs mul Γ ⊂ (ran Γ)[⟂] and thus the condition 4.8.3 implies that Γ is single-valued. Since the con-
dition 4.8.3 implies 4.1.2, quasi boundary triples are 𝐴𝐵-generalized boundary triples. Corollary 4.3 gives the following char-
acterization for quasi boundary triples.
Corollary 4.9. An 𝐴𝐵-generalized boundary triple {,Γ0,Γ1} for 𝐴∗ with the Weyl function𝑀 = 𝐸 +𝑀0(⋅) represented in
the form (4.2) is a quasi boundary triple (with single-valued Γ) for 𝐴∗ if and only if ran Γ is dense in ⊕, or equivalently,
dom𝐸∗ ∩ ker Im𝑀(𝜆)
(
= dom𝐸∗ ∩ ker Im𝑀0(𝜆)
)
= {0}, (4.11)
for some or, equivalently, for every 𝜆 ∈ ℂ ⧵ℝ.
Proof. Item (ii) of Corollary 4.3 shows that ran Γ is dense in  if and only if dom𝐸∗ ∩ ker 𝛾(𝜆) = {0} for some or, equiva-
lently, for every 𝜆 ∈ ℂ ⧵ℝ. This is equivalent to the conditions in (4.11), since ker 𝛾(𝜆) = ker Im𝑀(𝜆) = ker Im𝑀0(𝜆); see
Corollary 4.3. □
Remark 4.10. A connection between 𝐵-generalized boundary triples and quasi boundary triples can be found in [27, Theo-
rem 7.57], [70, Propositions 5.1, 5.3]. In fact, each of them is special case of Theorem 4.4. Moreover, it should be noted that
in the formulation of the converse part in [27, Theorem 7.57] one should use a 𝐵-generalized boundary pair {,Γ}, instead
of a 𝐵-generalized boundary triple
{,Γ0,Γ1}, since ker 𝛾(𝜆) = ker Im𝑀(𝜆) = 0 (𝑀 is strict) does not imply in general that
ker 𝛾(𝜆) = ker Im𝑀(𝜆) = ker Im𝑀0(𝜆) = 0, i.e. 𝑀0 ∈ [] as in the proof of Theorem 4.4 above: only the factor mapping
Γ∕mul Γ (see [7], [42, Eq. (2.15)]) becomes single-valued (equivalently the correspondingWeyl function is strict, cf. [25, Propo-
sition 4.7]). It should be also noted that a condition which is equivalent to (4.11) appears in [70, Section 5.1]; see also [69]. For
some further related facts, see Corollary 5.18 and Remark 5.20 in Section 5.
The next result describes a connection between 𝐵-generalized boundary pairs and ordinary boundary triples. In the special
case of 𝐵-generalized boundary triples the corresponding result is presented in [27, Theorem 7.24].
DERKACH ET AL. 1303
Theorem 4.11. Let {,Γ} be a 𝐵-generalized boundary pair for 𝐴∗ and let 𝑀(⋅) be the corresponding Weyl function. Then
there exists an ordinary boundary triple
{𝑠,Γ00,Γ01} with𝑠 = ran Im𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ, and operators 𝐸 = 𝐸∗ ∈ () and
𝐺 ∈ (,𝑠) with ker 𝐺 =  ⊖𝑠 such that (1.6) holds with 𝐺−1 standing for the inverse of 𝐺 as a linear relation. If𝑀0(⋅)
is the Weyl function corresponding to the ordinary boundary triple
{𝑠,Γ00,Γ01}, then
𝑀(𝜆) = 𝐺∗𝑀0(𝜆)𝐺 + 𝐸, 𝜆 ∈ 𝜌
(
𝐴0
)
.
Proof. The proof relies on [27, Theorem 7.24] and [26, Propositions 3.18, 4.1].
Let 𝐸 = Re𝑀(𝑖). Then by [26, Propositions 3.18] (cf. Lemma 3.12) the transform
Γ̃ =
{{
𝑓,
(
ℎ
−𝐸ℎ + ℎ′
)}
∶
{
𝑓, ℎ̂
}
∈ Γ
}
(4.12)
defines a new 𝐵-generalized boundary pair for 𝐴∗ with the Weyl function𝑀(⋅) − 𝐸 and the original 𝛾-field 𝛾(⋅) of {,Γ}.
Let 𝑃𝑠 be the orthogonal projection onto 𝑠 ∶= ran Im𝑀(𝜆). Then according to [26, Proposition 4.1] the transform
Γ(𝑠) ∶ ℌ2 →
(𝑠)2 given by
Γ(𝑠) =
{{
𝑓,
(
𝑘
𝑃𝑠𝑘
′
)}
∶
{
𝑓, ?̂?
}
∈ Γ̃, (𝐼 − 𝑃𝑠)𝑘 = 0
}
(4.13)
determines a 𝐵-generalized boundary pair
{𝑠,Γ(𝑠)} for (𝐴(𝑠))∗, where 𝐴(𝑠) is defined by
𝐴(𝑠) ∶= ker Γ(𝑠). (4.14)
The corresponding Weyl function and 𝛾-field are given by
𝑀 (𝑠)(𝜆) = 𝑃𝑠(𝑀(𝜆) − 𝐸)↾𝑠, 𝛾 (𝑠)(𝜆) = 𝛾(𝜆)↾𝑠.
Recall that ker(𝑀(𝜆) − 𝐸) = ker Im𝑀(𝜆) does not depend on 𝜆 ∈ ℂ ⧵ℝ. Consequently, 𝑀(𝜆) − 𝐸 = 𝑀 (𝑠)(𝜆)⊕ 0⊖𝑠 .
Since ker 𝛾
(
?̄?
)
= ker
(
𝑀
(
?̄?
)
− 𝐸
)
= ker 𝑃𝑠 one has ran 𝛾
(
?̄?
)∗
⊂ 𝑠 and it follows from Corollary 4.3 that ran Γ̃1 ⊂ 𝑠. There-
fore, (4.13) implies that 𝐴(𝑠) defined in (4.14) coincides with 𝐴: ker Γ(𝑠) = ker Γ = 𝐴. By construction𝑀 (𝑠)(⋅) ∈ 𝑠[𝑠] and
hence Γ(𝑠) is single-valued; i.e.
{𝑠,Γ(𝑠)0 ,Γ(𝑠)1 } is in fact a 𝐵-generalized boundary triple for 𝐴∗; cf. [25, Proposition 4.7].
One can now apply [27, Theorem 7.24] with 𝑅 = Re𝑀 (𝑠)(𝑖) = 0 and 𝐾 =
(
Im𝑀 (𝑠)(𝑖)
)1∕2
to conclude existence of an ordi-
nary boundary triple
{𝑠,Γ00,Γ01} with the Weyl function𝑀0(⋅) such that
Γ(𝑠)0 = 𝐾
−1Γ00, Γ
(𝑠)
1 = 𝐾Γ
0
1, and 𝑀
(𝑠)(𝜆) = 𝐾𝑀0(𝜆)𝐾, 𝜆 ∈ ℂ ⧵ℝ.
In particular, 𝑀(𝑖) = 𝐸 + 𝑖 𝐾2𝑃𝑠 and 𝑀(𝜆) = 𝐸 + 𝑃𝑠𝐾𝑀0(𝜆)𝐾𝑃𝑠. The statement follows by taking 𝐺 = 𝐾𝑃𝑠. Indeed, since
ran Γ̃1 ⊂ 𝑠 and mul Γ̃0 = ker Im𝑀(𝜆) = ker 𝑃𝑠 (see Lemma 3.6) (4.13) shows that domΓ(𝑠) = dom Γ̃ and
Γ̃ = Γ(𝑠) ⊕
{{
0̂,
(
𝑘
0
)}
∶ 𝑃𝑠𝑘 = 0
}
=
{{
?̂?,
(
𝑃−1𝑠 Γ
(𝑠)
0 ?̂?
𝑃𝑠Γ
(𝑠)
1 ?̂?
)}
∶ ?̂? ∈ domΓ(𝑠)
}
.
Finally, using 𝐺−1 = 𝑃−1𝑠 𝐾
−1 = 𝐾−1 ⊕
(
{0} × ker 𝑃𝑠
)
and (4.12) yields the formulas (1.6) and (1.7). □
The notion of an𝐴𝐵-generalized boundary pair introduced in Definition 4.1 appears to be useful in characterizing the class of
Nevanlinna functions with unbounded values (and multi-valued Nevanlinna families) whose imaginary parts generate closable
forms 𝜏𝑀(𝜆) = [(𝑀(𝜆)⋅, ⋅) − (⋅,𝑀(𝜆)⋅)]∕2𝑖 via (3.6) and whose closures are domain invariant. All such functions, after renor-
malization by a bounded operator𝐺 ∈ [], turn out to be Weyl functions of𝐴𝐵-generalized boundary triples, i.e., for a suitable
choice of 𝐺, 𝐺∗𝑀𝐺 is a function of the form (4.2): see Theorem 5.32 and Corollary 5.34 in Section 5.
4.2 A Kreı˘n type formula for 𝑨𝑩-generalized boundary triples
In this section a Kreı˘n type (resolvent) formula for 𝐴𝐵-generalized boundary triples will be presented. We refer to [27, Proposi-
tion 7.27] where a special case of 𝐵-generalized boundary triples was treated, and [11, 12] for a special case of quasi boundary
triples. The form of the formula as given in Theorem 4.12 below is new even in the standard case of ordinary boundary triples.
1304 DERKACH ET AL.
If 𝐴0 = ker Γ0 is selfadjoint, then it follows from the first von Neumann’s formula that for each 𝜆 ∈ 𝜌
(
𝐴0
)
the domain of Γ
can be decomposed as follows:
domΓ = 𝐴0 +̂
(
domΓ ∩ ?̂?𝜆(𝐴∗)
)
.
Now let Γ be single-valued and let Γ be decomposed as Γ =
{
Γ0,Γ1
}
. Let 𝐴 be an extension of 𝐴 which belongs to the domain
of Γ and let Θ be a linear relation in  corresponding to 𝐴:
Θ = Γ
(
𝐴
)
, 𝐴 ⊂ domΓ ⇔ 𝐴 = 𝐴Θ ∶= Γ−1(Θ), Θ ⊂ ran Γ. (4.15)
Theorem 4.12. Let 𝐴 be a closed symmetric relation, let Π = {,Γ0,Γ1} be an 𝐴𝐵-generalized boundary triple for 𝐴∗
with 𝐴0 = ker Γ0, and let 𝑀(⋅) and 𝛾(⋅) be the corresponding Weyl function and 𝛾-field, respectively. Then for any extension
𝐴Θ ∈ Ext𝐴 satisfying 𝐴Θ ⊂ domΓ the following Kreı˘n-type formula holds(
𝐴Θ − 𝜆
)−1 = (𝐴0 − 𝜆)−1 + 𝛾(𝜆)(Θ −𝑀(𝜆))−1𝛾(?̄?)∗, 𝜆 ∈ 𝜌(𝐴0). (4.16)
Here the inverses in the first and last terms are taken in the sense of linear relations.
The proof of this theorem is postponed until Section 5.2, where an analogous resolvent formula is proved for unitary boundary
triples. However, some remarks and consequences of Theorem 4.12 are in order already here.
Remark 4.13. We emphasize that in the Kreı˘n-type formula (4.16) it is not assumed that 𝜆 ∈ 𝜌
(
𝐴Θ
)
. In particular, 𝐴Θ − 𝜆 need
not be invertible; 𝐴Θ and Θ need not even be closed. Hence, even when Π =
{,Γ0,Γ1} is an ordinary boundary triple for 𝐴∗
the formula (4.16) uses only the assumption 𝜆 ∈ 𝜌
(
𝐴0
)
instead of the standard assumption that 𝜆 ∈ 𝜌
(
𝐴0
)
∩ 𝜌
(
𝐴Θ
)
.
The following statement is an immediate consequence of Theorem 4.12.
Corollary 4.14. Let the assumptions be as in Theorem 4.12 and let 𝜆 ∈ 𝜌(𝐴0). Then:
(i) ker
(
𝐴Θ − 𝜆
)
= 𝛾(𝜆)ker(Θ −𝑀(𝜆));
(ii) if (Θ −𝑀(𝜆))−1 is a bounded operator, then the same is true for
(
𝐴Θ − 𝜆
)−1;
(iii) if 0 ∈ 𝜌(Θ −𝑀(𝜆)) then 𝜆 ∈ 𝜌
(
𝐴Θ
)
.
5 SOME CLASSES OF UNITARY BOUNDARY TRIPLES AND WEYL
FUNCTIONS
5.1 Unitary boundary pairs and unitary colligations
Some formulas from Section 3 can be essentially improved when using the interrelations between unitary relations and unitary
colligations, see [10]. Let {,Γ} be a unitary boundary pair. As was shown in [25, Proposition 2.10] the linear relation, the
so-called main transform of Γ,
̃ ∶=
{{(
𝑓
ℎ
)
,
(
𝑓 ′
−ℎ′
)}
∶
{(
𝑓
𝑓 ′
)
,
(
ℎ
ℎ′
)}
∈ Γ
}
(5.1)
is selfadjoint in ℌ⊕. Denote its Cayley transform 𝐼 − 2𝑖(̃ + 𝑖)−1 by 𝜔(Γ):
𝜔(Γ) ∶=
{{(
𝑓 + 𝑖𝑓 ′
ℎ − 𝑖ℎ′
)
,
(
−𝑓 + 𝑖𝑓 ′
−ℎ − 𝑖ℎ′
)}
∶
{(
𝑓
𝑓 ′
)
,
(
ℎ
ℎ′
)}
∈ Γ
}
(5.2)
Then 𝜔(Γ) is the graph of a unitary operator  ∶( ℌ ) → ( ℌ ). The mapping 𝜔 ∶Γ → establishes a one-to-one correspon-
dence between the set of unitary boundary pairs and the set of unitary operators inℌ⊕.
The inverse transform Γ=𝜔−1( ) takes the form
Γ =
{{(
𝑔′ − 𝑔
𝑖(𝑔′ + 𝑔)
)
,
(
𝑢′ − 𝑢
−𝑖(𝑢′ + 𝑢)
)}
∶
{(
𝑔
𝑢
)
,
(
𝑔′
𝑢′
)}
∈ gr 
}
. (5.3)
DERKACH ET AL. 1305
As was shown in [29]𝑈 = 𝜔(Γ) is the Potapov–Ginzburg transform of Γ in the sense of [7]. Let us consider the unitary operator
 and the pair of Hilbert spaces ℌ and  as a unitary colligation (see [18]) written in the block form
 =
(
𝑇 𝐹
𝐺 𝐻
)
∈ (ℌ⊕), (5.4)
where 𝑇 ∈ (ℌ), 𝐹 ∈ (,ℌ), 𝐺 ∈ (ℌ,), and𝐻 ∈ (). Then the representation (5.3) for Γ takes the form
Γ =
{{(
(𝑇 − 𝐼)𝑔 + 𝐹𝑢
𝑖(𝑇 + 𝐼)𝑔 + 𝑖𝐹 𝑢
)
,
(
𝐺𝑔 + (𝐻 − 𝐼)𝑢
−𝑖𝐺𝑔 − 𝑖(𝐻 + 𝐼)𝑢
)}
∶ 𝑔 ∈ ℌ, 𝑢 ∈ 
}
. (5.5)
Since  = ( ∗)−1, then
 =
{{(
𝑇 ∗𝑔′ + 𝐺∗𝑢′
𝐹 ∗𝑔′ +𝐻∗𝑢′
)
,
(
𝑔′
𝑢′
)}
∶ 𝑔′ ∈ ℌ, 𝑢′ ∈ 
}
and hence Γ admits a dual representation
Γ =
{{(
(𝐼 − 𝑇 ∗)𝑔′ − 𝐺∗𝑢′
𝑖(𝐼 + 𝑇 ∗)𝑔′ + 𝑖𝐺∗𝑢′
)
,
(
−𝐹 ∗𝑔′ + (𝐼 −𝐻∗)𝑢′
−𝑖𝐹 ∗𝑔′ − 𝑖(𝐼 +𝐻∗)𝑢′
)}
∶ 𝑔
′ ∈ ℌ,
𝑢′ ∈ 
}
. (5.6)
Observe, that for each element
{
𝑓, ℎ̂
}
∈ Γ the vectors 𝑔, 𝑔′ ∈ ℌ and ℎ, ℎ′ ∈  are in fact uniquely determined in (5.5) and
(5.6). Let us collect some formulas concerning Γ and  which are immediate from (5.5) and (5.6) (see also [10]).
