Integr. Equ. Oper. Theory (2020) 92:42
https://doi.org/10.1007/s00020-020-02600-w
Published online October 3, 2020
c© The Author(s) 2020
Integral Equations
and Operator Theory
Generalized Schur–Nevanlinna functions
and their realizations
Lassi Lilleberg
Abstract. Pontryagin space operator valued generalized Schur functions
and generalized Nevanlinna functions are investigated by using discrete-
time systems, or operator colligations, and state space realizations. It is
shown that generalized Schur functions have strong radial limit values
almost everywhere on the unit circle. These limit values are contractive
with respect to the indefinite inner product, which allows one to general-
ize the notion of an inner function to Pontryagin space operator valued
setting. Transfer functions of self-adjoint systems such that their state
spaces are Pontryagin spaces, are generalized Nevanlinna functions, and
symmetric generalized Schur functions can be realized as transfer func-
tions of self-adjoint systems with Kre˘ın spaces as state spaces. A crite-
rion when a symmetric generalized Schur function is also a generalized
Nevanlinna function is given. The criterion involves the negative index
of the weak similarity mapping between an optimal minimal realization
and its dual. In the special case corresponding to the generalization of
an inner function, a concrete model for the weak similarity mapping can
be obtained by using the canonical realizations.
Mathematics Subject Classification. Primary: 47A48; Secondary: 47A56,
47B50, 93B28.
Keywords. Operator colligation, Passive system, Self-adjoint system,
Transfer function, Generalized Schur class, Generalized Nevanlinna class.
1. Introduction
Let U and Y be separable Pontryagin spaces with the same finite negative
index, and let L(U ,Y) be the class of bounded linear operators from U to Y.
An L(U ,Y)-valued function θ belongs to generalized Schur class Sκ(U ,Y), if
it is holomorphic at the origin and the Schur kernel
Kθ(w, z) =
1 − θ(z)θ∗(w)
1 − zw¯ , w, z ∈ ρ(θ), (1.1)
where θ∗(w) = (θ(w))∗, has κ negative squares. This means that for any
finite sets of points {w1, . . . , wn} ⊂ ρ(θ), where ρ(θ) is maximal domain of
42 Page 2 of 29 L. Lilleberg IEOT
analyticity of θ, and vectors {f1, . . . , fn} ⊂ Y, the Hermitian matrix
(〈Kθ(wj , wi)fj , fi〉Y
)n
i,j=1
, (1.2)
where 〈·, ·〉Y is the inner product of Y, has no more than κ negative eigen-
values, and there exists a matrix of the form (1.2) which has exactly κ neg-
ative eigenvalues. On the other hand, an L(U)-valued function θ, where U
is a Pontryagin space, belongs to generalized Nevanlinna class Nκ(U) if it
is meromorphic on C \ R, real, or symmetric, in a sense that θ(z) = θ#(z)
for every z ∈ ρ(θ), where θ#(z) is defined to be θ∗(z¯), and the Nevanlinna
kernel
Nθ(w, z) =
θ(z) − θ∗(w)
z − w¯ , w, z ∈ ρ(θ), (1.3)
has κ negative squares. If U and Y are Hilbert spaces, the classes S0(U ,Y)
and N0(U), which are denoted as S(U ,Y) and N(U), coincide with the or-
dinary Schur and Nevanlinna classes. That is, S(U ,Y) consists of L(U ,Y)-
valued functions holomorphic and bounded by one in D, and N(U) consists
of L(U)-valued functions holomorphic and symmetric in C\R such that their
imaginary parts are nonnegative in the upper half plane. The classes of gener-
alized Schur and Nevanlinna functions were first studied by Kre˘ın and Langer
in series of papers [26,27,27,29,30], first in the scalar case (U = Y = C) and
later in the operator valued case.
The study of the (generalized) Schur functions in infinite dimensional
spaces naturally leads to contractive operators and passive linear discrete-
time systems; or what is the same thing, contractive operator colligations.
An operator colligation
Σ = (TΣ;X ,U ,Y) (1.4)
consists of a Kre˘ın space X (the state space), Pontryagin spaces U (the in-
coming space) and Y (the outgoing space) with the same negative index, and
the system operator TΣ ∈ L(X ⊕ U ,X ⊕ Y), where the direct orthogonal
sum X ⊕ U or (XU
)
is with respect to the indefinite inner product. Here TΣ
is bounded and everywhere defined, and has the block representation of the
form
TΣ =
(
A B
C D
)
:
(X
U
)
→
(X
Y
)
, (1.5)
where A ∈ L(X ) is the main operator, and B ∈ L(U ,X ), C ∈ L(X ,Y), and
D ∈ L(U ,Y) . The colligation will be usually called as a system, since it can
be seen as a linear discrete-time system, and the system is identified with its
operator expression (1.5). The system Σ is passive (isometric, co-isometric,
conservative, self-adjoint), if the system operator TΣ in (1.5) is contractive
(isometric, co-isometric, unitary, self-adjoint) with respect to the indefinite
inner product. The transfer function of the system (1.5), or characteristic
function of the operator colligation, is defined by
θΣ(z) := D + zC(I − zA)−1B, (1.6)
IEOT Generalized Schur–Nevanlinna functions Page 3 of 29 42
whenever I − zA is invertible. Especially, θΣ is defined and holomorphic on a
neighbourhood of the origin. The values θΣ(z) are bounded operators from U
to Y. Conversely, if θ is an operator valued function, and the transfer function
of a system Σ coincides with it in a neighbourhood of the origin, then Σ is a
(scattering) realization of θ, and a realization problem for the operator valued
function θ analytic at the origin is to find system of the form (1.5) such that
its transfer function coincides with θ.
For ordinary Schur functions, this connection was discovered and stud-
ied, for instance, by Arov [7,8], de Branges and Rovnyak [17,18], Brodski˘i
[19] and Sz.-Nagy and Foias [36]. The standard Hilbert space theory of ordi-
nary Schur functions has a counterpart for the generalized Schur functions,
and this will led to replacing the Hilbert state space, or all of the spaces, by
Pontryagin, or in some cases, even by Kre˘ın spaces. In the case where U and
Y are Hilbert spaces, the generalized Schur class Sκ(U ,Y) and its connections
to unitary colligations were studied, for instance, by Dijksma, Langer and de
Snoo [22]. Arov’s approach to use passive systems was utilized by Saprikin
[34], Arov and Saprikin [13], Arov, Rovnyak and Saprikin [12] and by the
author in [31] to study the class Sκ(U ,Y) where U and Y are Hilbert spaces.
If U and Y are Pontryagin spaces with the same negative index, one encoun-
ters operator colligations, or systems, such that all the spaces are indefinite.
Theory of canonical isometric, co-isometric and conservative systems in that
case is considered, for instance, in [2,3,20,23], along with the other prop-
erties of the generalized Schur functions. Especially, symmetric generalized
Schur functions, with a little bit more general definition than in this paper,
were studied in [3]. The results about the unitary similarities between the
canonical realizations obtained therein will be used.
Theory of passive systems and generalized Schur functions in the case
where U and Y are Pontryagin spaces with the same negative index, was
studied by the author in [32]. On the other aspects, in the case where U and
Y are finite dimensional, the class Sκ(U ,Y) is closely related to generalized
Potapov class and generalized J -inner functions; see for instance [1,6,8,21,
37].
On the other hand, the generalized Nevanlinna functions have been
studied alongside with the Schur functions, mainly with scalar, matrix and
Hilbert space operator valued cases. Instead of unitary and contractive oper-
ators, the study of the generalized Nevanlinna functions involves dissipative
and self-adjoint linear operators and relations; see for instance [25,29].
The aim of this paper is to study connections of discrete-time systems,
transfer functions and operator valued analytic functions which are both
generalized Schur and generalized Nevanlinna functions for some indices, that
is, which belongs to the class Sκ1(U) ∩ Nκ2(U), where U is a Pontryagin
space. Before involving the realization theory, the structural properties of the
generalized Schur functions and generalized Nevanlinna functions are studied
by using the Potapov–Ginzburg transformation. Especially, in the case where
U and Y are finite dimensional anti-Hilbert spaces, the behaviour of the
functions in the classes Sκ1(U) and Nκ2(U) is reciprocal to the Hilbert space
case, see Corollary 2.3 and Proposition 2.6. Moreover, in Theorem 2.8, when U
42 Page 4 of 29 L. Lilleberg IEOT
and Y are Pontryagin spaces with the same negative indices, it will be proved
that for θ ∈ Sκ(U ,Y), the strong radial limit value θ(ζ) := limr→1− θ(rζ)
where ζ belongs to the unit circle T, exists almost everywhere (a.e.), and
their values are contractive with respect to the underlying indefinite inner
products. Theorem 2.8 also gives rise to a notion of a generalized J -inner
function in infinite dimensional spaces.