Proposition 5.1. Let {,Γ} be a unitary boundary pair for 𝐴∗ with Γ given by (5.5), and let 𝐴∗ = domΓ, 𝐴0 = ker Γ0. Then:
𝐴∗ = ran
(
𝑇 − 𝐼 𝐹
𝑖(𝑇 + 𝐼) 𝑖𝐹
)
= ran
(
(𝐼 − 𝑇 )∗ −𝐺∗
𝑖(𝐼 + 𝑇 )∗ 𝑖𝐺∗
)
,
mul𝐴∗ = (𝐼 − 𝑇 )−1ran𝐹 = (𝐼 − 𝑇 ∗)−1ran𝐺∗;
mul𝐴 = ker(𝐼 − 𝑇 ) = ker(𝐼 − 𝑇 ∗);
𝐴0 =
{(
(𝑇 − 𝐼)𝑔 + 𝐹𝑢
𝑖(𝑇 + 𝐼)𝑔 + 𝑖𝐹 𝑢
)
∶ 𝐺𝑔 + (𝐻 − 𝐼)𝑢 = 0, 𝑔 ∈ ℌ, 𝑢 ∈ 
}
=
{(
(𝐼 − 𝑇 ∗)𝑔′ − 𝐺∗𝑢′
𝑖(𝐼 + 𝑇 ∗)𝑔′ + 𝑖𝐺∗𝑢′
)
∶ 𝐹 ∗𝑔′ + (𝐻∗ − 𝐼)𝑢′ = 0, 𝑔′ ∈ ℌ, 𝑢′ ∈ 
} (5.7)
ran Γ0 = ran(𝐼 −𝐻) + ran𝐺 = ran(𝐼 −𝐻∗) + ran𝐹 ∗; (5.8)
mul Γ =
{(
(𝐻 − 𝐼)𝑢
−𝑖(𝐻 + 𝐼)𝑢
)
∶ 𝑢 ∈ ker 𝐹
}
=
{(
(𝐼 −𝐻∗)𝑢′
−𝑖(𝐼 +𝐻∗)𝑢′
)
∶ 𝑢′ ∈ ker 𝐺∗
}
,
in particular,
mul Γ = {0} ⇐⇒ ker 𝐹 = {0} ⇐⇒ ker 𝐺∗ = {0}.
The characteristic function (or transfer function) of the unitary colligation  (see [18])
𝜃(𝜁 ) = 𝐻 + 𝜁𝐺(𝐼 − 𝜁𝑇 )−1𝐹 (𝜁 ∈ 𝔻)
is holomorphic in 𝔻 and takes values in the set of contractive operators in.
1306 DERKACH ET AL.
Proposition 5.2. Let {,Γ} be a unitary boundary pair for 𝐴∗ with Γ given by (5.5), let 𝜆 ∈ ℂ+ and let 𝜁 = 𝜆−𝑖𝜆+𝑖 . Then the
𝛾-field admits the representations
𝛾(𝜆) =
{{
(𝜃(𝜁 ) − 𝐼)𝑢, (1 − 𝜁 )(𝐼 − 𝜁𝑇 )−1𝐹𝑢
}
∶ 𝑢 ∈ }, (5.9)
𝛾(?̄?) =
{{
(𝜃(𝜁 )∗ − 𝐼)𝑢, (1 − 𝜁 )(𝐼 − 𝜁𝑇 ∗)−1𝐺∗𝑢
}
∶ 𝑢 ∈ }, (5.10)
and its kernel does not depend on 𝜆 ∈ ℂ ⧵ℝ:
ker 𝛾(𝜆) = mul Γ0 = ker 𝛾
(
?̄?
)
. (5.11)
In particular,
𝛾(𝑖) = {{(𝐻 − 𝐼)𝑢, 𝐹 𝑢} ∶ 𝑢 ∈ }, 𝛾(−𝑖) = {{(𝐻∗ − 𝐼)𝑢, 𝐺∗𝑢} ∶ 𝑢 ∈ }. (5.12)
The Weyl function𝑀 corresponding to the boundary pair {,Γ} and the characteristic function 𝜃 are connected by
𝑀(𝜆) = 𝑖(𝐼 + 𝜃(𝜁 ))(𝐼 − 𝜃(𝜁 ))−1, 𝑀(?̄?) = −𝑖(𝐼 + 𝜃(𝜁 )∗)(𝐼 − 𝜃(𝜁 )∗)−1 (5.13)
If Γ is single-valued then dom 𝛾(𝜆) and dom 𝛾(?̄?) are dense in.
Proof. Since 𝜁 = 𝜆−𝑖
𝜆+𝑖 ∈ 𝔻 the operator (𝐼 − 𝜁𝑇 ) has a bounded inverse. Using the substitution 𝑔 = 𝑓 + 𝜁 (𝐼 − 𝜁𝑇 )
−1𝐹𝑢 one
can rewrite the expression (5.5) for Γ in the form{{(
(𝑇 − 𝐼)𝑓 + (1 − 𝜁 )(𝐼 − 𝜁𝑇 )−1𝐹𝑢
𝑖(𝑇 + 𝐼)𝑓 + 𝑖(1 + 𝜁 )(𝐼 − 𝜁𝑇 )−1𝐹𝑢
)
,
(
𝐺𝑓 + (𝜃(𝜁 ) − 𝐼)𝑢
−𝑖𝐺𝑓 − 𝑖(𝜃(𝜁 ) + 𝐼)𝑢
)}
∶𝑓 ∈ ℌ
𝑢 ∈ 
}
. (5.14)
Since 𝜆 = 𝑖 1+𝜁1−𝜁 the choice 𝑓 = 0 in (5.14) leads to
Γ↾ ?̂?𝜆 =
{{(
(1 − 𝜁 )(𝐼 − 𝜁𝑇 )−1𝐹𝑢
𝜆(1 − 𝜁 )(𝐼 − 𝜁𝑇 )−1𝐹𝑢
)
,
(
(𝜃(𝜁 ) − 𝐼)𝑢
−𝑖(𝜃(𝜁 ) + 𝐼)𝑢
)}
∶ 𝑢 ∈ 
}
and hence (5.9) and the first equalities in (5.12) and (5.13) follow.
Similarly, the substitution 𝑔′ = 𝑓 ′ + 𝜁
(
𝐼 − 𝜁𝑇 ∗
)−1
𝐺∗𝑢′ in (5.5) shows that the linear relation Γ coincides with the set of
vectors {(
(𝐼 − 𝑇 ∗)𝑓 ′ +
(
𝜁 − 1
)(
𝐼 − 𝜁𝑇 ∗
)−1
𝐺∗𝑢′
𝑖(𝐼 + 𝑇 ∗)𝑓 ′ + 𝑖
(
𝜁 + 1
)(
𝐼 − 𝜁𝑇 ∗
)−1
𝐺∗𝑢′
)
,
(
−𝐹 ∗𝑓 ′ + (𝐼 − 𝜃(𝜁 )∗)𝑢′
−𝑖𝐹 ∗𝑓 ′ − 𝑖(𝐼 + 𝜃(𝜁 )∗)𝑢′
)}
, (5.15)
where 𝑓 ′ ∈ ℌ, 𝑢′ ∈ . Hence with 𝑓 ′ = 0 one obtains from (5.15)
Γ↾ ?̂??̄? =
{{( (
𝜁 − 1
)(
𝐼 − 𝜁𝑇 ∗
)−1
𝐺∗𝑢′
?̄?
(
𝜁 − 1
)(
𝐼 − 𝜁𝑇 ∗
)−1
𝐺∗𝑢′
)
,
(
(𝐼 − 𝜃(𝜁 )∗)𝑢′
𝑖(𝐼 + 𝜃(𝜁 )∗)𝑢′
)}
∶ 𝑢′ ∈ 
}
. (5.16)
Now the formula (5.10) and the second equalities in (5.12) and (5.13) are implied by (5.16).
The equalities in (5.11) hold by the definition of the 𝛾-field and, in fact, are also clear from (5.9), (5.10), and the description
of mul Γ in Proposition 5.1.
Finally, if mul Γ = {0}, then using the fact that 𝛾(±𝑖) is single-valued, one concludes that
ker(𝐻 − 𝐼) ⊂ ker 𝐹 , ker(𝐻∗ − 𝐼) ⊂ ker 𝐺∗, (5.17)
and hence Proposition 5.1 shows that ker(𝐼 −𝐻) = ker(𝐼 −𝐻∗) = {0}. Therefore dom 𝛾(−𝑖) = ran(𝐼 −𝐻∗) and dom 𝛾(𝑖) =
ran(𝐼 −𝐻) are dense in . Equivalently, dom 𝛾(𝜆) is dense in  for all 𝜆 ∈ ℂ ⧵ℝ. □
Proposition 5.3. Let {,Γ} be a unitary boundary pair for 𝐴∗. Then the closure of ?̂?(𝜆) is given by
?̂?(𝜆) =
(
Γ0↾ ?̂?𝜆(𝐴∗)
)−1
, 𝜆 ∈ ℂ ⧵ℝ. (5.18)
DERKACH ET AL. 1307
In particular, ker ?̂?(𝜆) = mul Γ0, mul ?̂?(𝜆) =
(
ker Γ0
)
∩ ?̂?𝜆(𝐴∗), and
ran ?̂?(𝜆) =
(
domΓ0
)
∩ ?̂?𝜆(𝐴∗), dom ?̂?(𝜆) = Γ0
((
domΓ0
)
∩ ?̂?𝜆(𝐴∗)
)
.
Proof. By definition ?̂?(𝜆) =
(
Γ0↾ ?̂?𝜆(𝐴∗)
)−1
=
(
Γ0 ∩
(
?̂?𝜆(𝐴∗) ×))−1, which implies that
?̂?(𝜆)
−1
⊂ Γ0↾ ?̂?𝜆(𝐴∗), 𝜆 ∈ ℂ ⧵ℝ.
To prove the reverse inclusion, assume that
{
𝑓𝜆, ℎ
}
∈ Γ0 ∩
(
?̂?𝜆(𝐴∗) ×). With 𝜆 ∈ ℂ+ it follows from (5.14) that there are
sequences 𝑓𝑛 ∈ ℌ and 𝑢𝑛 ∈ , such that{(
(𝑇 − 𝐼)𝑓𝑛 + (1 − 𝜁 )(1 − 𝜁𝑇 )−1𝐹𝑢𝑛
𝑖(𝑇 + 𝐼)𝑓𝑛 + 𝑖(1 + 𝜁 )(1 − 𝜁𝑇 )−1𝐹𝑢𝑛
)
, 𝐺𝑓𝑛 + (𝜃(𝜁 ) − 𝐼)𝑢𝑛
}
ℌ2
←→
{(
𝑓𝜆
𝜆𝑓𝜆
)
, ℎ
}
.
This implies that (𝐼 − 𝜁𝑇 )𝑓𝑛
ℌ
←→ 0 and hence 𝑓𝑛
ℌ
←→ 0, since 𝜆 ∈ ℂ+ or, equivalently, 𝜁 ∈ 𝔻. Thus{(
(1 − 𝜁 )(1 − 𝜁𝑇 )−1𝐹𝑢𝑛
𝜆(1 − 𝜁 )(1 − 𝜁𝑇 )−1𝐹𝑢𝑛
)
, (𝜃(𝜁 ) − 𝐼)𝑢𝑛
}
ℌ2
←→
{(
𝑓𝜆
𝜆𝑓𝜆
)
, ℎ
}
,
which by (5.9) in Proposition 5.2 means that
{
𝑓𝜆, ℎ
}
∈ ?̂?(𝜆)
−1
.
Similarly, with ?̄? ∈ ℂ− it follows from (5.15) that for every
{
𝑓?̄?, ℎ
}
∈ Γ0 ∩
(
?̂??̄?(𝐴∗) ×) there exists a sequence 𝑢′𝑛 ∈ 
such that {(
(𝜁 − 1)(1 − 𝜁𝑇 )−1𝐺∗𝑢′𝑛
?̄?(𝜁 − 1)(1 − 𝜁𝑇 )−1𝐺∗𝑢′𝑛
)
, (𝐼 − 𝜃(𝜁 )∗)𝑢′𝑛
}
ℌ2
←→
{(
𝑓?̄?
?̄?𝑓?̄?
)
, ℎ
}
,
which by (5.10) in Proposition 5.2 means that
{
𝑓?̄?, ℎ
}
∈ ?̂?
(
?̄?
)−1
. This completes the proof of (5.18) and the remaining state-
ments follow easily from this identity. □
Corollary 5.4. Let {,Γ0,Γ1} be a unitary boundary triple for 𝐴∗ and let 𝑀(⋅) be the corresponding Weyl function. Then
the mapping Γ0 is closable if and only if for some, equivalently for every, 𝜆 ∈ ℂ ⧵ℝ the Weyl function satisfies the following
condition:
ℎ𝑛

←→ ℎ and Im
(
𝑀(𝜆)ℎ𝑛, ℎ𝑛
)
→ 0 (𝑛→ ∞) ⇐⇒ ℎ = 0. (5.19)
Proof. By Lemma 3.6𝑀(⋅) is an operator valued function with ker(𝑀(𝜆) −𝑀(𝜆)∗) = {0}. In this case (3.10) implies that(
𝜆 − ?̄?
)‖𝛾(𝜆)ℎ‖2ℌ = 2𝑖 Im (𝑀(𝜆)ℎ, ℎ) ,
ℎ ∈ dom𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ. From this formula it is clear that the condition (5.19) is equivalent to ker 𝛾(𝜆) = {0}. Therefore, the
result follows from Proposition 5.3. □
Clearly, the condition (5.19) is stronger than the condition (1.8) appearing in the definition of strict Nevanlinna functions.
If 𝑀(⋅) ∈ [] then the condition (5.19) simplifies to ker Im𝑀(𝜆) = {0}, i.e., for bounded Nevanlinna functions the con-
ditions (5.19) and (1.8) are equivalent. Hence, if
{,Γ0,Γ1} is a 𝐵-generalized boundary triple then Γ0 is closable by The-
orem 1.7. However, when 𝑀(⋅) is an unbounded Nevanlinna function, the condition in Corollary 5.4 need not be satisfied.
Example 5.19 shows that already for 𝑆-generalized boundary triples
{,Γ0,Γ1} the mapping Γ0 need not be closable.
The next result contains an essential improvement of Lemma 3.8 for unitary boundary pairs.
Theorem 5.5. Let {,Γ} be a unitary boundary pair for 𝐴∗, let its Potapov–Ginzburg transform = 𝜔(Γ) be given by (5.4),
let𝐻(𝜆) be defined by
𝐻(𝜆) ∶ ℎ→
{(
(𝐴0 − 𝜆)−1ℎ
ℎ + 𝜆(𝐴0 − 𝜆)−1ℎ
)}
,
1308 DERKACH ET AL.
see (3.11), and denote 𝜁 = 𝜆−𝑖
𝜆+𝑖 for 𝜆 ∈ ℂ+. Then:
(i) with 𝜆 ∈ ℂ+ the adjoint of the 𝛾-field is given by the formulas
𝛾
(
?̄?
)∗ = {{𝑔, 𝑣} ∶ (𝜃(𝜁 ) − 𝐼)𝑣 + (𝜁 − 1)𝐺(𝐼 − 𝜁𝑇 )−1𝑔 = 0, 𝑔 ∈ ℌ, 𝑣 ∈ }, (5.20)
𝛾(𝜆)∗ =
{
{𝑔′, 𝑣′} ∶ (𝜃(𝜁 )∗ − 𝐼)𝑣′ +
(
𝜁 − 1
)
𝐹 ∗
(
𝐼 − 𝜁𝑇 ∗
)−1
𝑔′ = 0, 𝑔′ ∈ ℌ, 𝑣′ ∈ }; (5.21)
(ii) for all 𝜆 ∈ ℂ+ ∪ ℂ− one has the equality
Γ𝐻(𝜆) =
(
0
𝛾
(
?̄?
)∗) +̂ ({0} × mul Γ),
and, in particular,
Γ1𝐻(𝜆) = 𝛾
(
?̄?
)∗ +̂ ({0} × mul Γ1),
and, furthermore, here ran
(
Γ1𝐻(𝜆)
)
= ran
(
Γ1𝐻
(
?̄?
))
does not depend on 𝜆 ∈ ℂ+ ∪ ℂ−;
(iii) the range of 𝛾(𝜆)∗ does not depend on 𝜆 ∈ ℂ+ ∪ ℂ−:
ran 𝛾
(
?̄?
)∗ = (𝐼 −𝐻)−1(ran𝐺) = (𝐼 −𝐻∗)−1(ran𝐹 ∗) = ran 𝛾(𝜆)∗; (5.22)
(iv) the multi-valued part of 𝛾(𝜆)∗ does not depend on 𝜆 ∈ ℂ+ ∪ ℂ−:
mul 𝛾
(
?̄?
)∗ = ker(𝐻 − 𝐼) = ker(𝐻∗ − 𝐼) = mul 𝛾(𝜆)∗.
In particular, if mul Γ1 = {0} then the equality ran
(
Γ1𝐻(𝜆)
)
= ran 𝛾
(
?̄?
)∗ holds for all 𝜆 ∈ ℂ+ ∪ ℂ−. In this case also
mul 𝛾(𝜆)∗ = {0} or, equivalently, the 𝛾-field 𝛾(𝜆) is a densely defined operator for all 𝜆 ∈ ℂ ⧵ℝ.
Proof.