In realization theory, the study of the class Sκ1(U)∩Nκ2(U), where U is
a Pontryagin space, naturally leads to self-adjoint systems. For the ordinary
Schur and Nevanlinna functions, these connections were studied by Arlinski˘ı,
Hassi and de Snoo in [4] and by Arlinski˘ı and Hassi in [5]. One of their main
results was that θ ∈ S(U) ∩ N(U), where U is a Hilbert space, if and only if
θ has a minimal passive self-adjoint realization of the form (1.5) such that
the state space is a Hilbert space [4, Theorem 5.4]. In the case θ ∈ Sκ1(U) ∩
Nκ2(U), one can obtain a similar realization which is self-adjoint, but not
passive in the general case; see Theorem 3.5, Remark 3.6 and Proposition 3.7.
On the other hand, every θ ∈ Sκ(U ,Y) has a minimal passive realization
Σ, and it can be chosen such that it is optimal or ∗-optimal [32, Theorem
3.5]; for the case where U and Y are Hilbert spaces, see also [34, Theorem
5.3]. For a symmetric θ ∈ Sκ(U), these realizations have special properties.
Namely, the dual system of the optimal minimal passive realization of θ is a
∗-optimal minimal passive realization of θ, and vice versa. One can form a
weak similarity mapping Z between those systems such that Z is everywhere
defined, contractive and self-adjoint. If θ has a meromorphic continuation to
C\R, then the negative index of the mapping Z with respect to the indefinite
inner product in question determines the number of the negative squares of
the Nevanlinna kernel (1.3); see Theorem 3.10. That is, the negative index of
the Z, which roughly speaking tells that how much Z behaves like a positive
operator with respect to the indefinite inner product in question, can be
used to determine whether θ is also a generalized Nevanlinna function. If, in
addition, the boundary values of θ on the unit disc T are unitary, then Z is
also unitary and can be represented in an explicit form by using the canonical
realizations from [2,3].
It is a classical problem to determine if an ordinary Schur function θ
can represented as a corner of a bi-inner dilation of the form
Θ =
(
θ θ2
θ3 θ4
)
; (1.7)
see, for an instance, [8,14]. Arlinski˘ı and Hassi showed in [5] that every
θ ∈ S(U) ∩ N(U), where U is a Hilbert space, has a bi-inner dilation, and
moreover, a dilation (1.7) can be chosen such that it is an ordinary Nevanlinna
function. In the last section of this paper, similar results will be obtained for
the subclasses of Sκ1(U)∩Nκ2(U), where U is a Pontryagin space. In particu-
lar, functions in Sκ1(U)∩Nκ2(U) with the property that their minimal passive
realizations are unitarily similar, always have a dilation with unitary bound-
ary values almost everywhere on T, and those functions in Sκ(U) ∩ Nκ(U)
which have a minimal passive self-adjoint realization, always have a dilation
Θ with unitary boundary values almost everywhere on T. Moreover, Θ can
IEOT Generalized Schur–Nevanlinna functions Page 5 of 29 42
be chosen such that it is a generalized Nevanlinna function with the index κ;
see Theorem 4.1.
2. Structural properties of the generalized Schur and
generalized Nevanlinna functions
When U and Y are Pontryagin spaces with the same negative index, the full
structure of the functions in Sκ(U ,Y) and Nκ(U) is somewhat more compli-
cated than in the better known Hilbert space case. For instance, when U and
Y are Hilbert spaces, Kre˘ın–Langer factorizations shows that a function in
Sκ(U ,Y) has exactly κ poles, counting multiplicities; see Lemma 2.5. This
does not hold anymore when the negative index of U and Y is not zero; a
function θ ∈ Sκ(U ,Y) may has any countable number of poles, see Corol-
lary 2.3 and Example 2.7 below. However, some properties of the function
θ in Sκ(U ,Y) or Nκ(U) can be analyzed by using a suitable transformation
θ 	→ θ′, where θ′ ∈ Sκ(U ′,Y ′) or Nκ(U ′) for some Hilbert spaces U ′ and Y ′.
In what follows, all notions of continuity and convergence are under-
stood to be with respect to the strong topology, which is induced by any
fundamental decomposition of the space in question. Let θ be an L(U ,Y)-
valued function holomorphic on a set ρ(θ), where U and Y are Pontryagin
spaces with the same negative index. Let U = U+ ⊕U− and Y = Y+ ⊕Y− be
some fixed fundamental decompositions of U and Y. Represent θ as
θ(z) =
(
θ11(z) θ12(z)
θ21(z) θ22(z)
)
:
(U+
U−
)
→
(Y+
Y−
)
, (2.1)
and define U ′ = U+ ⊕ |U−| and Y ′ = Y+ ⊕ |Y−|, where |U−| and |Y−| are
antispaces of U− and Y−. The antispace of an inner product space H is by
definition the space that coincides with H as a vector space and is endowed
with an inner product −〈·, ·〉H. Denote
{
σ : U− → |U−| , τ : Y− → |Y−|,
σ∗ = −σ−1 , τ∗ = −τ−1, (2.2)
for the identity mappings. The Potapov–Ginzburg transformation; see [2,
Sect. 4.3] and [15, Sect. 5.§1], of θ is then defined to be an L(U ′,Y ′)-valued
function
θP (z) =
(
θ11(z) − θ12(z)θ−122 (z)θ21(z) θ12(z)θ−122 (z)τ−1
−σθ−122 (z)θ21(z) σθ−122 (z)τ−1
)
=
(
θP 11(z) θP 12(z)
θP 21(z) θP 22(z)
)
,
(2.3)
whose domain ρ(θP ) consists of all the points z ∈ ρ(θ) such that θ22(z) is
invertible. A calculation shows that
θ(z) =
(
θP 11(z) − θP 12(z)θP −122 (z)θP 21(z) θP 12(z)θP −122 (z)σ
−τ−1θP −122 (z)θP 21(z) τ−1θP −122 (z)σ
)
(2.4)
42 Page 6 of 29 L. Lilleberg IEOT
holds for every z ∈ ρ(θP ). Note that the values of θP 22 are invertible whenever
they exist. Define, respectively, L(Y ′,Y) and L(U ′,U)-valued functions
Φ(z) =
(
IY+ −θ12(z)σ−1
0 −θ22(z)σ−1
)
and Ψ(z) =
(
IU+ −θ#21(z)τ−1
0 −θ#22(z)τ−1
)
. (2.5)
Proposition 2.1. Let U and Y be Pontryagin spaces with the same negative
index π ≥ 1, and let θ be an L(U ,Y)-valued function holomorphic on a set
ρ(θ) and meromorphic on a set D.
(i) If θP exists, it is meromorphic on D, and if θP is meromorphic on a set
DP , then so is θ.
(ii) The Potapov–Ginzburg transformation
(
θ#
)
P
of θ# is (θP )
#.
(iii) The identities
I − θ(z)θ∗(w) = Φ(z)(I − θP (z)θ∗P (w)
)
Φ∗(w) (2.6)
I − θ#(z)θ#∗(w) = Ψ(z)
(
I − θ#P (z)θ#P
∗
(w)
)
Ψ∗(w) (2.7)
θ(z) − θ(w¯) = Φ(z)(θP (z) − θP (w¯)
)
Ψ∗(w) (2.8)
θ#(z) − θ#(w¯) = Ψ(z)
(
θ#P (z) − θ#P (w¯)
)
Φ∗(w) (2.9)
hold whenever the corresponding functions are defined.
(iv) If θ ∈ Sκ(U ,Y), then ρ(θP ) is of the form ρ(θ) \Ξ, where Ξ contains at
most κ points.
(v) If θ ∈ Sκ(U ,Y) and θ−122 (0) exists, then θP ∈ Sκ(U ′,Y ′). If θP ∈
Sκ(U ′,Y ′) then θ ∈ Sκ(U ,Y).
(vi) If U = Y and θ is a symmetric function such that θ22 is invertible for
some α ∈ C \ R, then θ ∈ Nκ(U) if and only if θP ∈ Nκ(U ′).
(vii) If U = Y, θ = θ# and θ−122 (0) exists, then θ ∈ Sκ1(U) ∩ Nκ2(U) if and
only if θP ∈ Sκ1(U ′) ∩ Nκ2(U ′).
Proof. (i) Suppose θP exists, i.e. θ22 in decomposition (2.1) is invertible for
some point α ∈ ρ(θ). Since θ is meromorphic on D, so are all the entries in
(2.1). To prove that θP is meromorphic on D, it is now sufficient to show
that θ−122 is meromorphic on D, since then all the entries in (2.3) are mero-
morphic. To this end, note that the values of θ22 are operators between the
spaces with the same finite dimension. Therefore, θ22(z) can be identified as
a square matrix, and θ−122 (z) has a representation θ
−1
22 (z) =
cof(θ22(z))
det(θ22(z))
, where
det(θ22(z)) and cof(θ22(z)) are, respectively, the determinant and the cofac-
tor matrix of θ22(z). The function det(θ22) is not identically zero since θ22(α)
is invertible. Since θ22 is meromorphic on D, so are the functions det(θ22(z))
and cof(θ22(z)). It follows now that θ−122 exists and it is meromorphic on D,
and so is θP .
If θP is meromorphic on DP , by using the same argument as above, one
can show that θP −122 is meromorphic on DP , and then it follows from (2.4)
that θ is meromorphic on DP .
For the proof of (ii), (iii) and (iv), see [2, Lemmas 4.3.1 and 4.3.2 and
Theorem 4.3.3].
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(v) From the part (iv) it follows that θP exists. By (2.6), it holds
Kθ(w, z) = Φ(z)KθP (w, z)Φ
∗(w) (2.10)
for the Schur kernels Kθ and KθP of the form (1.1), whenever the functions
are defined. Let Ω be a region such that θ and θP both are holomorphic on
Ω. Then, the values of θ22 are bijective in Ω, and it easily follows from this
fact that Φ∗(w) is onto for every w ∈ Ω. Then it follows from (2.10) that KθP
restricted to Ω has the same number of negative squares than Kθ restricted to
Ω. Now an application of [2, Theorem 1.1.4] shows that unrestricted KθP and
Kθ have the same number of negative squares. Therefore, if θ ∈ Sκ(U ,Y) and
θ−122 (0) exists, θP is holomorphic at the origin and KθP has exactly κ negative
squares, so θP ∈ Sκ(U ′,Y ′). Conversely if θP ∈ Sκ(U ′,Y ′), the function θP
and then also θ are holomorphic at the origin, Kθ has exactly κ negative
squares, so θ ∈ Sκ(U ,Y).
(vi) It follows from the assumption U = Y that U ′ = Y ′, and the as-
sumption that θ22 is invertible for some point guarantees that θP exists.
Moreover, the function θP is also symmetric by part (ii). From these sym-
metry conditions it follows that σ = τ in (2.2) and Ψ(z) = Φ(z) in (2.5). By
(2.8), it then holds
Nθ(w, z) = Ψ(z)NθP (w, z)Ψ
∗(w)
for the Nevanlinna kernels Nθ and NθP of the form (1.3), whenever the func-
tions are defined. Now the same argument as used in the proof of part (iii)
shows that Nθ and NθP have the same number of negative squares. Moreover,
part (i) shows that if either θ or θP is meromorphic on C \ R, then so is the
other. The claim now follows.
(vii) This follows straightforwardly from the parts (v) and (vi). 
Remark 2.2. The assumption that θ−122 (0) exists in parts (v) and (vii) of
Proposition 2.1 is technical; it is needed because the generalized Schur func-
tion must be analytic at the origin. If θ ∈ Sκ(U ,Y) and θ22(0) is not invert-
ible, it follows from part (iv) that θ22(α) is invertible for some α ∈ D. The
conclusions of the part (v) of Proposition 2.1 then hold if θP (z) is replaced
by θP (η(z)), where η(z) = α−z1−α¯z ; see [2, Sect. 2.5 B]. The same is true in
the part (vii) of Proposition 2.1, if α ∈ (−1, 1), since then η(z¯) = η(z) and
θP (η(z)) = (θP (η(z)))∗. By part (iv), α can be chosen to be real.
In one dimensional cases, that is, when U = Y = −C, where −C is the
antispace of the complex numbers, the Potapov–Ginzburg transformation
reduces to transformation of the form θ 	→ θ−1.
Corollary 2.3. A function θ1 such that θ1(0) = 0 belongs to Sκ1(−C) if and
only if θ−11 ∈ Sκ1(C), and a function θ2 which is not identically zero belongs
to Nκ2(−C) if and only if θ−12 ∈ Nκ2(C). Moreover, a function θ such that
θ(0) = 0 belongs to Sκ1(−C)∩Nκ2(−C) if and only if θ−1 ∈ Sκ1(C)∩Nκ2(C).
Proof. The claims follow from parts (v)–(vii) of Proposition 2.1 by choosing
U = Y = −C, since then θP = θ−1 and U ′ = Y ′ = C. 
42 Page 8 of 29 L. Lilleberg IEOT
Remark 2.4. In Corollary 2.3, the roles of −C and C could be interchanged;
it still holds, if one replaces −C by C and C by −C. Moreover, Corollary 2.3
holds as stated, if one replaces the spaces −C and C, respectively, by −Cn and
C
n, and changes the assumptions “θ1 not identically zero” and “θ2(0) = 0”,
respectively, by “det(θ1) not identically zero” and “det(θ2(0)) = 0”. However,
in that case, the roles of −C and C could not be interchanged, since if n ≥ 2,
there are matrix functions in Sκ(Cn) such that their values are not invertible
anywhere on D.
When U and Y are Hilbert spaces, the class Sκ(U ,Y) has characteriza-
tions which do not involve the Schur kernel (1.1). For a proof of the following
lemma, combine [22, Proposition 7.11] and [2, Theorem 4.2.1].
Lemma 2.5. Let U and Y be Hilbert spaces, and let θ be an L(U ,Y)-valued
function holomorphic at the origin and meromorphic on D. Then the following
statements are equivalent:
(i) θ ∈ Sκ(U ,Y);
(ii) θ has finite pole multiplicity κ and
lim
r→1−
sup
|z|=r
‖θ(z)‖ ≤ 1
holds;
(iii) θ has factorizations of the form
θ(z) = θr(z)B−1r (z) = B
−1
l (z)θl(z),
where θr, θl ∈ S(U ,Y), Br and Br are Blaschke products of degree κ
with values, respectively, in L(U) and L(Y), such that Br(w)f = 0 and
θr(w)f = 0 for some w ∈ D only if f = 0, and B∗l (w)g = 0 and
θ∗l (w)g = 0 for some w ∈ D only if g = 0.
When U and Y are finite dimensional anti-Hilbert spaces with the same
negative index, i.e. U = Y = −Cn, the results of Lemma 2.5 have counter-
parts; in particular, the analog for Lemma 2.5(ii) will be stated and proved
in proposition below.
For a meromorphic function θ such that the values of θ are operators
between the spaces with the same finite dimension, z is called a zero of θ
if it is a pole of θ−1. If X and Y are Hilbert spaces, the lower bound of an
operator T : X → Y is a value L ≥ 0 satisfying ‖Tx‖Y ≥ L for all x ∈ X
such that ‖x‖X = 1. The operator T is called bounded below if a non-zero
lower bound exists, and the best possible choice of all the lower bounds, i.e.
the greatest one, is denoted as γ(T ).
Proposition 2.6. An n × n-matrix valued function θ meromorphic on D and
holomorphic at the origin belongs to Sκ(−Cn) if and only if θ has exactly κ
zeros in D, counting multiplicities, and
lim
r→1−
inf
|z|=r
γ(θ(z)) ≥ 1. (2.11)
where γ(θ(z)) is taken with respect to the usual norm of L(Cn).
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Proof. The values of θ can be considered as the operators in L(−Cn). Then,
the Potapov–Ginzburg transformation θP of θ is θ−1. Suppose θ ∈ Sκ(−Cn).