(i) Let 𝜆 ∈ ℂ+ and assume that {𝑔, 𝑣} ∈ 𝛾
(
?̄?
)∗
for some 𝑔 ∈ ℌ and 𝑣 ∈ . Then by (5.10) this means that((
1 − 𝜁
)(
𝐼 − 𝜁𝑇 ∗
)−1
𝐺∗𝑢, 𝑔
)
= ((𝜃(𝜁 )∗ − 𝐼)𝑢, 𝑣)
for all 𝑢 ∈  or, equivalently, (𝜃(𝜁 ) − 𝐼)𝑣 + (𝜁 − 1)𝐺(𝐼 − 𝜁𝑇 )−1𝑔 = 0. This proves the identity (5.20) in (ii) with 𝜆 ∈ ℂ+.
Similarly, using (5.9) it is seen that {𝑔′, 𝑣′} ∈ 𝛾(𝜆)∗
(
𝜆 ∈ ℂ+
)
is equivalent to
(𝜃(𝜁 )∗ − 𝐼)𝑣 +
(
𝜁 − 1
)
𝐹 ∗
(
𝐼 − 𝜁𝑇 ∗
)−1
𝑔′ = 0,
which proves (5.21).
(ii) By Lemma 3.8 (cf. also [27, Lemma 7.38]) the inclusions “⊂” hold in (ii). For the reverse inclusions “⊃” in (ii) it suffices
to prove the inclusion 𝛾
(
?̄?
)∗
⊂ Γ1𝐻(𝜆). For this purpose we first derive a formula for the mapping𝐻(𝜆) defined in (3.11)
analogous to what appears in Proposition 5.2. It follows from (5.14) and (5.15) (with 𝜆 ∈ ℂ+) that
𝐴0 − 𝜆 =
{(
(𝑇 − 𝐼)𝑓 + (1 − 𝜁 )(𝐼 − 𝜁𝑇 )−1𝐹𝑢
2𝑖
1−𝜁 (𝐼 − 𝜁𝑇 )𝑓
)
∶
𝑓 ∈ ℌ, 𝑢 ∈ 
𝐺𝑓 + (𝜃(𝜁 ) − 𝐼)𝑢 = 0
}
, (5.23)
𝐴0 − ?̄? =
{(
(𝐼 − 𝑇 ∗)𝑓 ′ +
(
𝜁 − 1
)(
𝐼 − 𝜁𝑇 ∗
)−1
𝐺∗𝑢′
−2𝑖
1−𝜁 (𝐼 − 𝜁𝑇
∗)𝑓 ′
)
∶
𝑓 ′ ∈ ℌ, 𝑢′ ∈ 
−𝐹 ∗𝑔′ + (𝐼 − 𝜃(𝜁 )∗)𝑢′
}
. (5.24)
In particular, using (5.23) and the substitution 𝑔 = 2𝑖(1 − 𝜁𝑇 )(1 − 𝜁 )−1𝑓 one obtains
ℎ ∶= (𝐴0 − 𝜆)−1𝑔 =
1 − 𝜁
2𝑖
(𝑇 − 𝐼)(𝐼 − 𝜁𝑇 )−1𝑔 + (1 − 𝜁 )(𝐼 − 𝜁𝑇 )−1𝐹𝑢,
DERKACH ET AL. 1309
where 𝑢 ∈  satisfies the equality
1 − 𝜁
2𝑖
𝐺(𝐼 − 𝜁𝑇 )−1𝑔 + (𝜃(𝜁 ) − 𝐼)𝑢 = 0,
or, equivalently,
(𝜃(𝜁 ) − 𝐼)(−2𝑖𝑢) + (𝜁 − 1)𝐺(𝐼 − 𝜁𝑇 )−1𝑔 = 0. (5.25)
Now denoting ℎ̂ =
(
ℎ
ℎ′
)
= 𝐻(𝜆)𝑔 ∈ 𝐴0 one concludes from (5.14) that
{
ℎ̂,−2𝑖𝑢
}
=
{
ℎ̂,−𝑖𝐺𝑓 − 𝑖(𝜃(𝜁 ) + 𝐼)𝑢
}
∈ Γ1,
so that {𝑔, 𝑣} ∈ Γ1𝐻(𝜆), where 𝑣 = −2𝑖𝑢 and 𝑔 satisfies (5.25) or, equivalently, 𝑓 = (1 − 𝜁 )(2𝑖(𝐼 − 𝜁𝑇 ))−1𝑔 satisfies
𝐺𝑓 + (𝜃(𝜁 ) − 𝐼)𝑢 = 0. It remains to compare (5.20) with (5.25) to conclude that the elements {𝑔, 𝑣} ∈ 𝛾
(
?̄?
)∗
in fact belong
to Γ1𝐻(𝜆), i.e., the inclusion 𝛾
(
?̄?
)∗
⊂ Γ1𝐻(𝜆) holds
(
𝜆 ∈ ℂ+
)
.
Similarly, one proves the inclusion 𝛾(𝜆)∗ ⊂ Γ1𝐻
(
?̄?
) (
𝜆 ∈ ℂ+
)
by means of (5.21), (5.15) and (5.24).
Finally, the equality ran
(
Γ1𝐻(𝜆)
)
= ran
(
Γ1𝐻
(
?̄?
))
and the independence from 𝜆 ∈ ℂ+ ∪ ℂ− holds by item (vi) of
Lemma 3.7.
(iii) Let 𝑢 ∈ (𝐻 − 𝐼)−1(ran𝐺 ∩ ran(𝐻 − 𝐼)) and 𝜆 ∈ ℂ+. Then (𝐻 − 𝐼)𝑢 ∈ ran𝐺 and hence
(𝜃(𝜁 ) − 𝐼)𝑢 = (𝐻 − 𝐼)𝑢 + 𝜁𝐺(𝐼 − 𝜁𝑇 )−1𝐹𝑢 ∈ ran𝐺. (5.26)
In view of (5.20) this proves that 𝑢 ∈ ran 𝛾
(
?̄?
)∗
. Conversely, if 𝑢 ∈ ran 𝛾
(
?̄?
)∗
then in view of (5.20) and (5.26)
(𝐻 − 𝐼)𝑢 ∈ ran𝐺, which proves the first equality in (5.22). Similarly, for 𝜆 ∈ ℂ− the last equality in (5.22) is implied
by (5.21).
Finally, the equality (𝐻 − 𝐼)−1ran𝐺 = (𝐻∗ − 𝐼)−1ran𝐹 ∗ is implied by the identities  ∗ =  ∗ = 𝐼ℌ⊕ . To see
this assume that 𝑢 ∈ (𝐻 − 𝐼)−1ran𝐺, i.e., that (𝐻 − 𝐼)𝑢 = 𝐺𝑔 for some 𝑔 ∈ ℌ. Then 𝐻∗(𝐻 − 𝐼)𝑢 = 𝐻∗𝐺𝑔 = −𝐹 ∗𝑇 𝑔
and using𝐻∗𝐻 = 𝐼 − 𝐹 ∗𝐹 one obtains (𝐼 −𝐻∗)𝑢 = 𝐹 ∗(𝐹𝑢 − 𝑇 𝑔). Hence (𝐻 − 𝐼)−1ran𝐺 ⊂ (𝐻∗ − 𝐼)−1ran𝐹 ∗ and the
reverse inclusion is proved similarly.
(iv) Part (i) shows that the adjoint of the 𝛾-field at 𝜆 = ±𝑖 is given by
𝛾(𝑖)∗ =
{
{𝑓, 𝑓 ′} ∶ (𝐻∗ − 𝐼)𝑓 ′ = 𝐹 ∗𝑓
}
, 𝛾(−𝑖)∗ =
{
{𝑓, 𝑓 ′} ∶ (𝐻 − 𝐼)𝑓 ′ = 𝐺𝑓
}
,
see also (5.12). On the other hand, mul 𝛾(𝜆)∗ = (dom 𝛾(𝜆))⟂ = (dom𝑀(𝜆))⟂ = mul𝑀(𝜆) does not depend on
𝜆 ∈ ℂ+ ∪ ℂ−. The identities in (iv) are now clear from the formulas in (i).
The last statement is obtained from (ii). □
Corollary 5.6. (Cf. [27].) Assume that Γ1 in Theorem 5.5 is a single-valued operator. Then the operator Γ1𝐻(𝜆) is bounded
or, equivalently, Γ1↾𝐴0 is bounded if and only if 𝐴0 is closed.
Proof. By Lemma 3.7 the operator Γ1𝐻(𝜆) is bounded if and only if the restriction Γ1↾𝐴0 is bounded. Since Γ1𝐻(𝜆) = 𝛾
(
?̄?
)∗
by Theorem 5.5, this mapping is closed and it follows from the closed graph theorem that Γ1↾𝐴0 is bounded if and only if its
domain dom
(
Γ1↾𝐴0
)
= 𝐴0 is closed. □
Remark 5.7. The unitary colligation { ,ℌ,,} from (5.4) is an operator formalization of a discrete time input/state/output
system
(
𝑥(𝑡 + 1)
𝑦(𝑡)
)
= 
(
𝑥(𝑡)
𝑢(𝑡)
)
(𝑡 ∈ ℕ) with the input 𝑢(𝑡) ∈  and the output 𝑦(𝑡) ∈ . The transfer function of this discrete
time input/state/output system coincides with the characteristic function 𝜃(𝑧) of the unitary colligation ( ,ℌ,,), see [5, 57].
1310 DERKACH ET AL.
Similarly, as was shown in [6, Theorem 5.35] any unitary boundary triple
{,Γ0,Γ1} with the extra properties ran Γ0 = ℌ
and mul ̃ = {0}, where ̃ (a skew-adjoint operator) is the analog of the main transform of Γ (see (5.1)), corresponds to some
impedance conservative continuous time input/state/output system
Σ =
{̃,ℌ,,} ∶ (?̇?(𝑡)
𝑦(𝑡)
)
= ̃
(
𝑥(𝑡)
𝑢(𝑡)
) (
𝑡 ∈ ℝ+
)
.
Realization problems for Schur functions via transfer functions of scattering conservative (and passive) continuous time
input/state/output systems were studied in [8, 9] and were motivated by the earlier works [65, 66]. On the other hand, con-
nections between general unitary boundary pairs and the notion of conservative state/signal system nodes, whose systematic
study was initiated in [5] (see also e.g. [53, 54]), have been established in [6, Theorem 5.34]. Moreover, the connection between
conservative state/signal system nodes and so-called Dirac structures can be found in [6, Proposition 5.38], while the connection
between Dirac structures and unitary boundary pairs is made explicit in [41].
5.2 A Kreı˘n type formula for unitary boundary triples
In this section Kreı˘n’s resolvent formula is extended to the setting of general unitary boundary triples. It is analogous to the
formula established in Section 4.2. Recall from [25] that for a unitary boundary triple the kernel 𝐴0 = ker Γ0 need not be
selfadjoint, it is in general only a symmetric extension of 𝐴 which can even coincide with 𝐴; see e.g. [25, Example 6.6]. For
simplicity the next result is formulated for nonreal points 𝜆 ∈ ℂ ⧵ℝ; these points are regular type points for 𝐴0.
As in Section 4.2, let𝐴 be an extension of𝐴which belongs to the domain of Γ and letΘ be a linear relation in corresponding
to 𝐴 via (4.15).
Theorem 5.8. Let𝐴 be a closed symmetric relation, letΠ = {,Γ0,Γ1} be a unitary boundary triple for𝐴∗ with𝐴0 = ker Γ0,
and let 𝑀(⋅) and 𝛾(⋅) be the corresponding Weyl function and 𝛾-field, respectively. Then for any linear relation Θ(⊂ ran Γ) in
 and the extension 𝐴Θ ∈ Ext𝐴 given by (4.15) the following equality holds(
𝐴Θ − 𝜆
)−1 − (𝐴0 − 𝜆)−1 = 𝛾(𝜆)(Θ −𝑀(𝜆))−1𝛾(?̄?)∗, 𝜆 ∈ ℂ ⧵ℝ, (5.27)
where the inverses in the first and last term are taken in the sense of linear relations.
Proof. We first prove the inclusion “⊂” in (5.27). Since𝐴0 is symmetric,
(
𝐴0 − 𝜆
)−1
is a bounded, in general nondensely defined,
operator for every fixed 𝜆 ∈ ℂ ⧵ℝ. Now assume that {𝑔, 𝑔′′} ∈
(
𝐴Θ − 𝜆
)−1 − (𝐴0 − 𝜆)−1. Then 𝑔 ∈ dom(𝐴Θ − 𝜆)−1 ∩
dom
(
𝐴0 − 𝜆
)−1
and {𝑔, 𝑔′} ∈
(
𝐴Θ − 𝜆
)−1
for some 𝑔′ ∈ ℌ, so that 𝑔′′ = 𝑔′ −
(
𝐴0 − 𝜆
)−1
𝑔. Hence 𝑔Θ ∶= {𝑔′, 𝑔 + 𝜆𝑔′} ∈
𝐴Θ ⊂ domΓ,
𝑔0 ∶=
{(
𝐴0 − 𝜆
)−1
𝑔,
(
𝐼 + 𝜆
(
𝐴0 − 𝜆
)−1)
𝑔
}
∈ 𝐴0 ⊂ domΓ, (5.28)
and
𝑔Θ − 𝑔0 =
{
𝑔′ −
(
𝐴0 − 𝜆
)−1
𝑔, 𝜆
(
𝑔′ −
(
𝐴0 − 𝜆
)−1
𝑔
)}
,
so that 𝑔Θ − 𝑔0 ∈ ?̂?𝜆(𝐴∗). Recall that ?̂?(𝜆) maps dom ?̂?(𝜆) onto ?̂?𝜆(𝐴∗) ⊂ domΓ and hence there exists 𝜑 ∈ dom ?̂?(𝜆) =
dom𝑀(𝜆) such that
𝑔Θ − 𝑔0 = ?̂?(𝜆)𝜑, Γ ?̂?(𝜆)𝜑 = {𝜑,𝑀(𝜆)𝜑}, (5.29)
see (3.2), (3.3); notice that 𝑀(𝜆) is an operator, since mul Γ = {0}. Clearly Γ0𝑔0 = 0 and according to Theorem 5.5 one has
Γ1𝑔0 = Γ1𝐻(𝜆)𝑔 = 𝛾
(
𝜆
)∗
𝑔, where 𝐻(𝜆) is defined by (3.11). Observe, that here 𝛾
(
?̄?
)∗
is an operator since 𝐻(𝜆) and Γ1 are
operators. Now it follows from (5.29) that{
0, 𝛾
(
?̄?
)∗
𝑔
}
+ {𝜑,𝑀(𝜆)𝜑} = Γ 𝑔0 + Γ ?̂?(𝜆)𝜑 = Γ 𝑔Θ ∈ Θ, (5.30)
see (4.15). Consequently, {
𝜑, 𝛾
(
?̄?
)∗
𝑔 +𝑀(𝜆)𝜑
}
∈ Θ and
{
𝜑, 𝛾
(
?̄?
)∗
𝑔
}
∈ Θ −𝑀(𝜆)
DERKACH ET AL. 1311
or, equivalently, {𝑔, 𝜑} ∈ (Θ −𝑀(𝜆))−1𝛾
(
?̄?
)∗
and hence (5.29) shows that{
𝑔, 𝑔′′
}
= {𝑔, 𝛾(𝜆)𝜑} ∈ 𝛾(𝜆)(Θ −𝑀(𝜆))−1𝛾
(
?̄?
)∗
,
which proves the first inclusion in (5.27).
To prove the reverse inclusion “⊃” in (5.27) assume that {𝑔, 𝑔′′} ∈ 𝛾(𝜆)(Θ −𝑀(𝜆))−1𝛾
(
?̄?
)∗
. Since dom(Θ −𝑀(𝜆)) ⊂
dom𝑀(𝜆) = dom 𝛾(𝜆) the assumption on {𝑔, 𝑔′′} means that for some 𝜑 ∈  one has {𝛾(?̄?)∗𝑔, 𝜑} ∈ (Θ −𝑀(𝜆))−1 and
{𝑔, 𝑔′′} = {𝑔, 𝛾(𝜆)𝜑} ∈ 𝛾(𝜆)(Θ −𝑀(𝜆))−1𝛾
(
?̄?
)∗
.
It follows from
{
𝜑, 𝛾
(
?̄?
)∗
𝑔 +𝑀(𝜆)𝜑
}
∈ Θ and (4.15) that Γ 𝑔Θ =
{
𝜑, 𝛾
(
?̄?
)∗
𝑔 +𝑀(𝜆)𝜑
}
for some 𝑔Θ ∈ 𝐴Θ. By Theo-
rem 5.5 Γ1𝐻(𝜆) = 𝛾
(
?̄?
)∗
, which shows that 𝑔 ∈ ran
(
𝐴0 − 𝜆
)
, 𝜆 ∈ ℂ ⧵ℝ; see (3.11). Now associate with 𝑔 the element
𝑔0 as in (5.28). Since Γ 𝑔0 =
{
0, 𝛾
(
?̄?