Then, by Proposition 2.1, θ−1 is meromorphic on D. It can be assumed that
θ−1 exists at the origin, since if not, one only has to consider θ−1(η(z))
as in Remark 2.2. Then θ−1 ∈ Sκ(Cn) by Proposition 2.1. It follows from
Lemma 2.5 that θ−1 has exactly κ poles in D, counting multiplicities, and it
holds
lim
r→1−
sup
|z|=r
‖θ−1(z)‖ ≤ 1. (2.12)
It follows now that θ has exactly κ zeros, counting multiplicities, in D, and
(2.11) holds.
Assume then that θ has κ zeros in D and (2.11) holds. It can be again
assumed that z = 0 is not a zero of θ. Then θ−1 is meromorphic on D
and holomorphic at the origin, it has κ poles and (2.12) holds. It follows
from Lemma 2.5 that θ−1 ∈ Sκ(Cn), and then by Proposition 2.1 that θ ∈
Sκ(−Cn). 
Lemma 2.5 and Proposition 2.6 show that when U and Y are definite,
that is, Hilbert spaces or anti-Hilbert spaces, functions in the class Sκ(U ,Y)
can have only finite number of poles or zeros, respectively. This does not hold
in general, when the spaces U and Y are indefinite. In that case, it is possible
that θ ∈ Sκ(U ,Y) has infinite number of zeros and poles, as Example 2.7
below shows. However, a function θ ∈ Sκ(U ,Y) still has some properties
similar to (2.11) or (2.12). Indeed, the radial limit values of θ ∈ Sκ(U ,Y)
exists a.e. on T, and they are contractive with respect to the indefinite inner
product of U and Y; see Theorem 2.8 below.
Example 2.7. Let b1 and b2 be scalar infinite Blaschke products such that
b2(0) = 0. Consider an L (C ⊕ −C)-valued function θ(z) =
(
b1(z) 0
0 b−12 (z)
)
.
A calculation shows that the Potapov–Ginzburg transformation θP of the
function θ is the L (C2)-valued function θP (z) =
(
b1(z) 0
0 b2(z)
)
. It easily follows
from Lemma 2.5 that θP ∈ S(C2), and then by Proposition 2.1 that θ ∈
S(C ⊕ −C). Moreover, θ has infinite number of zeros and poles.
Theorem 2.8. Let U and Y be Pontryagin spaces with the same negative in-
dex.
(i) If θ ∈ Sκ(U ,Y), then strong radial limit values limr→1− θ(rζ) exist for
a.e. ζ ∈ T, and the limit values are contractive with respect to the in-
definite inner products of U and Y.
(ii) If θ ∈ Sκ(U ,Y), strong radial limit values of the function θ are isometric
(co-isometric) a.e. on T if and only if strong radial limit values of θP
are isometric (co-isometric) a.e. on T.
Proof. (i)The Hilbert space case is known. For ordinary Schur functions, the
result is classical, see [36, Chapter V]. If κ > 0 and U and Y are Hilbert
spaces, θ has Kre˘ın–Langer factorizations of the form (2.5). Since inverse
Blaschke products are rational functions with unitary values everywhere on
T, the result now follows from the case κ = 0.
42 Page 10 of 29 L. Lilleberg IEOT
Assume then that the negative index of U and Y is not zero. By Propo-
sition 2.1, the Potapov–Ginzburg transform θP of θ exists. It can be assumed
that θ22(0) invertible; if not, one only need to consider θP (η(z)), where η is
as in Remark 2.2. By Proposition 2.1, θP ∈ Sκ(U ′,Y ′), and since U ′ and Y ′
are Hilbert spaces, θP is meromorphic on D, has strong contractive radial
limit values almost everywhere on T, and the same holds for the entries θP11 ,
θP12 , θP21 and θP22 in (2.3). By Lemma 2.5, θP has exactly κ poles in D,
counting multiplicities, and therefore θP22 has no more than κ poles in D. It
now follows again from Lemma 2.5 that θP22 ∈ Sκ′(|U−|, |Y−|), where κ′ ≤ κ.
Then, θP22 has the Kre˘ın–Langer factorization of the form
θP22 = B
−1θ0, (2.13)
where B−1 is an inverse Blaschke product and θ0 ∈ S0(|U−|, |Y−|). The values
of θP22 are operators between the spaces with same finite dimension, and they
can be identified with square matrices. Moreover, the values of θP22 are by
construction and Lemma 2.1 invertible at least on ρ(θ) \Ξ, where Ξ contains
at most κ points. Since the values of B−1 are invertible whenever they exists,
it follows that the values of θ0 are invertible on ρ(θ) \ Ξ. In particular, the
function det(θ0), is not identically zero. These facts combined with (2.13)
show that θ−1P22(z) has a representation
θ−1P22(z) =
cof(θP22(z))
det(θP22(z))
=
cof(θP22(z))
det(B−1(z)) det(θ0(z))
, (2.14)
where cof means the cofactor matrix. The function θP22 is meromorphic on D
and has strong contractive radial limit values a.e. on T, so clearly cof(θP22)
is meromorhic in D and has strong radial limit values a.e. on T. Since the
values of Blaschke product B−1 are unitary everywhere on the unit cirle,
|det(B−1(ζ))| = 1 for every ζ ∈ T. The values of θ0 are contractive every-
where on D, and therefore det(θ0) is bounded holomorphic function in D.
This implies that radial limit values of det(θ0) exist, and since det(θ0) is not
identically zero, the radial limit values also differ from zero a.e. on T. It now
follows from (2.14) that θ−1P22 is meromorphic on D and has radial limit values
a.e. on T. It has been proved that all the entries in the representation of θ in
(2.4) are meromorphic in D and have strong radial limit values a.e. on T, so
the same holds for θ. The fact that the radial limit values of θ are contractive
with the respect to the inner products of U and Y follows now easily from
the identity (2.6) in Proposition 2.1, since the radial limit values of θP are
contractive.
(ii) Consider the identities (2.6) and (2.7) from Proposition 2.1. The
claims follow from these identities if one proves that the strong radial limit
values of Φ and Ψ exist and are onto a.e. on T. It follows from the part (i) that
all the entries in the definition of Φ in (2.5) have strong radial limit values
a.e. on T, so the same holds for Φ. Since θ−122 = σ
−1θP 22τ and the strong
radial limit values of θP 22 exist a.e. on T, the strong radial limit values of
θ−122 also exist a.e. on T. Especially, the strong radial limit values of θ22 are
invertible a.e. on T. An easy calculation then shows that the strong radial
IEOT Generalized Schur–Nevanlinna functions Page 11 of 29 42
limit values of Φ are onto a.e. on T. Similar argument shows that the same
holds for Ψ, and the claims follow. 
In the special case where U = Y and U is finite dimensional, Theorem 2.8
above could be derived from [1, Theorem 6.8]. A function θ ∈ Sκ(U ,Y), where
U and Y are Hilbert spaces, is called inner (co-inner, bi-inner), if the radial
limit values of θ are isometric (co-isometric, unitary) a.e. on T. By using
a similar notion as in [1,6,8,21], a function θ ∈ Sκ(U ,Y), where U and Y
are Pontryagin spaces with the same negative index, is called a generalized
J -inner (co-J -inner, bi-J -inner ) function, if the radial limit values of θ
are isometric (co-isometric, unitary) a.e. on T, with respect to the inner
product of U and Y. Following [12]; see also [31, Sect. 4], the class Uκ(U ,Y) is
defined to be the subclass of the generalized bi-J -inner functions in Sκ(U ,Y).
The class Uκ(U ,U) is written as Uκ(U). For a symmetric function, it is
evident that if the radial values are isometric or co-isometric a.e., they are
also unitary.
3. Linear systems, self-adjoint realizations and similarity
mappings in state spaces
If needed, the colligation, or the system, of the form (1.4) will be written as
Σ = (A,B,C,D;X ,U ,Y). Often in this paper, U = Y and it will be then
written Σ = (TΣ;X ,U). In what follows, unless otherwise stated, the state
space X and the spaces U and Y are assumed to be Pontryagin spaces, which
will be indicated by the notation Σ = (TΣ;X ,U ,Y;κ) where κ is reserved for
the negative index of X . Note that the common negative index of U and Y
is not assumed to be related to κ. The adjoint or dual of the system Σ is the
system Σ∗ such that its system operator is the indefinite adjoint T ∗Σ of TΣ.
That is, Σ∗ = (T ∗Σ;X ,Y,U). In this paper, all the adjoints are with respect
to the indefinite inner products in question. The identity θΣ∗(z) = θΣ#(z)
holds for the transfer function θΣ∗ of the dual system Σ∗.
The following subspaces
X c := span {ranAnB : n = 0, 1, . . .} (3.1)
X o := span {ranA∗nC∗ : n = 0, 1, . . .} (3.2)
X s := span {ranAnB, ranA∗mC∗ : n,m = 0, 1, . . .}, (3.3)
are called, respectively, controllable, observable and simple subspaces. The
system is said to be controllable (observable, simple) if X c = X (X o =
X ,X s = X ) and minimal if it is both controllable and observable. When
Ω  0 is some symmetric neighbourhood of the origin, that is, z¯ ∈ Ω when-
ever z ∈ Ω, then also
X c = span {ran (I − zA)−1B : z ∈ Ω} (3.4)
X o = span {ran (I − zA∗)−1C∗ : z ∈ Ω} (3.5)
X s = span {ran (I − zA)−1B, ran (I − wA∗)−1C∗ : z, w ∈ Ω} (3.6)
42 Page 12 of 29 L. Lilleberg IEOT
In the case where all the spaces are Hilbert spaces, it is well known; see
for instance [8, Proposition 8], that the transfer function of the passive system
is an ordinary Schur function. In general case where X , U and Y are Pon-
tryagin spaces such that U and Y have the same negative index, the transfer
function of the passive system Σ = (TΣ;X ,U ;κ) is a generalized Schur func-
tion, with the index not larger that the negative index of the state space [32,
Proposition 2.4]. Conversely, every θ ∈ Sκ(U ,Y) has a realization of the form
(1.5), and the realization can be chosen such that it is controllable isometric
(observable co-isometric, simple conservative, minimal passive) [2, Chapter
2], [32, Lemma 2.8]. Any two controllable isometric (observable co-isometric,
simple conservative) realizations of θ ∈ Sκ(U ,Y) are unitarily similar [2,
Theorem 2.1.3]. Two given realizations Σ1 = (A1, B1, C1,D1;X1,U ,Y;κ1)
and Σ2 = (A2, B2, C2,D2;X2,U ,Y;κ2) of the same L(U ,Y)-valued function
θ analytic at the origin are called unitarily similar if D1 = D2 and there
exists a unitary operator U : X1 → X2 such that
A1 = U−1A2U, B1 = U−1B2, C1 = C2U. (3.7)
Unitary similarity preserves dynamical properties of the system and also the
spectral properties of the main operator. Moreover, it easily follows that if the
realizations are unitarily similar, their state spaces have the same negative
index.
The realizations Σ1 and Σ2 above are said to be weakly similar if D1 =
D2 and there exists an injective closed densely defined possible unbounded
linear operator Z : X1 → X2 with the dense range such that
ZA1x = A2Zx, C1x = C2Zx, x ∈ D(Z), and ZB1 = B2, (3.8)
where D(Z) is the domain of Z. It is known that two minimal realizations
of θ ∈ Sκ(U ,Y), or more generally, any L(U ,Y)-valued function holomorphic
at the origin, are weakly similar; see [32, Proposition 2.2], [31, Theorem 2.5]
and [35, p. 702].
For a generalized Nevanlinna function θ ∈ Nκ(U) in the special case
where U is a Hilbert space, the realization of θ usually means a representation
of the form
θ(z) = θ(z0)∗ + (z − z0)Γ∗(I + (z − z0)(H − z)−1)Γ, (3.9)
such that X is a Pontryagin space, Γ ∈ L(U ,X ), H is a self-adjoint linear
relation in X and z0 is some fixed point in ρ(H) ∩ C+, where ρ(H) is the
field of regularity of H. In fact, θ is a generalized Nevanlinna function if and
only if it has a representation of the form (3.9) [25,29]. The realization can
be chosen such that the negative index of X coincides with the index κ of
θ ∈ Nκ(U), and it holds
X = span {(I + (z − z0)(H − z)−1)Γu : z ∈ ρ(H), u ∈ U
}
.
In that case, the realization is unique up to unitary equivalence.
In general, a function θ ∈ Nκ(U) is not necessary holomorphic at the
origin, and therefore it cannot be realized in the form (1.6). However, a self-
adjoint system with a Pontryagin state space always induces some generalized
Nevanlinna function.
IEOT Generalized Schur–Nevanlinna functions Page 13 of 29 42
Proposition 3.1. Let Σ = (A,B,B∗,D;X ,U ;κ) be a self-adjoint system. Then
the transfer function θ of Σ belongs to the generalized Nevanlinna class Nκ′
(U), where κ′ is the dimension of a maximal negative subspace of
span{ran (I − zA)−1B : z ∈ Ω}, (3.10)
where Ω is some sufficiently small symmetric neighbourhood of the origin.
Proof. Since Σ is self-adjoint, A and D must be self-adjoint operators, U = Y,
θ(z) = θ#(z), and B∗ = C. Then the spaces (3.1)–(3.3) coincide. It follows
from [15, Corollary 3.15, pp. 106] that the non-real spectrum of A consists of
not more than 2κ (counting multiplicities) eigenvalues situated symmetrically
with respect to the real axis. Since (I − zA)−1 exists whenever 1/z is in
the resolvent set ρ(A) of A, it follows that θ(z) = D + zB∗(I − zA)−1B is
meromorphic on C\R with at most 2κ non-real poles. By using the resolvent
identity; cf. also [2, Theorem 1.2.4], and the fact that the system operator is
self-adjoint, one deduces that the Nevanlinna kernel of θ can be represented
as
Nθ(w, z) =
θ(z) − θ∗(w)
z − w¯ = B
∗(I − zA∗)−1(I − w¯A)−1B. (3.11)
Therefore, it follows from [2, Lemma 1.1.1’] that the number of negative
eigenvalues of the Gram matrix of the form
(〈Nθ(wj , wi)fj , fi〉U
)n
i,j=1
=
(〈
(I − w¯jA)−1Bfj , (I − w¯iA)−1Bfi
〉
X
)n
i,j=1
,
where fi ∈ U and wi ∈ C \ R, i = 1, . . . , n, coincides with the dimension
of a maximal negative subspace of the span of {(I − w¯iA)−1Bfi}ni=1. It now
follows that the Nevanlinna kernel Nθ has κ′ negative squares, where κ is
the dimension of a maximal negative subspace of (3.10), and the proof is
complete. 
By using the fact that the transfer function of the passive system (1.5)
is a generalized Schur function with the index not larger than the negative
index of the state space of Σ, it follows from Proposition 3.1 that the transfer
function of a passive self-adjoint system is both a generalized Schur function
and a generalized Nevanlinna function. Moreover, if U is a Hilbert space,
the negative indices coincide. Some further machinery from the Kre˘ın space
operator theory will be needed to prove this.
Let X be a Kre˘ın space. The negative index ind−(H), with respect to
the inner product of X , of the bounded self-adjoint operator H ∈ L(X ) is
defined to be the supremum of all positive integers n such that there exists
an invertible and nonpositive matrix of the form
(〈Hxj , xi〉X
)n
i,j=1
, where
{xk}nk=1 ⊂ X . If such a matrix does not exists for any n, then ind−(H) is
defined to be zero. In that case, the operator H is nonnegative with respect
to the inner product of X .. In general, the negative index of the self adjoint
operator measures how much the operator behaves like a positive operator.
For an arbitrary T ∈ L(X ,Y), the operator T ∗T is a bounded self adjoint
operator in L(X ), and it is easy to deduce that T is contractive if and only
if ind−(I − T ∗T ) = 0.
42 Page 14 of 29 L. Lilleberg IEOT
It well known; cf. [15, Theorem 3.4 on p. 267.], [23, Lecture 2], that
every bounded linear operator between Kre˘ın spaces can be dilated to unitary
operator. In this paper, the following version of that result, which can be
derived from [23, Theorems 2.3 and 2.4], is needed.