)∗
𝑔
}
and Γ ?̂?(𝜆)𝜑 = {𝜑,𝑀(𝜆)𝜑} we conclude that (5.30) is satisfied. Therefore,
𝑔0 + ?̂?(𝜆)𝑔 − 𝑔Θ ∈ ker Γ = 𝐴 and thus 𝑔0 + ?̂?(𝜆)𝜑 ∈ 𝐴Θ or, equivalently,{
𝑔,
(
𝐴0 − 𝜆
)−1
𝑔 + 𝛾(𝜆)𝜑
}
∈
(
𝐴Θ − 𝜆
)−1
.
Hence,
{𝑔, 𝑔′′} = {𝑔, 𝛾(𝜆)𝜑} ∈
(
𝐴Θ − 𝜆
)−1 − (𝐴0 − 𝜆)−1.
This proves the reverse inclusion in (5.27) and completes the proof. □
It is useful to make some further comments on the formula (5.27).
Remark 5.9.
(i) Again notice the generality of the formula (5.27); in particular, as in Theorem 4.12 𝜆 need not belong to 𝜌
(
𝐴Θ
)
.
(ii) A careful look at the above proof shows that the key elements which in addition to the general properties of 𝛾-fields and
Weyl functions of isometric boundary triples are used in the proof are the following two requirements: (1) the equality
Γ1𝐻(𝜆) = 𝛾
(
?̄?
)∗
, so that Γ1𝑔0 = 𝛾
(
?̄?
)∗
𝑔 when 𝑔 and 𝑔0 are connected by (5.28); (2) 𝛾(𝜆) (hence also𝑀(𝜆)) is a densely
defined operator or, equivalently, 𝛾
(
?̄?
)∗
is a closed operator. Hence, the formula (5.27) in Theorem 4.12 remains valid for
isometric boundary triples which satisfy these two additional properties.
(iii) In the formula (5.27) the operator
(
𝐴0 − 𝜆
)−1
cannot be shifted to the right hand side without loosing the stated equality.
Indeed, in that case only the following inclusion remains valid:(
𝐴Θ − 𝜆
)−1
⊃
(
𝐴0 − 𝜆
)−1 − 𝛾(𝜆)(Θ −𝑀(𝜆))−1𝛾(?̄?)∗.
Namely, by the equality Γ1𝐻(𝜆) = 𝛾
(
?̄?
)∗
one has ran
(
𝐴0 − 𝜆
)
= dom 𝛾
(
?̄?
)∗
and thus the term
(
𝐴0 − 𝜆
)−1
can be shifted
to the right side of (5.27) without changing the domain on the right side. However, in this case the range of the right side
belongs to the span dom𝐴0 +𝔑𝜆(𝐴∗) and for general unitary boundary triples this would restrict the choice of 𝐴Θ; recall
that for a unitary boundary triple 𝐴0 need not be even essentially selfadjoint, one can even have 𝐴0 = 𝐴.
By considering the multi-valued parts we obtain the following statement for the point spectrum of 𝐴Θ from Theorem 5.8.
Corollary 5.10. With the assumptions in Theorem 5.8 one has 𝜆 ∈ 𝜎𝑝(𝐴Θ) if and only if 0 ∈ 𝜎𝑝
(
Θ −𝑀(𝜆)
)
, in which case
ker
(
𝐴Θ − 𝜆
)
= 𝛾(𝜆)ker(Θ −𝑀(𝜆)), 𝜆 ∈ ℂ ⧵ℝ.
We are now ready to prove also Theorem 4.12 from Section 4.2.
Proof of Theorem 4.12. By assumption Π =
{,Γ0,Γ1} is an 𝐴𝐵-generalized boundary triple for 𝐴∗. Hence, mul Γ = 0 and
according to Theorem 4.2 (iv) this implies that Γ1𝐻(𝜆) = 𝛾
(
?̄?
)∗
and, moreover, 𝛾(?̄?)∗ is a bounded everywhere defined operator.
Thus, from part (ii) in Remark 5.9 one concludes that the formula (5.27) holds. Furthermore, for an 𝐴𝐵-generalized boundary
triple 𝐴0 is selfadjoint. Thus dom
(
𝐴0 − 𝜆
)−1 = ℌ (𝜆 ∈ ℂ ⧵ℝ) and the formula (5.27) is equivalent to the formula (4.16) in
Theorem 4.12. □
1312 DERKACH ET AL.
5.3 𝑺-generalized boundary triples
Here we extend Definition 1.11 to the case of boundary pairs.
Definition 5.11. A unitary boundary pair {,Γ} is said to be an 𝑆-generalized boundary pair, if 𝐴0 is a selfadjoint linear
relation in ℌ.
In the following proposition some special boundary triples/pairs are characterized in terms of their Potapov–Ginzburg trans-
form.
Proposition 5.12. Let {,Γ} be a unitary boundary pair, let  = 𝜔(Γ) be its Potapov–Ginzburg transform given by (5.2)
and (5.4), and let 𝐴∗ = domΓ, 𝐴0 = ker Γ0. Then:
(i)
{,Γ0,Γ1} is an ordinary boundary triple if and only if
ran𝐺 =  ⇐⇒ ran𝐹 ∗ = ;
(ii)
{,Γ0,Γ1} is a 𝐵-generalized boundary triple if and only if{
ker 𝐹 = {0},
ran(𝐼 −𝐻) =  ⇐⇒
{
ker 𝐺∗ = {0},
ran(𝐼 −𝐻∗) =  ;
(iii) {,Γ} is a 𝐵-generalized boundary pair if and only if
Γ0|?̂?𝑖 =  ⇐⇒ ran(𝐼 −𝐻) =  ⇐⇒ ran(𝐼 −𝐻∗) = ;
(iv) Γ0 is surjective if and only if
ran(𝐼 −𝐻) + ran𝐺 =  ⇐⇒ ran(𝐼 −𝐻∗) + ran𝐹 ∗ = ;
(v) {,Γ} is an 𝑆-generalized boundary pair if and only if
ran𝐺 ⊂ ran(𝐼 −𝐻) and ran𝐹 ∗ ⊂ ran (𝐼 −𝐻∗).
Proof. The statements (i)–(iii) can be found in [10, Proposition 5.9, Corollaries 5.11 and 5.12].
(iv) This is implied by (5.8).
(v) This statement follows from the equalities
𝐴0 − 𝑖 =
{(
(𝑇 − 𝐼)𝑔 + 𝐹𝑢
2𝑖𝑔
)
∶ 𝐺𝑔 + (𝐻 − 𝐼)𝑢 = 0, 𝑔 ∈ ℌ, 𝑢 ∈ 
}
,
𝐴0 + 𝑖 =
{(
(𝐼 − 𝑇 ∗)𝑔′ − 𝐺∗𝑢′
2𝑖𝑔′
)
∶ 𝐹 ∗𝑔′ + (𝐻∗ − 𝐼)𝑢′ = 0, 𝑔′ ∈ ℌ, 𝑢′ ∈ 
}
which, in turn, are implied by (5.7). □
Remark 5.13. An example of a unitary boundary triple {,Γ}, such that 𝐴0 is selfadjoint and Γ0 is not surjective is presented
in [25, Example 6.6]. Observe also that𝐴0 is a maximal symmetric operator if at least one of the conditions ran𝐺 ⊂ ran (𝐼 −𝐻)
or ran𝐹 ∗ ⊂ ran(𝐼 −𝐻∗) is satisfied.
The statement (v) in Proposition 5.12 is closely related to the early work of Calkin on existence of maximal symmetric
extensions 𝐴 contained in the domain of a reduction operator for 𝐴∗ (meaning here domΓ); cf. [21, Theorems 4.8, 4.11, 4.12].
His results are described in modern terms in [42, Theorems 2.26, 2.27] by means of an angular representation for 𝐴0.
The following lemma shows that the conditions (iv) and (v) in Proposition 5.12 are not unrelated.
Lemma 5.14. Let  be a unitary colligation of the form (5.4). Then the following conditions are equivalent:
(i) ran(𝐼 −𝐻) + ran𝐺 = ;
(ii) ran(𝐼 −𝐻∗) + ran𝐹 ∗ = ;
DERKACH ET AL. 1313
(iii) ran(𝐼 −𝐻) = ;
(iv) ran(𝐼 −𝐻∗) = .
Proof. The equivalence of (i) and (ii) is implied by (5.8).
Since ran (𝐼 −𝐻) ⊆ ran(𝐼 −𝐻) + ran𝐺 and ran(𝐼 −𝐻∗) ⊆ ran(𝐼 −𝐻∗) + ran𝐹 ∗ it remains to prove the implications (i)
⇒ (iii) and (ii)⇒ (iv).
Assume that ran(𝐼 −𝐻) + ran𝐺 = . Then [35, Theorem 2.2] and the identity𝐻𝐻∗ + 𝐺𝐺∗ = 𝐼 yield
ran(𝐼 −𝐻) + ran𝐺 = ran
(
((𝐼 −𝐻)(𝐼 −𝐻∗))1∕2
)
+ ran
(
(𝐺𝐺∗)1∕2
)
= ran
(
((𝐼 −𝐻)(𝐼 −𝐻∗) + 𝐺𝐺∗)1∕2
)
= ran
(
(𝐼 − 2Re𝐻 +𝐻𝐻∗ + 𝐺𝐺∗)1∕2
)
= ran
(
(𝐼 − Re𝐻)1∕2
)
.
This implies the equality ran(𝐼 − Re𝐻) =  and hence −𝐼 ≤ Re𝐻 ≤ 𝑞𝐼 for some 𝑞 < 1. Therefore, the numerical range of𝐻
is contained in the half-planeRe 𝑧 ≤ 𝑞 and hence 1 ∈ 𝜌(𝐻). This proves (iii). The implication (ii)⇒ (iv) is proved similarly. □
Corollary 5.15. If Π = {,Γ0,Γ1} is a unitary boundary triple with ran Γ0 = , then 𝐴0 = 𝐴∗0 and Π is necessarily a
𝐵-generalized boundary triple.
Remark 5.16. If
{,Γ0,Γ1} is an ordinary boundary triple, then Γ and, consequently, Γ0 and Γ1 are surjective. Hence,𝐴0 = 𝐴∗0
and 𝐴1 = 𝐴∗1. This conclusion can be made directly also from Proposition 5.12. Indeed, the assumption ran𝐺 =  implies
0 ∈ 𝜌(𝐺𝐺∗). In view of the identity 𝐺𝐺∗ = 𝐼 −𝐻𝐻∗ this implies 1 ∈ 𝜌(𝐻𝐻∗) and hence 1 ∈ 𝜌(𝐻). By Proposition 5.12 (v)
this condition yields 𝐴0 = 𝐴∗0.
We are now ready to prove Theorem 1.12 in a more general setting, where {,Γ} is an arbitrary unitary boundary pair. It
gives a complete characterization of the Weyl functions 𝑀(⋅) of 𝑆-generalized boundary pairs. In its present general form it
completes and extends [25, Theorem 4.13] and [27, Theorem 7.39].
Theorem 5.17. Let Π = {,Γ} be a unitary boundary pair and let 𝑀(⋅) and 𝛾(⋅) be the corresponding Weyl family and the
𝛾-field. Then the following statements are equivalent:
(i) 𝐴0 is selfadjoint, i.e. Π is an 𝑆-generalized boundary pair;
(ii) 𝐴∗ = 𝐴0 +̂ ?̂?𝜆 and 𝐴∗ = 𝐴0 +̂ ?̂?𝜇 for some (equivalently for all) 𝜆 ∈ ℂ+ and 𝜇 ∈ ℂ−;
(iii) ran Γ0 = dom𝑀(𝜆) = dom𝑀(𝜇) for some (equivalently for all) 𝜆 ∈ ℂ+ and 𝜇 ∈ ℂ−;
(iv) 𝛾(𝜆) and 𝛾(𝜇) are bounded for some (equivalently for all) 𝜆 ∈ ℂ+ and 𝜇 ∈ ℂ−;
(v) dom𝑀(𝜆) = dom𝑀
(
𝜆
)
and Im𝑀op(𝜆) is bounded for some (equivalently for all) 𝜆 ∈ ℂ+;
(vi) The Weyl family𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ, admits the representation
𝑀(𝜆) = 𝐸 +𝑀0(𝜆), (5.31)
where 𝐸 = 𝐸∗ is a selfadjoint relation in  and𝑀0 ∈ [0], with 0 = dom𝐸.
Proof. (i)⇔ (ii) This equivalence and the independence from 𝜆 ∈ ℂ+ and 𝜇 ∈ ℂ− is proved in [25, Theorem 4.13].
(i) ⇔ (iii) This can also be obtained from [25, Theorem 4.13], but we present here a different proof. Indeed, it follows
from (5.14) that for all 𝜆 ∈ ℂ+ and 𝜁 =
𝜆−𝑖
𝜆+𝑖
ran Γ0 = ran𝐺 + ran(𝜃(𝜁 ) − 𝐼). (5.32)
If𝐴0 = 𝐴∗0 then by Proposition 5.12 ran𝐺 ⊂ ran (𝐼 −𝐻) and (5.23) (see the proof of Theorem 5.5) yields ran𝐺⊂ ran(𝜃(𝜁 ) − 𝐼).
By (5.32), (5.9), and dom 𝛾(𝜆) = dom𝑀(𝜆) one obtains
ran Γ0 = ran(𝜃(𝜁 ) − 𝐼) = dom𝑀(𝜆) for all 𝜆 ∈ ℂ+.
Similarly, it follows from (5.32) and (5.24) in the proof of Theorem 5.5 that
ran Γ0 = ran𝐹 ∗ + ran (𝜃(𝜁 )∗ − 𝐼) = dom𝑀
(
?̄?
)
for all 𝜆 ∈ ℂ+.
1314 DERKACH ET AL.
Conversely, if for some 𝜆 ∈ ℂ+ one has ran Γ0 = dom𝑀(𝜆) = dom 𝛾(𝜆), then (5.8) implies, in particu-
lar, that ran𝐺⊂ ran(𝜃(𝜁 ) − 𝐼). Hence, it follows from (5.23) that ran
(
𝐴0 − 𝜆
)
= ℌ. Similarly the identities
ran Γ0 = dom𝑀
(
?̄?
)
=dom 𝛾
(
?̄?
)
imply that ran
(
𝐴0 − ?̄?
)
= ℌ and, thus, 𝐴0 = 𝐴∗0.
(i)⇒ (iv) This implication was proved in Theorem 4.2 (iv), (v).
(iv)⇒ (i) If some 𝛾(𝜆) ∶ dom 𝛾(𝜆)→ ℌ is bounded then dom 𝛾(𝜆)∗ = ℌ. Then by Theorem 5.5
ran
(
𝐴0 − ?̄?
)
= domΓ1𝐻
(
?̄?
)
= dom 𝛾(𝜆)∗ = ℌ.
Similarly if 𝛾(𝜇) is bounded then ran
(
𝐴0 − ?̄?
)
= ℌ. Thus, 𝐴0 is a selfadjoint relation in ℌ.
(iv)⇒ (v), (vi) Consider the decomposition (2.3)𝑀(𝜆) = gr𝑀op(𝜆)⊕𝑀∞ of the Weyl family𝑀(𝜆) with the operator part
𝑀op ∈ (0), where 0 = dom𝑀(𝜆). As was already shown, now 𝐴0 = 𝐴∗0 and dom𝑀op(𝜆) = ran Γ0 for all 𝜆 ∈ ℂ ⧵ℝ.
It follows from the equality 𝑀op(𝜆)∗ = 𝑀op
(
?̄?
)
that the operator 𝐸0 = Re𝑀op
(
𝜆0
)(
𝜆0 ∈ ℂ+
)
is selfadjoint with the domain
dom𝐸0 = ran Γ0. Moreover, since the operator 𝛾(𝜆) is bounded for all 𝜆 ∈ ℂ ⧵ℝ it follows from the equality (3.6) that the
operator
Im𝑀op
(
𝜆0
)
= Im 𝜆0𝛾
(
𝜆0
)∗
𝛾
(
𝜆0
)
is also bounded in 0 and hence the operator 𝑀op(𝜆) − 𝐸0 is bounded in 0 at 𝜆0. Therefore, its closure, denoted now by
𝑀0(𝜆), is bounded in 0 at 𝜆0 and then also for all 𝜆 ∈ ℂ ⧵ℝ; see e.g. [26, Proposition 4.18], [28, Theorem 3.9]. Finally, by
setting 𝐸 = 𝐸0 ⊕𝑀∞ one arrives at (5.31).
Finally, the implication (vi)⇒ (v) is clear and (v)⇒ (iv) (for 𝜇 = ?̄?) follows easily from (3.6). □
Theorem 5.17 implies Theorem 1.12. In the case that Γ is single-valued𝑀(𝜆) is an operator valued Nevanlinna function with
ker Im𝑀(𝜆) = ker(𝑀(𝜆) −𝑀(𝜆)∗) = {0}, i.e.,𝑀(⋅) ∈ 𝑠(); see (1.8) and Lemma 3.6.
Corollary 5.18. Let {,Γ} be an 𝑆-generalized boundary pair with the Weyl family 𝑀(⋅) = 𝐸 +𝑀0(⋅) as in Theorem 5.17.
Then ran Γ is dense in  ×, i.e., Γ defines an 𝑆-generalized boundary triple if and only if 𝐸 (= Re𝑀(𝜇)) is a selfadjoint
operator and
dom𝐸 ∩ ker 𝛾(𝜆) = 𝐸 ∩ ker Im𝑀0(𝜆) = {0}, 𝜆 ∈ ℂ ⧵ℝ. (5.33)
Proof. This follows from Lemma 3.6 and Corollary 4.3. □
Corollary 5.18 can be used to give an example of an 𝑆-generalized boundary triple
{,Γ0,Γ1} such that the mapping Γ0 is
not closable; cf. Corollary 5.4.
Example 5.19. Let 𝑀0(⋅) ∈ [] be a Nevanlinna function such that ker Im𝑀0(𝜆) is nontrivial and let 𝐸 be an unbounded
selfadjoint operator in  with dom𝐸 ∩ ker Im𝑀0(𝜆) = {0}. Then the function
𝑀(𝜆) = 𝐸 +𝑀0(⋅), 𝜆 ∈ ℂ ⧵ℝ,
is a domain invariant Nevanlinna function. It follows from Corollary 5.18 and Theorems 1.10, 1.12 that 𝑀(⋅) can be realized
as the Weyl function of some 𝑆-generalized boundary triple
{,Γ0,Γ1}. However, Im(𝑀(𝜆)ℎ, ℎ) = Im (𝑀0(𝜆)ℎ, ℎ) with
ℎ ∈ dom𝑀(𝜆) does not satisfy the condition (5.19) in Corollary 5.4, since the kernel ker Im𝑀(𝜆) = ker Im𝑀0(𝜆) is nontrivial
by construction.
Remark 5.20. Observe that in Theorems 1.12 and 5.17 the function 𝑀0(⋅) can be considered as the closure of 𝑀(⋅) − 𝐸. In
Theorem 5.17𝑀(⋅) is an operator valued function if and only if 𝐸 is an operator. By Corollary 5.18 even in this case Γ can still
be multi-valued if the kernel ker𝑀0(𝜆) or ker Im𝑀0(𝜆) = ker 𝛾(𝜆) is nontrivial and the condition (5.33) is violated. In fact,
any Nevanlinna function with bounded values in and ker Im𝑀0(𝜆) ≠ {0} combined with an unbounded selfadjoint operator
𝐸 in satisfying the condition (5.33) is associated with an 𝑆-generalized boundary triple {,Γ0,Γ1} with the Weyl function
𝑀 = 𝐸 +𝑀0(⋅). If such a function𝑀 is regularized by subtracting the unbounded constant operator 𝐸, the function𝑀0(⋅) =
𝑀(⋅) − 𝐸 corresponds to an 𝐴𝐵-generalized boundary triple
{, Γ̃0, Γ̃1} whose range ran Γ̃ is not dense in 2. In particular,
Γ̃ whose Weyl function is the regularized function𝑀(⋅) − 𝐸 is not a quasi boundary triple. The closure𝑀0(⋅) of𝑀(⋅) − 𝐸 is
the Weyl function of the closure of Γ̃ which in this case is always a (multi-valued) 𝐵-generalized boundary pair. An example of
an 𝑆-generalized boundary triple with ker Im𝑀0(𝜆) ≠ {0} satisfying the property (5.33) appears in [14, Proposition 2.17].
DERKACH ET AL. 1315
5.4 𝑬𝑺-generalized boundary triples and form domain invariance
Recall, see Definition 1.13, that a unitary boundary triple
{,Γ0,Γ1} for 𝐴∗ is called 𝐸𝑆-generalized, if the extension 𝐴0 is
essentially selfadjoint in ℌ.
As the main result of this section it will be shown that the class of Weyl functions of 𝐸𝑆-generalized boundary triples
coincides with the class of form domain invariant Nevanlinna functions.
Definition 5.21. A Nevanlinna function𝑀 ∈ () is said to be form domain invariant in ℂ+(ℂ−), if the quadratic form 𝔱𝑀(𝜆)
in  generated by the imaginary part of𝑀(𝜆) via
𝔱𝑀(𝜆)[𝑢, 𝑣] ∶=
1
𝜆 − ?̄?
[(𝑀(𝜆)𝑢, 𝑣) − (𝑢,𝑀(𝜆)𝑣)],
is closable for all 𝜆 ∈ ℂ+(ℂ−) and the closure of the form 𝔱𝑀(𝜆) has a constant domain. A Nevanlinna family 𝑀 ∈ ̃() is
said to be form domain invariant in ℂ+(ℂ−), if its operator part𝑀op(⋅) in the decomposition (2.3) is form domain invariant in
ℂ+(ℂ−).
The following two lemmas are preparatory for the main result.
Lemma 5.22. Let {,Γ0,Γ1} be a unitary boundary triple. Then the following statements are equivalent:
(i) ran
(
𝐴0 − 𝜆
)
is dense in ℌ for some or, equivalently, for every 𝜆 ∈ ℂ+(ℂ−);
(ii) 𝛾
(
?̄?
)
admits a single-valued closure 𝛾
(
?̄?
)
for some or, equivalently, for every 𝜆 ∈ ℂ+(ℂ−);
(iii) the form 𝔱
𝑀(𝜆) is closable for some or, equivalently, for every 𝜆 ∈ ℂ+(ℂ−).
Proof. (i)⇔ (ii) By Theorem 5.5 for every 𝜆 ∈ ℂ+(ℂ−)
dom 𝛾
(
?̄?
)∗ = dom(Γ1𝐻(𝜆)) = ran(𝐴0 − 𝜆).
Therefore, 𝛾
(
?̄?
)
admits a single-valued closure for 𝜆 ∈ ℂ+ (ℂ−) if and only if ran
(
𝐴0 − 𝜆
)
is dense in ℌ.
(ii)⇔ (iii) The equality (3.6) gives the following representation for 𝔱𝑀(𝜆):
𝔱𝑀(𝜆)[𝑢, 𝑣] = (𝛾(𝜆)𝑢, 𝛾(𝜆)𝑣)ℌ.
It is well-known (see e.g. [45, Chapter VI]) that the form (𝛾(𝜆)𝑢, 𝛾(𝜇)𝑣)ℌ is closable precisely when the operator 𝛾(𝜆) is
closable. □
Lemma 5.23. Let {,Γ0,Γ1} be an 𝐸𝑆-generalized boundary triple. Then:
(i) ker Γ0 = 𝐴0 is selfadjoint and the domain of Γ0 admits the decomposition
domΓ0 = 𝐴0 +̇
(
domΓ0 ∩ ?̂?𝜆(𝐴∗)
)
= 𝐴0 +̇ ran ?̂?(𝜆), 𝜆 ∈ ℂ ⧵ℝ; (5.34)
(ii) 𝛾(𝜆) admits a single-valued closure 𝛾(𝜆) for every 𝜆 ∈ ℂ ⧵ℝ;
(iii) the closure of the 𝛾-field satisfies
ran Γ0 = dom 𝛾(𝜆) = dom 𝛾(𝜇), 𝜆, 𝜇 ∈ ℂ ⧵ℝ; (5.35)
(iv) 𝛾(𝜆) and 𝛾(𝜇) are connected by
𝛾(𝜆) =
[
𝐼 + (𝜆 − 𝜇)
(
𝐴0 − 𝜆
)−1]
𝛾(𝜇), 𝜆, 𝜇 ∈ ℂ ⧵ℝ. (5.36)
Proof.
(i) Since the closed linear relation Γ0 has a closed kernel, one has𝐴0 ⊂ ker Γ0. Since𝐴0 is selfadjoint, the first von Neumann’s
formula shows that 𝐴∗ = 𝐴0 +̇ ?̂?𝜆(𝐴∗) for all 𝜆 ∈ ℂ ⧵ℝ. Consequently,
𝐴0 ⊂ domΓ0 ⊂ 𝐴0 +̇ ?̂?𝜆(𝐴∗), 𝜆 ∈ ℂ ⧵ℝ,
1316 DERKACH ET AL.
and this implies the first equality in (5.34). The second equality in (5.34) holds by Proposition 5.3. Finally, according
to Proposition 5.3 ker Γ0 ∩ ?̂?𝜆(𝐴∗) = mul ?̂?(𝜆) = {0}, since 𝛾(𝜆) or, equivalently, ?̂?(𝜆) is closable by Lemma 5.22. Since
𝐴0 ⊂ ker Γ0, the identity ker Γ0 ∩ ?̂?𝜆(𝐴∗) = {0} combinedwith the first equality in (5.34) implies the equality𝐴0 = ker Γ0.
(ii) The statement (ii) is implied by Lemma 5.22.
(iii) Since 𝐴0 is selfadjoint, the defect subspaces of 𝐴 are connected by
𝔑𝜆(𝐴∗) =
[
𝐼 + (𝜆 − 𝜇)
(
𝐴0 − 𝜆
)−1]
𝔑𝜇(𝐴∗), 𝜆, 𝜇 ∈ ℂ ⧵ℝ.
Hence, if 𝑓𝜆 =
[
𝐼 + (𝜆 − 𝜇)
(
𝐴0 − 𝜆
)−1]
𝑓𝜇, then 𝑓𝜇 =
{
𝑓𝜇, 𝜇𝑓𝜇
}
∈ ?̂?𝜇(𝐴∗) precisely when
𝑓𝜆 =
{
𝑓𝜆, 𝜆𝑓𝜆
}
= 𝑓𝜇 + (𝜆 − 𝜇)𝐻(𝜆)𝑓𝜇 ∈ ?̂?𝜆(𝐴∗), (5.37)
where 𝐻(𝜆)𝑓𝜇 =
{(
𝐴0 − 𝜆
)−1
𝑓𝜇,
(
𝐼 + 𝜆
(
𝐴0 − 𝜆
)−1)
𝑓𝜇
}
∈ 𝐴0. Since 𝐴0 ⊂ domΓ0, it follows from (5.37) that
𝑓𝜇 ∈ dom
(
Γ0
)
∩ ?̂?𝜇(𝐴∗) if and only if 𝑓𝜆 ∈ dom
(
Γ0
)
∩ ?̂?𝜆(𝐴∗) and{
𝑓𝜇, ℎ
}
∈ Γ0 ∩
(
?̂?𝜇(𝐴∗)⊕) ⇔ {𝑓𝜆, ℎ} ∈ Γ0 ∩ (?̂?𝜆(𝐴∗)⊕)
for some ℎ ∈ . Now, using (i) and Proposition 5.3 one gets
dom ?̂?(𝜆) = Γ0
(
domΓ0 ∩ ?̂?𝜆(𝐴∗)
)
= ran Γ0 = Γ0
(
domΓ0 ∩ ?̂?𝜇(𝐴∗)
)
= dom ?̂?(𝜇)
Clearly dom ?̂?(𝜆) = dom 𝛾(𝜆), 𝜆 ∈ ℂ ⧵ℝ, and hence (iii) is proved.
(iv) The proof of (iii) shows that
{
ℎ, 𝑓𝜇
}
∈ ?̂?(𝜇) if and only if
{
ℎ, 𝑓𝜆
}
∈ ?̂?(𝜆). Consequently,
{
ℎ, 𝑓𝜇
}
∈ 𝛾(𝜇) if and
only if
{
ℎ, 𝑓𝜆
}
=
{
ℎ,
[
𝐼 + (𝜆 − 𝜇)
(
𝐴0 − 𝜆
)−1]
𝑓𝜇
}
∈ 𝛾(𝜆) and, since 𝛾(𝜇) and 𝛾(𝜆) are operators, this means that (5.36) is
satisfied. □
Theorem 5.24. LetΠ = {,Γ0,Γ1} be a unitary boundary triple for𝐴∗ and let𝑀 and 𝛾(⋅) be the corresponding Weyl function
and the 𝛾-field. Then the following statements are equivalent:
(i) ran
(
𝐴0 − 𝜆
)
is dense in ℌ for some or, equivalently, for every 𝜆 ∈ ℂ+(ℂ−);
(ii) 𝛾(𝜆) admits a single-valued closure 𝛾(𝜆) for one 𝜆 ∈ ℂ+(ℂ−) with a domain dense in ;
(iii) 𝛾(𝜆) admits a single-valued closure 𝛾(𝜆) for every 𝜆 ∈ ℂ+(ℂ−) which is domain invariant with a constant domain dense
in ;
(iv) the form 𝔱𝑀(𝜆) is closable for one 𝜆 ∈ ℂ+(ℂ−);
(v) the Weyl function𝑀 belongs to𝑠() and is form domain invariant in ℂ+(ℂ−).
In particular, if the statements (i)–(v) are satisfied both in ℂ+ and ℂ− then Π is an 𝐸𝑆-generalized boundary triple and the
Weyl function𝑀 is form domain invariant with
dom 𝔱𝑀(𝜆) = dom 𝛾(𝜆) = ran Γ0, 𝜆 ∈ ℂ ⧵ℝ. (5.38)
Proof. The equivalence (i) ⇔ (ii) is obtained from Lemma 5.22. The fact that the domain of 𝛾(𝜆) is dense in  follows from
Proposition 5.2.
The equivalences (i)⇔ (iv), (v) and (ii)⇔ (iii) follow from Lemmas 5.22 and 5.23.
In particular, Lemma 5.22 shows that the form 𝔱𝑀(𝜆) is closable for some (and then for every) 𝜆 ∈ ℂ+ and for some (and then
for every) 𝜇 ∈ ℂ− if and only if 𝐴0 is essentially selfadjoint. In this case the closure of the form 𝔱𝑀(𝜆) is given by
𝔱𝑀(𝜆)[𝑢, 𝑣] =
(
𝛾(𝜆)𝑢, 𝛾(𝜆)𝑣
)
ℌ, (5.39)
in particular, dom 𝔱𝑀(𝜆) = dom 𝛾(𝜆). According to Lemma 5.23 this domain does not depend on 𝜆 ∈ ℂ ⧵ℝwhen𝐴0 is essentially
selfadjoint. The last equality in (5.38) is obtained from (5.35). □
DERKACH ET AL. 1317
Remark 5.25. Let
{,Γ0,Γ1} be an 𝐸𝑆-generalized boundary triple, and assume that (𝛼, 𝛽) ⊂ 𝜌(𝐴0). Then:
(i) for every 𝜇 ∈ (𝛼, 𝛽) 𝛾(𝜇) admits a single-valued closure 𝛾(𝜇) such that (5.35) and (5.36) hold for all 𝜆, 𝜇 ∈ (ℂ ⧵ℝ) ∪ (𝛼, 𝛽);
(ii) for every 𝜇 ∈ (𝛼, 𝛽) and 𝑢, 𝑣 ∈  there exists a limit
𝔱𝑀(𝜇)[𝑢, 𝑣] = lim
𝜈↓0
𝔱𝑀(𝜇+𝑖𝜈)[𝑢, 𝑣] =
(
𝛾(𝜇)𝑢, 𝛾(𝜇)𝑣
)
 .
The proof of the first statement is precisely the same as the proof of Lemma 5.23. The statement (ii) is implied by the equality
(5.39), and the continuity of 𝛾(𝜇)𝑢 with respect to 𝜇 ∈ (𝛼, 𝛽); see (5.36).
The assumption that 𝛾(𝜆) admits a single-valued closure for some 𝜆 ∈ ℂ− does not imply that 𝛾(𝜇) admits a single-valued
closure for some 𝜇 ∈ ℂ+. In particular, for a maximal symmetric relation 𝐴 the following extreme situation holds.
Proposition 5.26. Let Π = {,Γ} be a unitary boundary pair for 𝐴∗, let 𝛾(⋅) be the corresponding 𝛾-field, and assume that 𝐴
is maximal symmetric with 𝑛−(𝐴) = 0 and 0 < 𝑛+(𝐴) ≤ ∞. Then 𝛾(𝜆) is a bounded operator (in fact a zero operator) for every
𝜆 ∈ ℂ−, while 𝛾(𝜆) is a singular operator with mul 𝛾(𝜆) = 𝔑𝜆 for every 𝜆 ∈ ℂ+.
Proof. First recall that for every closed symmetric relation 𝐴 there is a unitary boundary pair for 𝐴∗; see [25, Proposition 3.7].
By definition 𝛾(𝜆) is a single-valued operator and it is known that ran 𝛾(𝜆) = 𝔑𝜆 for every 𝜆 ∈ ℂ ⧵ℝ; see [25, Lemma 2.14].