Theorem 3.2. Suppose that A ∈ L(X1,X2) where X1 and X2 are Pontryagin
spaces with the same negative index. Then there exist Kre˘ın spaces DA and
DA∗ with
ind−(I − A∗A) = ind−(DA) = ind−(DA∗) = ind−(I − AA∗),
and linear operators DA ∈ L(DA,X1) and DA∗ ∈ L(DA∗ ,X2) with zero ker-
nels and a linear operator L ∈ L(DA,DA∗) such that it holds
I − A∗A = DAD∗A, I − AA∗ = DA∗D∗A∗ .
Furthermore, the operator
UA :=
(
A DA∗
D∗A −L∗
)
:
( X1
DA∗
)
→
(X2
DA
)
(3.12)
is unitary. Moreover, if ind−(I − A∗A) = ind−(I − AA∗) is finite, then UA
is essentially unique.
The operator UA in Theorem 3.2 is called as a Julia operator of A, the
operators DA and DA∗ are called, respectively, defect operators of A and
A∗, and the spaces DA and DA∗ are called, respectively, defect spaces of A
and A∗. In general, any bounded operator V with the zero kernel is called
as a defect operator of A if it holds I − A∗A = V V ∗. Julia operator of A is
essentially unique, if for any other Julia operator
U ′A =
(
A DA∗
′
DA
′∗ −L′∗
)
:
( X1
DA∗
′
)
→
( X2
DA
′
)
,
of A, there exists unitary operators H1 : DA∗ → DA∗ ′ and H2 : DA → DA′
such that
DA∗ = DA∗ ′H1, DA = DA′H2, H1L = L′H2.
If θ is the transfer function of the system (1.5), the Schur kernel of the
form (1.1) can be represented as a sum of two kernels. This can be done by
using the defect operators of the system operator and its adjoint. A special
case, where the system is passive, i.e. the system operator is contractive, is
proved in [32, Lemma 2.4]; see also the proof of [34, Theorem 2.2]. The proofs
given therein can be applied word by word to get the next result, since the
existence of defect operator is guaranteed by Theorem 3.2. Therefore, the
proof will not be repeated here.
Lemma 3.3. Let Σ = (A,B,C,D;X ,U ,Y;κ) be a system with the transfer
function θ. Denote the system operator of Σ as T. If
DT =
(
DT,1
DT,2
)
: DT →
(X
U
)
DT ∗ =
(
DT ∗,1
DT ∗,2
)
: DT ∗ →
(X
Y
)
, (3.13)
IEOT Generalized Schur–Nevanlinna functions Page 15 of 29 42
are defect operators of T and T ∗, respectively, then the identities
IY − θ(z)θ∗(w) = (1 − zw¯)G(z)G∗(w) + ψ(z)ψ∗(w), (3.14)
IU − θ∗(w)θ(z) = (1 − zw¯)F ∗(w)F (z) + ϕ∗(w)ϕ(z), (3.15)
with
G(z) = C(IX − zA)−1, ψ(z) = DT ∗,2 + zC(IX − zA)−1DT ∗,1 , (3.16)
F (z) = (IX − zA)−1B, ϕ(z) = D∗T,2 + zD∗T,1(IX − zA)−1B, (3.17)
hold for every z and w in a sufficiently small symmetric neighbourhood of the
origin.
The system (1.5) can be expanded to a larger system such that the
state space and the main operator will not change. This expansion is called
an embedding. The embedding of the system (1.5) is any system determined
by the system operator
TΣ˜ =
(
A B˜
C˜ D˜
)
=
⎛
⎝
A
(
B B1
)
(
C
C1
) (
D D12
D21 D22
)
⎞
⎠ :
⎛
⎝
X(U
U ′
)
⎞
⎠ →
⎛
⎝
X(Y
Y ′
)
⎞
⎠,
where U ′ and Y ′ are Hilbert spaces. The transfer function of the embedded
system is
θΣ˜(z)=
(
D + zC(IX − zA)−1B D12 + zC(IX − zA)−1B1
D21 + zC1(IX − zA)−1B D22 + zC1(IX − zA)−1B1
)
=
(
θΣ(z) θ12(z)
θ21(z) θ22(z)
)
,
where θΣ is the transfer function of the original system.
Proposition 3.4. If Σ = (A,B,B∗,D;U ;κ) is a passive self-adjoint system,
its transfer function θ belongs to Sκ1(U) ∩ Nκ2(U), where κ1 ≤ κ2 and κ2 is
the dimension of a maximal negative subspace of
span{ran (I − zA)−1B : z ∈ Ω},
where Ω is a sufficiently small symmetric neighbourhood of the origin . More-
over, if U is a Hilbert space, then κ1 = κ2.
Proof. It follows from Proposition 3.1 that θ ∈ Nκ2(U). Moreover, since Σ
is passive, θ is also a generalized Schur function with the negative index κ1,
which is not larger than the negative index κ of the state space X . By using
Lemma 3.3, the equation (3.14) and a result from [2, Theorem 1.5.5], it follows
that κ1 ≤ κ′1 + κ′2, where κ′1 and κ′2 are the negative indices of the kernels
(1 − zw¯)−1 (ψ(z)ψ∗(w)) and G(z)G∗(w) in (3.14), respectively. Since Σ is self-
adjoint system, A = A∗ and C = B∗. Then the same argument as in the proof
of Proposition 3.1 shows that κ′2 = κ2. Since Σ is passive, the system operator
of T and its adjoint T ∗ are contractive. Therefore, ψ∗(w) is an operator in
a Hilbert space DT ∗ , and it follows that the kernel (1 − zw¯)−1 (ψ(z)ψ∗(w))
has no negative square; for details, see the proof of [32, Proposition 2.4].
Therefore κ1 ≤ κ2.
Assume then that U is a Hilbert space. Denote the system operator of
Σ as T. Theorem 3.2 guarantees the existence of the defect operator DT of
42 Page 16 of 29 L. Lilleberg IEOT
T with the properties described therein. Since T = T ∗ is contractive, the
domain DT of DT is a Hilbert space. Moreover, (3.12) shows that it holds
(
T DT
) (
T DT
)∗ = TT ∗ + DTD∗T = T
2 + DTD∗T = I.
That is, the operator
(
T DT
)
is co-isometric. Therefore the system Σ˜ with
the system operator
TΣ˜ =
(
A
(
B DT,1
)
B∗
(
D DT,2
)
)
:
⎛
⎝
X( U
DT
)
⎞
⎠ →
(X
U
)
is a co-isometric embedding of Σ. The transfer function of Σ˜ is given by
θ˜ =
(
D DT,2
)
+ zB∗(I − zA∗)−1 (B DT,1
)
=
(
θ ψ
)
(3.18)
where ψ is defined as in (3.16). Since the system Σ is self-adjoint, the identity
(3.11) holds. By applying (3.14) from Lemma 3.3, it follows that
Kθ˜(w, z) = B
∗(I − zA∗)−1(I − w¯A)−1B = Nθ(w, z).
Since Nθ has κ2 negative squares, so has Kθ˜, and therefore θ˜ is a generalized
Schur function with the index κ2. The first identity in (3.18) shows that the
total number of poles, counting multiplicities, of θ˜ and θ are equal. It then
follows from Lemma 2.5 that θ˜ and θ have the same index, and the proof is
complete. 
Theorem 3.5. If U is a Hilbert space, then θ ∈ Sκ1(U)∩Nκ2(U) has a minimal
self-adjoint realization Σ = (A,B,B∗,D;X ,U ;κ2).
Proof. Define θˇ(z) = −θ (1/z)+θ(0). Then θˇ clearly is meromorphic on C\R,
analytic at the infinity with θˇ(∞) = 0 and it holds θˇ(z) = θˇ#(z). Moreover,
for any finite set of points {w1, . . . , wn} ⊂ ρ(θˇ), and vectors {f1, . . . , fn} ⊂ U ,
it holds
(〈Nθˇ(wj , wi)fj , fi〉U
)n
i,j=1
=
⎛
⎝
〈
θ∗
(
1
wj
)
− θ
(
1
wi
)
wi − wj fj , fi
〉
U
⎞
⎠
n
i,j=1
=
⎛
⎝
〈
θ
(
1
wi
)
− θ∗
(
1
wj
)
1
wi
− 1wj
wifj , wjfi
〉
U
⎞
⎠
n
i,j=1
=
(〈
Nθ
(
1
wj
,
1
wi
)
wifj , wjfi
〉
U
)n
i,j=1
=
(〈
Nθ
(
w′j , w
′
i
)
f ′j , f
′
i
〉
U
)n
i,j=1
.