Hence the statement in the lower half-planeℂ− is clear. In particular, one has ker 𝛾(𝜆) = dom 𝛾(𝜆) and consequently ran 𝛾(𝜆)∗ =
mul 𝛾(𝜆)∗, 𝜆 ∈ ℂ−. On the other hand, by Theorem 5.5 ran 𝛾(𝜆)∗ andmul 𝛾(𝜆)∗ do not depend on 𝜆 ∈ ℂ ⧵ℝ. Consequently, the
equality ran 𝛾(𝜆)∗ = mul 𝛾(𝜆)∗ holds also for every 𝜆 ∈ ℂ+. Then equivalently dom 𝛾(𝜆) = ker 𝛾(𝜆), which shows that 𝛾(𝜆) is a
singular operator with mul 𝛾(𝜆) = ran 𝛾(𝜆) = 𝔑𝜆 for every 𝜆 ∈ ℂ+. □
Observe that in Proposition 5.26 the corresponding Weyl function𝑀 is actually domain invariant in each half-plane ℂ+ and
ℂ−, while it is neither domain nor form domain invariant inℂ ⧵ℝ; see Proposition 3.11 (i). For an explicit example demonstrating
Proposition 5.26 we refer to [25, Example 6.7], where 𝐴 is the minimal differential operator generated in ℌ = 𝐿2(0,∞) by the
differential expression 𝑖𝐷.
Remark 5.27.
(a) If 𝐴+ is a maximal symmetric operator inℌwith 𝑛−
(
𝐴+
)
= 0 and 0 < 𝑛+
(
𝐴+
) ≤ ∞ and 𝐴− is a maximal symmetric oper-
ator in ℌ with 𝑛+(𝐴−) = 0 and 0 < 𝑛−(𝐴−) ≤ ∞, then 𝐴 = 𝐴+ ⊕𝐴− is a symmetric operator in ℌ⊕ℌ with defect num-
bers
{
𝑛+
(
𝐴+
)
, 𝑛−
(
𝐴−
)}
. Moreover, if Π± =
{±,Γ±} is a unitary boundary pair for 𝐴∗± then clearly the orthogonal sum
Π+ ⊕ Π− =
{+ ⊕−,Γ+ ⊕ Γ−} is a unitary boundary pair for 𝐴∗. Moreover, the corresponding 𝛾-field is
𝛾(𝜆) = 𝛾+(𝜆)⊕ 𝛾−(𝜆), 𝜆 ∈ ℂ ⧵ℝ. Now Proposition 5.26 shows that 𝛾(𝜆) is not closable for any 𝜆 ∈ ℂ ⧵ℝ. Hence, there
exists symmetric operators 𝐴 with arbitrary deficiency indices 𝑛±(𝐴) and a unitary boundary pair for 𝐴∗ such that the
corresponding 𝛾-field 𝛾(𝜆) is not closable for any 𝜆 ∈ ℂ ⧵ℝ. This also holds in the case of equal deficiency indices
0 < 𝑛−(𝐴−) = 𝑛+
(
𝐴+
) ≤ ∞. However, in this case the boundary pair Π for 𝐴∗ is not minimal in general.
(b) Let 𝐴+ be a maximal symmetric operator in ℌ with 𝑛−
(
𝐴+
)
= 0 and 𝑛+
(
𝐴+
)
= 1 and let 𝐴− be a symmetric operator
in ℌ with equal deficiency indices 𝑛+(𝐴−) = 𝑛−(𝐴−) = ∞. Then 𝐴 = 𝐴+ ⊕𝐴− is a symmetric operator in ℌ⊕ℌ with
equal deficiency indices 𝑛+(𝐴) = 𝑛−(𝐴) = ∞. Moreover, if Π− = {−,Γ−} is an ordinary boundary triple for 𝐴∗− then the
corresponding 𝛾-field 𝛾−(𝜆) is a bounded operator in (−,𝔑𝜆(𝐴∗−)). Considering the boundary pair Π+ ⊕ Π− for 𝐴∗
one concludes from Proposition 5.26 that the corresponding 𝛾-field 𝛾(𝜆) = 𝛾+(𝜆)⊕ 𝛾−(𝜆) is a bounded operator for every
𝜆 ∈ ℂ−, while 𝛾(𝜆) is not closable for any 𝜆 ∈ ℂ+.
In the next example we present a unitary boundary triple whose Weyl function is form domain invariant but not domain
invariant.
Example 5.28. Let 𝜑(⋅) be a scalar Nevanlinna function and  = 𝐿2(0,∞). Define an operator valued function 𝐺𝜑(⋅)
𝐺𝜑(𝜆)𝑢 = −𝑖
𝑑2𝑢
𝑑𝑥2
, dom
(
𝐺𝜑(𝜆)
)
=
{
𝑢 ∈ 𝑊 22
(
ℝ+
)
∶ 𝑢′(0) + 𝑖𝜑(𝜆)𝑢(0) = 0
}
, 𝜆 ∈ ℂ+.
1318 DERKACH ET AL.
Clearly, 𝐺𝜑(𝜆) is densely defined, 𝜌
(
𝐺𝜑(𝜆)
) ≠ ∅ for each 𝜆 ∈ ℂ+ and the family 𝐺𝜑(⋅) is holomorphic in ℂ+ in the resolvent
sense. Now consider the form generated by the imaginary part of 𝐺𝜑(𝜆). Integrating by parts one obtains
𝔱𝜑(𝜆)[𝑢] ∶= Im
(
𝐺𝜑(𝜆)𝑢, 𝑢
)
= ∫ℝ+ |𝑢′(𝑥)|2 𝑑𝑥 + Im𝜑(𝜆)|𝑢(0)|2,
where 𝑢 ∈ dom 𝔱𝜑(𝜆) = dom
(
𝐺𝜑(𝜆)
)
. Hence the form 𝔱𝜑(𝜆) is nonnegative and𝐺𝜑(𝜆) is𝑚-dissipative for each 𝜆 ∈ ℂ+. Moreover,
𝐺𝜑(⋅) ∈ 𝑅𝑠() since ker 𝔱𝜑(𝜆) = {0}. Therefore, by Theorem 1.10, there exists a certain unitary boundary triple such that the
corresponding Weyl function coincides with 𝐺𝜑(⋅).
Notice that the form 𝔱𝐺𝜑(𝜆)
(
𝜆 ∈ ℂ+
)
associated with𝐺𝜑(𝜆) in (1.14) coincides with 𝔱𝜑(𝜆) up to an inessential renormalization
by Im 𝜆. Clearly, the form 𝔱𝜑(𝜆) is closable with the closure given by
𝔱𝜑(𝜆)[𝑢] = ∫ℝ+ |𝑢′(𝑥)|2 𝑑𝑥 + Im𝜑(𝜆)|𝑢(0)|2, dom 𝔱𝜑(𝜆) = 𝑊 12 (ℝ+).
Thus, the form domain dom
(
𝔱𝜑(𝜆)
)
= 𝑊 12
(
ℝ+
)
does not depend on 𝜆 ∈ ℂ+ while the domain dom𝐺𝜑(𝜆) does, i.e. 𝐺𝜑 satisfies
Assumption 1.15. The operator associated with the form 𝔱𝜑(𝜆) is given by
𝐺𝜑,𝐼 (𝜆)𝑢 = −
𝑑2𝑢
𝑑𝑥2
, dom
(
𝐺𝜑,𝐼 (𝜆)
)
=
{
𝑢 ∈ 𝑊 22 (ℝ+) ∶ 𝑢
′(0) = (Im𝜑(𝜆))𝑢(0)
}
.
The operator 𝐺𝜑,𝐼 can be treated as the imaginary part of the unbounded operator 𝐺𝜑.
A simple example of a unitary boundary triple whose Weyl function is form domain invariant and 𝛾-field is unbounded can
be obtained as follows (see also [25, Example 6.5]).
Example 5.29. Let𝐻 be a nonnegative selfadjoint operator in the Hilbert space ℌ with ker𝐻 = {0}. Let
𝐴∗ = ran𝐻1∕2 × dom𝐻1∕2, so that 𝐴 ∶= (𝐴∗)∗ = {0, 0} and (𝐴)∗ = ℌ2,
and define
Γ0𝑓 = 𝐻−1∕2𝑓, Γ1𝑓 = 𝐻1∕2𝑓 ′; 𝑓 = {𝑓, 𝑓 ′}, 𝑓 ∈ ran𝐻1∕2, 𝑓 ′ ∈ dom𝐻1∕2.
Then
{
ℌ,Γ0,Γ1
}
is a unitary boundary triple for 𝐴∗ = 𝐴∗. Indeed, Green’s identity (1.1) is satisfied, and ran Γ is dense in
ℌ2. Moreover, it is straightforward to check that Γ is closed, since 𝐻1∕2 is selfadjoint and, in particular, closed. Observe, that
𝑓𝜆 ∶=
{
𝑓𝜆, 𝜆𝑓𝜆
}
∈ 𝐴∗ if and only if 𝑓𝜆 = 𝐻1∕2𝑘 and 𝜆𝑓𝜆 = 𝐻−1∕2𝑔, with 𝑘 ∈ dom𝐻1∕2 and 𝑔 ∈ ran𝐻1∕2, are connected by
𝐻−1∕2𝑔 = 𝜆𝐻1∕2𝑘. Then 𝑘 ∈ dom𝐻 and
Γ0𝑓𝜆 = 𝑘, Γ1𝑓𝜆 = 𝜆𝐻𝑘.
These formulas imply that 𝛾(𝜆) = 𝐻1∕2 and 𝑀(𝜆) = 𝜆𝐻 , 𝜆 ∈ ℂ. In particular, the Weyl function is a Nevanlinna function.
According to [25, Proposition 3.6] this implies that Γ is in fact 𝐽ℌ-unitary.
Note that𝑀(𝜆) and its inverse are domain invariant, but in general unbounded Nevanlinna functions with unbounded imag-
inary parts. Clearly, 𝐴0 = ker Γ0 = {0} × dom𝐻1∕2 and 𝐴1 = ker Γ1 = ran𝐻1∕2 × {0} are essentially selfadjoint and 𝐴0 is
selfadjoint (𝐴1 selfadjoint) if and only if 𝑀(𝜆) = 𝜆𝐻 (−𝑀(𝜆)−1 = −𝜆−1𝐻−1, respectively) is a Nevanlinna function with
bounded values (cf. Theorem 1.12).
In this example the Weyl function is also domain invariant. In fact, domain invariance of a Nevanlinna function 𝑀 implies
its form domain invariance.
Proposition 5.30. Let 𝑀 be a Nevanlinna function in the Hilbert space . If the equality dom𝑀(𝜆) = dom𝑀(?̄?) holds for
some 𝜆 ∈ ℂ ⧵ℝ, then𝑀 is form domain invariant.
In particular, if𝑀 is domain invariant, then it is also form domain invariant.
Proof. If dom𝑀(𝜆) = dom𝑀
(
?̄?
)
for some 𝜆 ∈ ℂ ⧵ℝ, then one can write
𝔱𝑀(𝜆)[𝑢, 𝑣] =
(
𝑀(𝜆) −𝑀(𝜆)∗
𝜆 − ?̄?
𝑢, 𝑣
)

= (𝛾(𝜆)𝑢, 𝛾(𝜆)𝑣)ℌ, 𝑢, 𝑣 ∈ dom𝑀(𝜆).
DERKACH ET AL. 1319
Hence, the operator 𝑁(𝜆) ∶= 𝑀(𝜆)−𝑀(𝜆)
∗
𝜆−?̄? is nonnegative and densely defined in  ⊖mul𝑀(𝜆). Therefore, the form 𝔱𝑀(𝜆) is
closable for 𝜆 ∈ ℂ ⧵ℝ; see [45]. By applying the same reasoning to ?̄? it is seen that also the form 𝔱𝑀(?̄?) is closable. Now by
applying Lemma 5.22 it is seen that 𝐴0 is essentially selfadjoint and hence by Theorem 5.24𝑀 is form domain invariant. □
The converse statement does not hold. In fact, in [28] an example of a form domain invariant Nevanlinna function is con-
structed, such that the domains of𝑀(𝜆) and𝑀(𝜇) have a zero intersection:
dom𝑀(𝜆) ∩ dom𝑀(𝜇) = {0} for all 𝜆, 𝜇 ∈ ℂ+, 𝜆 ≠ 𝜇.
Remark 5.31. A unitary boundary pair {,Γ} for 𝐴∗ is said to be 𝐸𝑆-generalized if 𝐴0 = 𝐴∗0. 𝐸𝑆-generalized boundary pairs
can be characterized by the following equivalent conditions:
(i) for every 𝜆 ∈ ℂ ⧵ℝ, 𝛾(𝜆) admits a single-valued closure 𝛾(𝜆) with a constant domain;
(ii) theWeyl family𝑀 ∈ () is form domain invariant, i.e. its operator part𝑀op(⋅) in the decomposition (2.3) is form domain
invariant.
Notice, that in the case when (i)–(ii) are in force and mul Γ is nontrivial it may happen that the domain of the form
𝔱𝑀op(𝜆)[𝑢, 𝑣] =
(
𝛾(𝜆)𝑢, 𝛾(𝜆)𝑣
)
ℌ is not dense in , 𝜆 ∈ ℂ ⧵ℝ; for an example involving differential operators; see [29, Exam-
ple 5.40].
5.5 Renormalizations of form domain invariant Nevanlinna functions
The next theorem shows that form domain invariant Nevanlinna functions𝑀 in can be renormalized with a bounded operator
𝐺 such that the renormalized function 𝐺∗𝑀𝐺 becomes domain invariant.
Theorem 5.32. Let {,Γ0,Γ1} be a unitary boundary triple for 𝐴∗ with the 𝛾-field 𝛾(⋅) and the Weyl function𝑀 , and assume
that 𝐴0 = ker Γ0 is essentially selfadjoint. Then:
(1) There exists a bounded operator 𝐺 in with ran𝐺 = dom 𝛾(𝜆), 𝜆 ∈ ℂ ⧵ℝ, and ker 𝐺 = {0}, such that(
Γ̃0
Γ̃1
)
= clos
(
𝐺−1 0
0 𝐺∗
)(
Γ0
Γ1
)
defines an 𝐴𝐵-generalized boundary pair
{, Γ̃} for 𝐴∗.
(2) The corresponding Weyl function𝑀 is domain invariant and it is given by
𝑀(𝜆) = 𝐸 +𝑀0(𝜆),
where 𝐸 is a closed densely defined symmetric operator in  and 𝑀0(⋅) is the restriction of a Nevanlinna function
𝑀0(⋅) ∈ [] onto the domain dom𝐸.
(3) Furthermore, 𝐺∗𝑀(𝜆)𝐺 is also a Weyl function of a closed 𝐴𝐵-generalized boundary pair and it satisfies
𝐺∗𝑀(𝜆)𝐺 = 𝐸0 +𝑀0(𝜆) ⊂ 𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ,
where 𝐸0 ⊂ 𝐸 is a closed densely defined symmetric restriction of 𝐸.
Proof. The proof is divided into five steps.
1. Construction of a bounded operator 𝐺 with the properties
ker 𝐺 = {0}, ran𝐺 = dom 𝛾(𝜇) and dom 𝛾(𝜇)𝐺 = , for some 𝜇 ∈ ℂ ⧵ℝ.
Since 𝐴0 is essentially selfadjoint, 𝛾(𝜆) is closable and the dense subspace 0 = dom 𝛾(𝜆) of  does not depend on
𝜆 ∈ ℂ ⧵ℝ; see Theorem 5.24. Since 0 is an operator range there exists a bounded selfadjoint operator 𝐺 = 𝐺∗ with
ran𝐺 = 0 and ker 𝐺 = {0}; for instance, one can fix 𝜇 ∈ ℂ ⧵ℝ and then take 𝐺 = (𝛾(𝜇)∗𝛾(𝜇) + 𝐼)−1∕2. Namely,
1320 DERKACH ET AL.
dom 𝛾(𝜇) = dom𝑀(𝜇) is dense in, since mul Γ = {0} by assumption, and hence 𝛾(𝜇)∗𝛾(𝜇) is a selfadjoint operator satis-
fying dom 𝛾(𝜇) = dom
(
𝛾(𝜇)∗𝛾(𝜇)
)1∕2
= dom
(
𝛾(𝜇)∗𝛾(𝜇) + 𝐼
)1∕2
. With this choice of 𝐺 the domain of 𝛾(𝜇)𝐺 is dense in
since dom 𝛾(𝜇) is a core for the form 𝔱𝑀(𝜇) and due to the equality dom
(
𝔱𝑀(𝜇) + 𝐼
)
= ran𝐺 one concludes that dom 𝛾(𝜇) is
also a core for the operator 𝐺−1 =
(
𝛾(𝜇)∗𝛾(𝜇) + 𝐼
)1∕2
.
2. Construction of an isometric boundary triple
{,Γ𝐺0 ,Γ𝐺1 } such that the corresponding 𝛾-field 𝛾𝐺(𝜆) is a bounded densely
defined operator.