The identity above yields that Nθ and Nθˇ have the same number of negative
squares, and therefore θˇ ∈ Nκ1(U). Since θ is holomorphic at the origin, it has
the Neumann series of the form θ(z) =
∑∞
n=0 θnz
n, for every z ∈ Ω, where
IEOT Generalized Schur–Nevanlinna functions Page 17 of 29 42
Ω is a sufficiently small symmetric neighbourhood of the origin. Therefore
limz→0 z−1 (θ(z) − θ(0)) = θ1, and also
lim
z→∞ zθˇ(z) = − limz→∞ z
(
θ
(
1
z
)
− θ(0)
)
= − lim
z′→0
z
′−1 (θ (z′) − θ(0)) = −θ1.
This implies y limy→∞
∣
∣〈θˇ(iy)f, f
〉
U
∣
∣ < ∞ for every f ∈ U . Therefore, θˇ has
the realization of the form (3.9) which reduces to θˇ(z) = B̂∗(Â − z)−1B̂,
where B̂ ∈ L(U ,X ) and  is a self-adjoint operator in a Pontryagin space X̂
with the negative index κ2 [28], [33, pp. 348–349]. But then θ can be realized
as
θ(z) = D − B̂∗
(
 − 1
z
)−1
B̂ = D + zB̂∗
(
I − zÂ
)−1
B̂,
where D = θ(0) = θ∗(0). That is, Σ̂ = (Â, B̂, B̂∗,D; X̂ ,U ;κ2) is a self-adjoint
realization of θ. It follows from Proposition 3.1 that the dimension of the
maximal negative subspace of span{ran (I − zÂ)−1B̂ : z ∈ Ω} := S, where Ω
is some sufficiently small symmetric neighbourhood of the origin, is κ2, the
negative index of X̂ . Then, the closure X̂ c of S must be a regular subspace
of X̂ . Therefore, X̂ = X̂ c ⊕
(
X̂ c
)⊥
, and
(
X̂ c
)⊥
is a Hilbert subspace of
X̂ . Since Σ is self-adjoint system, the spaces X̂ c, X̂ o and X̂ s coincide. These
facts and (3.1) imply that the system operator T̂ of Σ̂ can be represented as
T̂ =
⎛
⎝
(
A1 0
0 A
) (
0
B
)
(
0 B∗
)
D
⎞
⎠ :
⎛
⎜
⎝
(
(X̂ c)⊥
X̂ c
)
U
⎞
⎟
⎠→
⎛
⎜
⎝
(
(X̂ c)⊥
X̂ c
)
U
⎞
⎟
⎠ . (3.19)
Define X = X̂ c. A calculation shows that ÂnB̂ = AnB for every n ∈ N0 =
{0, 1, 2, . . .} and therefore span {ranAnB : n = 0, 1, . . .} = X . That is, Σ =
(A,B,C,D;X ,U ;κ1) is a self-adjoint minimal realization of θ. 
Remark 3.6. The realization Σ in Theorem 3.5 is not shown to be passive.
In the case where U is a Hilbert space and θ ∈ S(U) ∩ N(U), that is, when θ
is an ordinary Schur and Nevanlinna function, it is known from [4, Theorem
5.1] that there exists a minimal self-adjoint passive realization Σ of θ. In
general, if θ ∈ Sκ1(U)∩Nκ2(U), where U is Pontryagin space, it follows from
Proposition 3.4 that a self-adjoint minimal realization Σ = (TΣ;X ,U ;κ) of
θ can be passive only if κ1 ≤ κ2 = κ, and in the case where U is a Hilbert
space, only if κ1 = κ2 = κ.
The conditions of Theorem 3.5 that U is a Hilbert space there and
θ ∈ Nκ2(U) can be relaxed slightly; with a cost of weakened conclusions.
Proposition 3.7. Let U be a Pontryagin space. Then a symmetric function
θ ∈ Sκ(U) has a self-adjoint realization Σ = (A,B,C,D;X ,U) where X is a
Kre˘ın space.
Proof. Let Σ1 = (A,B,C,D;X1,U ;κ) be a simple conservative realization
of θ. Since θ is symmetric in sense θ(z) = θ#(z), it holds D = D∗ and the
dual system Σ∗1 = (A
∗, C∗, B∗,D;X1,U ;κ) of Σ1 is also a simple conservative
42 Page 18 of 29 L. Lilleberg IEOT
realization of θ. Therefore Σ1 and Σ∗1 are unitarily similar, that is, there exists
a unitary mapping J : X1 → X1 such that, see (3.7),
A = J∗A∗J, JB = C∗, C = B∗J, J−1 = J∗. (3.20)
The letter J is used, because the operator J is now also self-adjoint in X1.
Indeed, let N ∈ N0. Easy calculations show that it holds
J
(
N∑
n=0
AnB
)
=
N∑
n=0
A
∗n
C∗ =
N∑
n=0
A
∗n
J∗B = J∗
(
N∑
n=0
AnB
)
, (3.21)
and similarly
J
(
N∑
n=0
A
∗n
C∗
)
= J∗
(
N∑
n=0
A
∗n
C∗
)
. (3.22)
Since Σ1 is simple, it follows from (3.21) and (3.22) that J and J∗ coincide
on a dense lineal of X1, and then by continuity, everywhere. That is, J is
unitary and self-adjoint. Now introduce the inner product space X , which
coincides with X1 as a vector space but which is endowed with the inner
product 〈x, y〉X = 〈Jx, y〉X1 . Then X is a Kre˘ın space. Moreover, it holds
〈Ax, y〉X = 〈JAx, y〉X1 = 〈x,A∗Jy〉X1 = 〈x, JAy〉X1 = 〈x,Ay〉X ,
〈Bu, x〉X = 〈JBu, x〉X1 = 〈u,B∗Jx〉U = 〈u,B∗Jx〉U = 〈u,Cx〉U .
This implies that A is self-adjoint in the Kre˘ın space X , and the adjoint of B :
U → X is C viewed as operator from X to U . Then, A, B, C and their adjoints
all are everywhere defined, and therefore bounded also with respect to the
topology induced by X [16, Chapter VI 2]. Define Σ = (A,B,C,D;X ,U).
Then, Σ is a self-adjoint realization of θ, and the proof is complete. 
In Proposition 3.7, the self-adjoint realization Σ = (TΣ;X ,U) with the
Kre˘ın space X was constructed from a simple conservative realization Σ1 =
(TΣ1 ;X1,U ;κ). If X is a Pontryagin space with the negative index κ′′, it
follows from Proposition 3.1 that the transfer function θ ∈ Sκ(U) belongs
also in the class Nκ′(U), where κ′ ≤ κ′′. One might conjecture that this
happens for every θ ∈ Sκ(U) ∩ Nκ′(U). However, Theorem 3.8 below shows
that this is not true; it happens only when θ ∈ Uκ(U)∩Nκ′(U). That is, the
values of θ must also be unitary for all but finitely many points on the unit
circle T; see the page 11.
Theorem 3.8. Let θ ∈ Sκ1(U) be symmetric in a sense θ(z) = θ#(z), and let
Σ1 = (A,B,C,D;X1,U ;κ1) be a simple conservative realization of θ. Con-
struct the Kre˘ın space X and the self-adjoint realization Σ=(A,B,C,D;X ,U)
by using the method given in the proof of Proposition 3.7. Then θ ∈ Uκ1(U)∩
Nκ2(U) if and only if X is a Pontryagin space with the negative index κ2.
Proof. Let ∗ and [∗] refer, respectively, to the adjoint with respect to the
inner product of X1 and X .
⇐: Suppose that X is a Pontryagin space with the negative index κ2.
Let J be the unitary similarity mapping used in Proposition 3.7 with the
IEOT Generalized Schur–Nevanlinna functions Page 19 of 29 42
properties (3.20). Then, since
〈A∗x, y〉X = 〈JA∗x, y〉X1 = 〈x,AJy〉X1 = 〈x, JA∗y〉X1 = 〈x,A∗y〉X ,
A and A∗ are both self-adjoint operators with respect to the inner product
of the Pontryagin space X , and it follows from [15, Corollary 3.15, pp. 106]
that the non-real spectra of A and A∗ consist only of finitely many points.