Introduce the transform
{,Γ𝐺0 ,Γ𝐺1 } of the boundary triple {,Γ0,Γ1} by setting(
Γ𝐺0
Γ𝐺1
)
=
(
𝐺−1 0
0 𝐺∗
)(
Γ0
Γ1
)
, (5.40)
where 𝐺 has the properties stated above. The block operator is isometric
(
in the Kreı˘n space
(2, 𝐽)) and hence Γ𝐺
is isometric as a composition of isometric mappings; i.e. Γ𝐺 satisfies Green’s identity (3.1) (Assumption 3.1.2). Since(
Γ0↾ ?̂?𝜆(𝐴∗)
)
= ?̂?(𝜆)−1 one has
Γ0?̂?𝜆(𝐴∗) = dom 𝛾(𝜆) ⊂ dom 𝛾(𝜆) = ran𝐺,
which implies that ran ?̂?(𝜆) = ?̂?𝜆(𝐴∗) = ?̂?𝜆(𝐴∗) ∩ domΓ ⊂ domΓ𝐺 (here 𝐴∗ = domΓ), and hence ?̂?𝜆
(
𝐴∗
𝐺
)
= ?̂?𝜆(𝐴∗).
Moreover, it is clear that ker Γ𝐺0 = ker Γ0 = 𝐴0 is essentially selfadjoint. Since the closure of 𝐴0 +̂ ?̂?𝜆(𝐴∗) is
𝐴0 +̂ ?̂?𝜆(𝐴∗) = 𝐴∗ one gets domΓ𝐺 = 𝐴∗ (Assumption 3.1.1). The corresponding 𝛾-field is given by
𝛾𝐺(𝜆) =
(
Γ𝐺0 ↾ ?̂?𝜆
(
𝐴∗
𝐺
))−1
= 𝛾(𝜆)𝐺, 𝜆 ∈ ℂ ⧵ℝ;
see Lemma 3.12. Since 𝛾(𝜆) is closable and 𝛾(𝜆)𝐺 ⊂ 𝛾(𝜆)𝐺, it follows from ran𝐺 = dom 𝛾(𝜆) that the closed operator 𝛾(𝜆)𝐺
is everywhere defined and, hence, bounded by the closed graph theorem. Thus also 𝛾(𝜆)𝐺 is a bounded operator with bounded
closure 𝛾(𝜆)𝐺 ⊂ 𝛾(𝜆)𝐺.
Next recall the operator 𝐻(𝜆), 𝜆 ∈ ℂ ⧵ℝ, from (3.11); see also Lemma 3.7. Since 𝐴0 is essentially selfadjoint,
dom
(
Γ1𝐻(𝜆)
)
= dom𝐻(𝜆) = ran
(
𝐴0 − 𝜆
)
is dense inℌ. Since ker Γ𝐺0 = 𝐴0 ⊂ domΓ
𝐺
1 andmul Γ
𝐺
1 = {0}, it follows from
Theorem 5.5 that
Γ𝐺1 𝐻(𝜆) = 𝐺
∗Γ1𝐻(𝜆) = 𝐺∗𝛾
(
?̄?
)∗
⊂
(
𝛾
(
?̄?
)
𝐺
)∗
, 𝜆 ∈ ℂ ⧵ℝ. (5.41)
By the construction of 𝐺 the domain of 𝛾(𝜇)𝐺 is dense in  for some 𝜇 ∈ ℂ ⧵ℝ. Therefore, (5.41) implies that Γ𝐺1 𝐻(?̄?)
is a bounded densely defined operator for some 𝜇 ∈ ℂ ⧵ℝ and, since 𝐴0 is essentially selfadjoint, Lemma 3.7 shows that
Γ𝐺1 𝐻(𝜆) is bounded and densely defined for all 𝜆 ∈ ℂ ⧵ℝ.
3. Verification of (1): Now consider the closure Γ̃ of Γ𝐺 in (5.40). It is shown below that ker Γ̃0 = 𝐴0, which means that
ker Γ̃0 is selfadjoint (Assumption 1.13.1), since 𝐴0 is essentially selfadjoint by assumption. By construction Γ𝐺 is defined
via the transform Γ𝐺 =
{
𝐺−1Γ0, 𝐺∗Γ1
}
of
{
Γ0,Γ1
}
. It follows from Lemma 3.8 (see also Theorem 5.5) that the graph of
Γ𝐺 contains all elements of the form{
𝑓, ?̂?
}
=
{
𝐻(𝜆)𝑘𝜆,
(
0
𝐺∗𝛾(?̄?)∗𝑘𝜆
)}
+
{(
𝛾(𝜆)𝐺ℎ
𝜆𝛾(𝜆)𝐺ℎ
)
,
(
ℎ
𝐺∗𝑀(𝜆)𝐺ℎ
)}
, (5.42)
where 𝑘𝜆 ∈ ran (𝐴0 − 𝜆) and ℎ ∈ dom𝐺∗𝑀(𝜆)𝐺 = dom 𝛾(𝜆)𝐺, 𝜆 ∈ ℂ ⧵ℝ. Let ℎ̂ = {ℎ, ℎ′} ∈ 𝐴0 and let 𝑘 ∈ ran
(
𝐴0 − 𝜆
)
be such that ℎ̂ = 𝐻(𝜆)𝑘, where 𝐻(𝜆) corresponds to the graph of 𝐴0; see (3.11). Moreover, let 𝑘𝑛 ∈ ran
(
𝐴0 − 𝜆
)
be a
sequence such that 𝑘𝑛 → 𝑘 as 𝑛→ ∞. Then 𝐻(𝜆)𝑘𝑛 → ℎ̂ ∈ 𝐴0, since 𝐻(𝜆) is bounded. Moreover, by boundedness of
Γ𝐺1 𝐻(𝜆) = 𝐺
∗𝛾
(
?̄?
)∗
ℎ̂𝑛 = Γ𝐺𝐻(𝜆)𝑘𝑛 =
{
0, 𝐺∗𝛾(?̄?)∗𝑘𝑛
}
→ {0, 𝑔}, 𝑔 ∈ .
Since Γ̃ is closed, it follows that
{
ℎ̂; {0, 𝑔}
}
∈ Γ̃ which shows that ℎ̂ ∈ ker Γ̃0. Hence, 𝐴0 ⊂ ker Γ̃0 and since ker Γ̃0 is
symmetric this implies that ker Γ̃0 = 𝐴0 = 𝐴∗0.
DERKACH ET AL. 1321
Since domΓ𝐺 = 𝐴∗, the closure Γ̃ has dense domain in 𝐴∗ (Assumption 1.9.1). Clearly, dom𝐺∗𝑀(𝜇)𝐺 = dom 𝛾(𝜇)𝐺 ⊂
ran Γ𝐺0 and hence the ranges of Γ
𝐺
0 and Γ̃0 are dense in  (Assumption 1.8.1). Furthermore, Γ̃ as the closure of Γ𝐺 is also
isometric, i.e., Green’s formula (3.1) holds for Γ̃ (Assumption 1.5.1). According to Definition 4.1 this means that Γ̃ is an
𝐴𝐵-generalized boundary pair for 𝐴∗.
4. Verification of (2): The form of the Weyl function𝑀(𝜆) = 𝐸 +𝑀0(𝜆) of Γ̃ is obtained from Theorem 4.2. Furthermore, by
Theorem 4.4 Γ̃ is closed if and only if 𝐸 is closed or, equivalently,𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ, is closed.
5. Verification of (3): Since Γ𝐺1 𝐻(𝜆) = 𝐺
∗𝛾
(
?̄?
)∗
and 𝛾𝐺(𝜆) ∶= 𝛾(𝜆)𝐺 are bounded and densely defined for each 𝜆 ∈ ℂ ⧵ℝ, if
follows from (5.42) that Γ̂𝐺 defined as{{
𝐻(𝜆)𝑘𝜆,
( 0
𝐺∗𝛾
(
?̄?
)∗
𝑘𝜆
)}
+
{(
𝛾𝐺(𝜆)ℎ
𝜆𝛾𝐺(𝜆)ℎ
)
,
(
ℎ
𝑀𝐺(𝜆)ℎ
)}
∶ 𝑘𝜆 ∈ ran
(
𝐴0 − 𝜆
)
ℎ ∈ dom𝑀𝐺(𝜆)
}
satisfies Γ̂𝐺 ⊂ Γ̃; here 𝛾𝐺(𝜆) and𝑀𝐺(𝜆) ∶= 𝐺∗𝑀(𝜆)𝐺 are the 𝛾-field and theWeyl function of Γ𝐺. Notice that𝐴0 ⊂ dom Γ̂𝐺
and, as shown above, ran Γ̂𝐺0 ⊃ dom 𝛾(𝜇)𝐺 is dense in (Assumption 1.5.2-1.5.3). Due to Γ̂𝐺 ⊂ Γ̃ also Green’s identity (3.1)
is satisfied (Assumption 1.5.1). Therefore, Γ̂𝐺 is also an𝐴𝐵-generalized boundary pair whoseWeyl function is clearly𝑀𝐺(𝜆),
which is closed. Now by Theorem 4.4 the 𝐴𝐵-generalized boundary pair Γ̂𝐺 is also closed and, since Γ̂𝐺 ⊂ Γ̃, one has
𝐺∗𝑀(𝜆)𝐺 ⊂ 𝑀(𝜆) = 𝐸 +𝑀0(𝜆), 𝜆 ∈ ℂ ⧵ℝ.
Now 𝐺∗𝑀(𝜆)𝐺 as a closed restriction of 𝐸 +𝑀0(𝜆) is of the form 𝐺∗𝑀(𝜆)𝐺 = 𝐸0 +𝑀0(𝜆), 𝜆 ∈ ℂ ⧵ℝ, where 𝐸0 is a
closed densely defined restriction of 𝐸; cf. Theorem 4.2. This proves the last statement.
□
Theorem 5.32 remains valid for all form domain invariant Nevanlinna functions 𝑀(⋅) ∈ () that need not be strict. The
only essential difference appearing in the proof of Theorem 5.32 in this case is that ker 𝛾(𝜆) = mul Γ0 (see (5.11)) is nontrivial,
and then also, ker 𝛾𝐺(𝜆) = ker 𝛾(𝜆)𝐺 is nontrivial. Notice that even if ker 𝛾(𝜆) = {0} (i.e.𝑀(⋅) ∈ 𝑠()) then the 𝛾-field ?̃?(𝜆)
as well as its closure ?̃?(𝜆) = 𝛾𝐺(𝜆) can have a nontrivial kernel. This explains why the constructed boundary pair Γ̃ can in general
be multi-valued even if the original boundary triple Γ =
{
Γ0,Γ1
}
is single-valued.
Theorem 5.32 combined with the next lemma yields an explicit representation for the class of form domain invariant Nevan-
linna functions as well as form domain invariant Nevanlinna families.
Lemma 5.33. Let 𝐺 be a bounded operator in the Hilbert space  with ker 𝐺 = ker 𝐺∗ = {0}, let 𝐻 be a closed symmetric
densely defined operator on  and let𝑀0(⋅) ∈ []. Then the function
𝑀(𝜆) = 𝐺−∗
(
𝐻 +𝑀0(𝜆)
)
𝐺−1, 𝜆 ∈ ℂ ⧵ℝ,
is form domain invariant if and only if for some, equivalently for every, 𝜆 ∈ ℂ ⧵ℝ
𝔇𝜆 ∶=
{
ℎ ∈  ∶ (Im𝑀0(𝜆)) 12ℎ ∈ ran𝐺∗} is dense in .
Proof. To calculate the form 𝔱𝑀(𝜆) let 𝜆 ∈ ℂ ⧵ℝ be fixed and let 𝑢, 𝑣 ∈ dom𝑀(𝜆). Then 𝑢, 𝑣 ∈ dom𝐺−1 and hence
𝔱𝑀(𝜆)[𝑢, 𝑣] =
1
𝜆 − ?̄?
[((
𝐻 +𝑀0(𝜆)
)
𝐺−1𝑢, 𝐺−1𝑣
)
−
(
𝐺−1𝑢,
(
𝐻 +𝑀0(𝜆)
)
𝐺−1𝑣
)]
= 1
Im 𝜆
((
Im𝑀0
)
(𝜆)𝐺−1𝑢, 𝐺−1𝑣
)
= 1
Im 𝜆
((
Im𝑀0(𝜆)
) 1
2𝐺−1𝑢,
(
Im𝑀0(𝜆)
) 1
2𝐺−1𝑣
)
,
where symmetry of𝐻 has been used. This form is closable precisely when the operator
(
Im𝑀0(𝜆)
) 1
2𝐺−1 is closable or, equiv-
alently, its adjoint
((
Im𝑀0(𝜆)
) 1
2𝐺−1
)∗
= 𝐺−∗
(
Im𝑀0(𝜆)
) 1
2 is densely defined. Since𝔇𝜆 = dom𝐺−∗
(
Im𝑀0(𝜆)
) 1
2 , the clos-
ability of 𝔱𝑀(𝜆) is equivalent for𝔇𝜆 to be dense in .
1322 DERKACH ET AL.
To prove that this criterion does not depend on 𝜆 ∈ ℂ ⧵ℝ consider𝑀0(⋅) as the Weyl function of some 𝐵-generalized bound-
ary pair (,Γ). Let 𝛾0(⋅) be the corresponding 𝛾-field and let 𝐴0 = ker Γ0 be the associated selfadjoint operator. Then the form
𝔱𝑀(𝜆)[𝑢, 𝑣] can be also rewritten in the form
𝔱𝑀(𝜆)[𝑢, 𝑣] =
(
𝛾0(𝜆)𝐺−1𝑢, 𝛾0(𝜆)𝐺−1𝑣
)
and hence the form 𝔱𝑀(𝜆)[𝑢, 𝑣] is closable if and only if 𝛾0(𝜆)𝐺−1 is a closable operator. Now for any 𝜆, 𝜇 ∈ ℂ ⧵ℝ one has(
𝐼 + (𝜆 − 𝜇)
(
𝐴0 − 𝜆
)−1)
𝛾0(𝜇)𝐺−1 = 𝛾0(𝜆)𝐺−1,
and since 𝐼 + (𝜆 − 𝜇)
(
𝐴0 − 𝜆
)−1
boundedwith bounded inverse, one concludes that 𝛾0(𝜇)𝐺−1 is closable exactlywhen 𝛾0(𝜆)𝐺−1
is closable and that the closures are connected by(
𝐼 + (𝜆 − 𝜇)
(
𝐴0 − 𝜆
)−1)
𝛾0(𝜇)𝐺−1 = 𝛾0(𝜆)𝐺−1.
Therefore, if 𝔱𝑀(𝜇)[𝑢, 𝑣] is closable for some 𝜇 ∈ ℂ ⧵ℝ then 𝔱𝑀(𝜆)[𝑢, 𝑣] is closable for all 𝜆 ∈ ℂ ⧵ℝ and the form domains of
these closures coincide. This completes the proof. □
Proposition 5.34. Let𝑀 be a strict form domain invariant operator valued Nevanlinna function in the Hilbert space. Then
there exist a bounded operator 𝐺 ∈ [] with ker 𝐺 = ker 𝐺∗ = {0}, a closed symmetric densely defined operator 𝐸 in, and
a bounded Nevanlinna function𝑀0(⋅) ∈ [] with the property
 = clos𝔇𝜆 ∶= clos
{
ℎ ∈  ∶ (Im𝑀0(𝜆)) 12ℎ ∈ ran𝐺∗}, 𝜆 ∈ ℂ ⧵ℝ, (5.43)
such that𝑀(⋅) admits the representation
𝑀(𝜆) = 𝐺−∗
(
𝐸 +𝑀0(𝜆)
)
𝐺−1, 𝜆 ∈ ℂ ⧵ℝ. (5.44)
Conversely, every Nevanlinna function 𝑀(⋅) of the form (5.44) is form domain invariant in ℂ ⧵ℝ, whenever 𝐸 ⊂ 𝐸∗ and
𝐺 ∈ (), ker 𝐺 = ker 𝐺∗ = {0}, and𝑀0(⋅) ∈ [] satisfy the condition (5.43).
Proof. Let the Nevanlinna function 𝑀 ∈ () be realized as the Weyl function of some boundary pair {,Γ} (see Theo-
rem 5.24, [25, Theorem 3.9]). Since𝑀 is form domain invariant, 𝐴0 is essentially selfadjoint by Theorem 5.24. Since𝑀 is an
operator valued Nevanlinna function, one can apply Theorem 5.32 (see also the discussion after Theorem 5.32), which shows
that the inclusion 𝐺∗𝑀(𝜆)𝐺 ⊂ 𝐸 +𝑀0(𝜆) holds for every 𝜆 ∈ ℂ ⧵ℝ. This implies that
𝑀(𝜆) = 𝐺−∗𝐺∗𝑀(𝜆)𝐺𝐺−1 ⊂ 𝐺−∗
(
𝐸 +𝑀0(𝜆)
)
𝐺−1, (5.45)
where 𝐺 is a bounded operator with ker 𝐺 = ker 𝐺∗ = {0} (cf. proof of Theorem 5.32 where ran𝐺 =  by construction).
Clearly, the function 𝐺−∗
(
𝐸 +𝑀0(𝜆)
)
𝐺−1 is dissipative for 𝜆 ∈ ℂ+ and accumulative for 𝜆 ∈ ℂ−. Since 𝑀 is Nevanlinna
function, it is m-dissipative in ℂ+ and m-accumulative in ℂ−. Therefore, the inclusion in (5.45) prevails as an equality. Since
𝑀(⋅) is form domain invariant Lemma 5.33 shows that the condition (5.43) holds for every 𝜆 ∈ ℂ ⧵ℝ.
Conversely, if𝑀(⋅) is a Nevanlinna function of the form (5.44), where𝐸,𝐺 and𝑀0(⋅) are as indicated and the condition (5.43)
holds for some 𝜆 ∈ ℂ ⧵ℝ, then by Lemma 5.33𝑀(⋅) is form domain invariant and the condition holds for every 𝜆 ∈ ℂ ⧵ℝ. □
Remark 5.35. As to the renormalization in Theorem 5.32 we do not know if the renormalized function 𝑀(⋅) = 𝐸 +𝑀0(⋅)
belongs to the class of Nevanlinna functions.
However, the representation of𝑀(⋅) in Proposition 5.34 combined with 𝐸 ⊂ 𝐸∗ leads to
𝑀(𝜆) = 𝑀
(
?̄?
)∗
⊃ 𝐺−∗
(
𝐸∗ +𝑀0(𝜆)
)
𝐺−1 ⊃ 𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ.
Hence,𝑀(𝜆) can also be represented with 𝐸∗ instead of 𝐸 as follows:
𝑀(𝜆) = 𝐺−∗
(
𝐸∗ +𝑀0(𝜆)
)
𝐺−1, 𝜆 ∈ ℂ ⧵ℝ.
DERKACH ET AL. 1323
In particular, if 𝐸 is any maximal symmetric extension of 𝐸 then one has also the representation
𝑀(𝜆) = 𝐺−∗
(
𝐸 +𝑀0(𝜆)
)
𝐺−1, 𝜆 ∈ ℂ ⧵ℝ.
Remark 5.36. The result in Proposition 5.34 remains valid also for form domain invariant Nevanlinna families. In this case there
exist a bounded operator 𝐺 ∈ [] with ker 𝐺 = ker 𝐺∗ = mul𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ, a closed symmetric densely defined operator
𝐸 in, and a Nevanlinna function𝑀0(⋅) ∈ [] satisfying (5.43), such that
𝑀(𝜆) = 𝐺−∗
(
𝐸 +𝑀0(𝜆)
)
𝐺−1, 𝜆 ∈ ℂ ⧵ℝ.
To see this, decompose 𝑀(𝜆) = gr𝑀op(𝜆) +𝑀∞, where 𝑀∞ = {0} × mul𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ, see (2.3). Now as in the proof
of Proposition 5.34 the operator part 𝑀op(𝜆) admits the representation 𝑀op(𝜆) = 𝐺−∗0
(
𝐸 +𝑀0(𝜆)
)
𝐺−10 with some operator
𝐺0 ∈ [0] in 0 =  ⊖mul𝑀(𝜆) with ker 𝐺0 = ker 𝐺∗0 = {0}. The desired representation of 𝑀 is obtained by letting 𝐺 to
be the zero continuation of 𝐺0 from 0 to  = 0 ⊕mul𝑀(𝜆).
The next example contains a wide class of 𝐸𝑆-generalized boundary triples and demonstrates the regularization procedure
formulated in Theorem 5.32.
Example 5.37. Let Π0 = {,Γ00,Γ01} be an ordinary boundary triple for 𝐴∗ with 𝐴00 = ker Γ00, 𝐴01 = ker Γ01, let𝑀0(⋅) and 𝛾0(⋅)
be the corresponding Weyl function and the 𝛾-field, and let 𝐺 ∈ () with ker 𝐺 = ker 𝐺∗ = {0}. Then the transform(
Γ0
Γ1
)
=
(
𝐺 0
0 𝐺−∗
)(Γ00
Γ01
)
, (5.46)
defines an 𝐸𝑆-generalized boundary triple Π =
{,Γ0,Γ1} for 𝐴∗. Indeed, since 𝐺 ∈ () the transform 𝑉 in (3.22) is
unitary in the Kreı˘n space
{2, 𝐽} and it follows from [26, Theorem 2.10 (ii)] that the composition Γ = 𝑉 ◦Γ0 is unitary. By
Lemma 3.12 one has ker Γ = 𝐴 and, since Γ is unitary, 𝐴∗ ∶= domΓ is dense in 𝐴∗. Since Π0 is an ordinary boundary triple, × {0} ⊂ ran Γ0 and hence one concludes from (5.46) that
ran Γ0 = ran𝐺, 𝐴0 ∶= ker Γ0 = 𝐴00 ∩ 𝐴∗.
Consequently, ran Γ0 is dense in  and 𝐴0 is essentially selfadjoint. Moreover, 𝐴1 ∶= ker Γ1 = ker Γ01 = 𝐴01 and ran Γ1 =
dom𝐺∗ = : this means that the transposed boundary triple {,Γ1,−Γ0} is 𝐵-generalized. Observe, that 𝐴0 is selfadjoint if
and only if ran𝐺 =  or, equivalently, when Π is an ordinary boundary triple for 𝐴∗, too.
Next the form domain of the Weyl function𝑀 is calculated. By Lemma 3.12𝑀(⋅) = 𝐺−∗𝑀0(⋅)𝐺−1 and 𝛾(⋅) = 𝛾0(⋅)𝐺−1. Let
𝜆 ∈ ℂ ⧵ℝ be fixed and let 𝑢, 𝑣 ∈ dom𝑀(𝜆). Then
𝔱𝑀(𝜆)[𝑢, 𝑣] =
1
𝜆 − ?̄?
[(
𝐺−∗𝑀0(𝜆)𝐺−1𝑢, 𝑣
)
−
(
𝑢, 𝐺−∗𝑀0(𝜆)𝐺−1𝑣
)]
= 1
𝜆 − ?̄?
[(
𝑀0(𝜆)𝐺−1𝑢, 𝐺−1𝑣
)
−
(
𝑀0(𝜆)∗𝐺−1𝑢, 𝐺−1𝑣
)]
=
(
𝛾0(𝜆)𝐺−1𝑢, 𝛾0(𝜆)𝐺−1𝑣
)
.
Since Π0 is an ordinary boundary triple, 𝛾0(𝜆) ∶  → ker(𝐴∗ − 𝜆) is bounded and surjective, i.e., the inverse of this mapping is
also bounded. Hence 𝛾0(𝜆)𝐺 is closed, when considered on its natural domain dom 𝛾0(𝜆)𝐺−1 = ran𝐺 (⊃ dom𝑀(𝜆)). Therefore,
the closure of the form 𝔱𝑀(𝜆) is given by
𝔱𝑀(𝜆)[𝑢, 𝑣] =
(
𝛾(𝜆)𝐺−1𝑢, 𝛾(𝜆)𝐺−1𝑣
)
, 𝑢, 𝑣 ∈ ran𝐺.
In particular,𝑀(𝜆) is a form domain invariant Nevanlinna function whose form domain is equal to ran𝐺. Since 𝐺 is bounded,
one can use 𝐺 to produce a regularized function𝑀 :
𝑀 = 𝐺∗𝑀𝐺 = 𝐺∗
(
𝐺−∗𝑀0(⋅)𝐺−1
)
𝐺 = 𝑀0(𝜆),
so that𝑀 coincides with the Nevanlinna function𝑀0(⋅) which belongs to the class𝑢[].
1324 DERKACH ET AL.
It is emphasized that when𝐺 is not surjective, the form domain invariant function𝑀(⋅) = 𝐺−∗𝑀0(⋅)𝐺−1 need not be domain
invariant. In fact, in [28] an example of a form domain invariant Nevanlinna function𝑀 was given, such that
dom𝑀(𝜆) ∩ dom𝑀(𝜇) = {0}, 𝜆 ≠ 𝜇 (𝜆, 𝜇 ∈ ℂ ⧵ℝ),
and the corresponding regularized function𝑀 therein still belongs to the class𝑢[].
In Example 5.37 the boundary triple Π is 𝐸𝑆-generalized while the transposed boundary triple Π⊤ ∶=
{,Γ1,−Γ0} is
𝐵-generalized. Therefore, according to [27, Theorem 7.24] there exist an ordinary boundary triple Π̃0 and operators 𝑅 = 𝑅∗,
𝐾 ∈ (), ker𝐾 = ker𝐾∗ = {0}, such thatΠ⊤ is the transform (1.6) of Π̃0. Recall that one can take e.g.𝑅 = Re( −𝑀(𝑖)−1),
𝐾 =
(
Im
(
−𝑀(𝑖)−1
))1∕2
. In particular, this yields the following connections between the associated Weyl functions:
−𝑀−1(⋅) = 𝐾∗𝑀0(⋅)𝐾 + 𝑅.
In particular, with 𝑅 = 0 one obtains𝑀(⋅) = 𝐾−1
(
−𝑀0(⋅)−1
)
𝐾−∗ and here −𝑀0(⋅)−1 ∈ 𝑢[].
Together with Example 5.37 this characterizes those 𝐸𝑆-generalized boundary triples Π for 𝐴∗ whose transposed boundary
triple Π⊤ is 𝐵-generalized.
Recall that Weyl functions of 𝑆-generalized boundary pairs are domain invariant, but converse does not hold (explicit exam-
ples can be found in Part II). As shown in the next proposition a domain invariant Nevanlinna function can always be renormalized
by means of a fixed bounded operator to a Nevanlinna function belonging to the class[].
Proposition 5.38. Let𝑀(⋅) be a domain invariant operator valued Nevanlinna function in the Hilbert space . Moreover, let
𝐺 with ker 𝐺 = ker 𝐺∗ = {0} be a bounded operator in  such that ran𝐺 = dom𝑀(𝜆), 𝜆 ∈ ℂ ⧵ℝ. Then the renormalized
function
𝑀𝐺(𝜆) = 𝐺∗𝑀(𝜆)𝐺, 𝜆 ∈ ℂ ⧵ℝ, (5.47)
is a Nevanlinna function in the class[]. Moreover,𝑀𝐺(⋅) ∈ 𝑠[] precisely when𝑀(⋅) ∈ 𝑠().
Proof. By assumptions the equality dom𝐺∗𝑀(𝜆)𝐺 = dom𝑀(𝜆)𝐺 =  holds for all 𝜆 ∈ ℂ ⧵ℝ. Consequently, the adjoint
𝑀𝐺(𝜆)∗ is a closed operator and in view of
𝑀𝐺(𝜆)∗ = (𝐺∗𝑀(𝜆)𝐺)∗ ⊃ 𝐺∗𝑀
(
?̄?
)
𝐺
one has dom𝑀𝐺(𝜆)∗ =. Therefore, the equality 𝑀𝐺(𝜆)∗ =𝐺∗𝑀(?̄?)𝐺 holds for all 𝜆 ∈ ℂ ⧵ℝ. Now clearly Im𝑀𝐺(𝜆)=
𝐺∗Im𝑀(𝜆)𝐺, which implies that𝑀𝐺 ∈ [] and also proves the last statement. □
The assumption ran𝐺 = dom𝑀(𝜆) in Proposition 5.38 (or more generally the inclusion dom𝑀(𝜆) ⊂ ran𝐺) guarantees that
𝑀(⋅) can be recovered from𝑀𝐺(⋅) in (5.47) similarly as was done in Proposition 5.34:
𝑀(𝜆) = 𝐺−∗𝐺∗𝑀(𝜆)𝐺𝐺−1 = 𝐺−∗𝑀𝐺(𝜆)𝐺−1, 𝜆 ∈ ℂ ⧵ℝ.
5.6 An example on renormalization
The following example demonstrates renormalization of an unbounded form domain invariant Nevanlinna function. In this
example the real part of𝑀(𝑖) is strongly subordinated with respect to its imaginary part. In this case the renormalized function
𝑀(⋅) is a Nevanlinna function in the class[].
Example 5.39. Let 𝑆 be a positively definite closed symmetric operator in , so that 𝑆 ≥ 𝜀𝐼 . Let
𝑀(𝑧) = 𝑧𝑆∗𝑆 + 𝑆, dom𝑀(𝑧) = dom𝑆∗𝑆, 𝑧 ∈ ℂ.
Replacing if necessary 𝑆 by 𝑆 + 𝑎𝐼 we can assume that 𝜀 > 1. First notice that
‖𝑓‖2 ≤ 𝜀−2‖𝑆𝑓‖2 = 𝜀−2(𝑆∗𝑆𝑓, 𝑓 ) ≤ 𝜀−2‖𝑆∗𝑆𝑓‖ ⋅ ‖𝑓‖, 𝑓 ∈ dom𝑆∗𝑆,
i.e. ‖𝑆∗𝑆𝑓‖ ≥ 𝜀2‖𝑓‖. It follows that 𝑆 is strongly subordinated with respect to 𝑆∗𝑆, i.e.
‖𝑆𝑓‖2 = (𝑆𝑓, 𝑆𝑓 ) = (𝑆∗𝑆𝑓, 𝑓 ) ≤ ‖𝑆∗𝑆𝑓‖ ⋅ ‖𝑓‖ ≤ 𝜀−2‖𝑆∗𝑆𝑓‖2, 𝑓 ∈ dom𝑆∗𝑆.
DERKACH ET AL. 1325
Since dom𝑆∗𝑆 ⊂ dom𝑆 ⊂ dom𝑆∗, one easily proves that 𝑆∗ is also strongly subordinated with respect to 𝑆∗𝑆. Now, these
inequalities imply that both operators 𝑆∕𝑧 and 𝑆∗∕𝑧 are also strongly subordinated to 𝑆∗𝑆 for |𝑧| ≥ 1. Therefore,
𝑀(𝑧)∗ = (𝑧𝑆∗𝑆 + 𝑆)∗ = 𝑧𝑆∗𝑆 + 𝑆∗ = 𝑧𝑆∗𝑆 + 𝑆 = 𝑀(𝑧).
Since 𝑀(⋅) is dissipative in ℂ+, it follows that 𝑀(𝑧) is m-dissipative for 𝑧 ∈ ℂ+, |𝑧| ≥ 1, and m-accumulative for 𝑧 ∈ ℂ−,|𝑧| ≥ 1. In turn, the latter implies that𝑀(𝑧) being holomorphic and dissipative is 𝑚-dissipative for each 𝑧 ∈ ℂ+. Summing up
we conclude that𝑀(⋅) is an entire Nevanlinna function with values in ().
Furthermore,
𝔱𝑀(𝑧)(𝑓, 𝑔) =
(𝑀(𝑧)𝑓, 𝑔) − (𝑓,𝑀(𝑧)𝑔)
𝑧 − ?̄?
= (𝑆𝑓, 𝑆𝑔), 𝑓 , 𝑔 ∈ dom𝑆∗𝑆, 𝑧 ∈ ℂ.
The form is closable because so is the operator 𝑆. Taking the closure we obtain the closed form 𝔱𝑀(𝑧)(𝑓, 𝑔) = (𝑆𝑓, 𝑆𝑔),
𝑓 , 𝑔 ∈ dom𝑆, 𝑧 ∈ ℂ, with constant domain. In other words, 𝑀(⋅) is a form domain invariant Nevanlinna function and the
(selfadjoint) operator associated with 𝔱𝑀(𝑧) in accordance with the second representation theorem is (𝑆∗𝑆)1∕2.
Now consider the renormalization of 𝑀(⋅) as in Theorem 5.32. The operator 𝐺 = (𝑆∗𝑆)−
1
2 is bounded and
ran𝐺 = dom 𝔱𝑀(𝑧). Moreover, 𝐺∗(𝑆∗𝑆)𝐺 = 𝐼↾ dom (𝑆∗𝑆)
1
2 and 𝐺∗𝑆𝐺 = 𝐺∗𝑈 , where 𝑈 ∶ ran (𝑆∗𝑆)
1
2 =  → ran𝑆 is the
(partial) isometry from the polar decomposition 𝑆 = 𝑈 (𝑆∗𝑆)
1
2 . Consequently, 𝐶 ∶= 𝐺∗𝑆𝐺 is a bounded selfadjoint operator
in. By Theorem 5.32 one has𝑀(𝑧) ⊃ clos(𝐺∗𝑀(𝑧)𝐺) = 𝑧𝐼 + 𝐶 . Thus,𝑀(𝑧) = 𝑧𝐼 + 𝐶 is a Nevanlinna function in the class
[].
Some modifications of this example can be found in [29].
ACKNOWLEDGEMENTS
The research was partially supported by a grant from the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of
Science and Letters. V. D. and M. M. gratefully also acknowledge support by the University of Vaasa. The research of V. D.
was also supported by Ministry of Education and Science of Ukraine (projects 0118U003138, 0118U00206), a Volkswagen
Foundation grant and a Fulbright grant. The research of M. M. has been prepared with the support of the “RUDN University
Program 5-100”.
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How to cite this article: Derkach V, Hassi S, Malamud M. Generalized boundary triples, I. Some classes of isometric
and unitary boundary pairs and realization problems for subclasses of Nevanlinna functions.Mathematische Nachrichten.
2020;293:1278–1327. https://doi.org/10.1002/mana.201800300