Then, (I −ζA)−1 and (I −ζA∗)−1 exist for all but finitely many ζ ∈ T. Since
Σ1 is conservative, the system operator T of Σ is unitary, and therefore the
defect spaces of T and T ∗ in (3.13), are zero spaces. By using (3.15) from
Lemma 3.3, it can be now deduced that
I − θ∗(ζ)θ(ζ) = (1 − ζζ)B∗(I − ζA∗)−1(I − ζA)−1B = 0
for all but finitely many ζ ∈ T, which shows that θ = θ# ∈ Uκ1(U). Choose
some fundamental decomposition of U , and consider the Potapov–Ginzburg
transformation θP as in (2.3). It can be assumed that θP is holomorphic at
the origin, since if not, one only has to consider θP (η(z)) as in Remark 2.2.
Then by Proposition 2.1 and Theorem 2.8, θP is symmetric and the radial
limit values of θP are unitary a.e. on T. It follows then from [12, Theorems
9.4 and 10.2]; see also [31, Theorem 4.4], that all simple passive realizations
of θP are conservative and minimal. Then it follows from [2, Theorems 2.1.3
and 4.3.3] that all simple conservative realizations of θ are minimal. There-
fore Σ1 is minimal, which implies that Σ is also minimal, since the norms
of spaces X1 and X are equivalent. Therefore, Σ is a minimal self-adjoint
system with a Pontryagin state space, and it follows from Proposition 3.1
that θ is a generalized Nevanlinna function whose negative index coincides
with the index of the maximal negative subspace of the space of the form
(3.10). But since Σ is minimal, the space (3.10) is dense in X , and by [16,
Theorem 1.4 on p. 185], it contains a maximal uniformly negative subspace
of X . It follows that θ ∈ Nκ2(U).
⇒: Let θ ∈ Uκ1(U)∩Nκ2(U). By using the Potapov–Ginzburg transfor-
mation similarly as above, it can be deduced that Σ1 and Σ are both minimal.
By using a similar argument as in the proof Proposition 3.1, one deduces that
the Nevanlinna kernel of θ can be represented as
Nθ(w, z)= C(I − zA)−1(I − w¯A)−1B
= B[∗](I − zA[∗])−1(I − w¯A)−1B. (3.23)
Then the matrix of the form
(〈Nθ(wj , wi)fj , fi〉U
)n
i,j=1
=
(〈
(I − w¯jA)−1Bfj , (I − w¯iA)−1Bfi
〉
X
)n
i,j=1
,
such that fi ∈ U , wi ∈ Ω, i = 1, . . . , n, where Ω is a sufficiently small symmet-
ric neighbourhood of the origin, is a Gram matrix. Since θ ∈ Nκ2(U), the ker-
nel Nθ has κ2 negative squares. These facts combined with [2, Lemma 1.1.1’]
imply that there exists a finite sequence {(I −wiA)−1Bfi}ni=1 ⊂ X of vectors
such that the linear span of {(I − wiA)−1Bfi}ni=1 contains a κ2-dimensional
anti-Hilbert subspace of X . Therefore, ind−X is at least κ2. Suppose that
ind−X > κ2. Then there exists a finite sequence {xi}κ′i=1 ⊂ X of linearly inde-
pendent negative vectors such that κ′ > κ2. Since Σ is minimal, span{ran (I−
42 Page 20 of 29 L. Lilleberg IEOT
zA)−1B : z ∈ Ω} := S, where Ω is a sufficiently small symmetric neighbour-
hood of the origin, is dense in X . Then, each xi can be approximated as
closely as desired by a vector x′i from S. By choosing a good enough approx-
imation x′i for each xi, one obtains κ
′ negative linearly independent vectors
x′1, . . . , x
′
κ′ ∈ S. That is, S contains k′-dimensional anti-Hilbert subspace,
and by using a similar argument as in the proof of Proposition 3.1, it can
be shown that the kernel Nθ has at least κ′ > κ2 negative squares. This
contradicts the assumption that θ ∈ Nκ2(U). Therefore, ind−X must be κ2,
and the proof is complete. 
A simple conservative realization of θ ∈ Sκ(U ,Y) is unique up to uni-
tary similarity, and therefore the results of Theorem 3.8 do not depend on
the choice of Σ and J. The concrete models for J can be obtained by us-
ing the canonical realizations of θ. If U and Y are Pontryagin spaces with
the same negative index, for an L(U ,Y)-valued function θ holomorphic on a
neighbourhood Ω of the origin, the kernel (1.1) has κ negative squares if and
only if the related L(Y ⊕ U)-valued kernel
Dθ(w, z) =
⎛
⎜
⎝
Kθ(w, z)
θ(z) − θ(w)
z − w
θ#(z) − θ#(w)
z − w Kθ#(w, z)
⎞
⎟
⎠ , w, z ∈ Ω, (3.24)
has κ negative squares; see [2, Theorem 2.5.2]. The Pontryagin space gener-
ated by the kernel (3.24), that is, the completion of the space
span
{
Dθ(w, z)
(
y
u
)
: w ∈ Ω, y ∈ Y, u ∈ U
}
, (3.25)
where Dθ(w, z)
(
y
u
)
is treated as a function of z, is denoted by D(θ). The
function space D(θ) is continuously contained in H(θ) ⊕ H(θ#), where H(θ)
is the Pontryagin space generated by the kernel (1.1). The spaces H(θ) and
D(θ) can be chosen as state spaces of, respectively, an observable co-isometric
realization and a simple conservative realization of θ ∈ Sκ(U ,Y). For h ∈
H(θ), (h
k
) ∈ D(θ) and u ∈ U , define respectively
⎧
⎨
⎩
A1 : h(z) 	→ h(z) − h(0)
z
, B1 : u 	→ θ(z) − θ(0)
z
u,
C1 : h(z) 	→ h(0), D : u 	→ θ(0)u,
(3.26)
and
⎧
⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
A2 :
(
h(z)
k(z)
)
	→
(
h(z) − h(0)
z
zk(z) − θ#(z)h(0)
)
, B2 :u 	→
(
θ(z) − θ(0)
z
u
(
IU − θ#(z)θ#∗(0)
)
u
)
,
C2 :
(
h
k
)
	→ h(0), D : u 	→ θ(0)u.
(3.27)
Then Σ1=(A1, B1, C1,D,H(θ),U ,Y,κ) and Σ2=(A2, B2, C2,D,D(θ),U ,Y,κ)
are, respectively, an observable co-isometric realization of θ, and a simple
conservative realization of θ; for the proof, see [2, Theorems 2.2.1 and 2.3.1].
IEOT Generalized Schur–Nevanlinna functions Page 21 of 29 42
These systems are called, respectively, the canonical co-isometric realization
and the canonical unitary (or conservative) realization of θ. Any observable
co-isometric realization of θ ∈ Sκ(U ,Y) is unitarily similar with the system
Σ1, and any simple conservative realization is unitarily similar with Σ2.
Suppose next the simple conservative realization of the symmetric func-
tion θ ∈ Sκ1(U) in Theorem 3.8 is chosen to be the canonical unitary real-
ization. Then it can be derived from [3, Theorem 3.6] that the self-adjoint
unitary similarity J is of the form J = ĴD(θ), where
Ĵ =
(
0 I
I 0
)
:
(H(θ)
H(θ)
)
→
(H(θ)
H(θ)
)
. (3.28)
In addition, if also θ ∈ Uκ1(U), it has been shown in the proof of Theorem 3.8
that all co-isometric observable or isometric controllable realizations of θ are
minimal conservative. Therefore it can be assumed that Σ1 in Theorem 3.8
is the canonical co-isometric realization. In that case, it can be derived from
[3, Corollary 3.7] that J is the closure of a linear relation Λ defined by
Λ
(
∑
i
Kθ(wi, z)fi
)
=
∑
i
Nθ(wi, z)fi,
where Kθ(wi,z)fi ∈H(θ) and Nθ(wi,z)fi ∈H(θ) are treated as a function of z.
Proposition 3.9. Let U be Pontryagin space and let θ be a symmetric L(U)-
valued function holomorphic at the origin and meromorphic on D ∪ C \ R.
Then the following statements are equivalent:
(i) θ has a minimal conservative self-adjoint realization Σ such that the
state space of Σ is a Hilbert space;
(ii) θ ∈ U(U) ∩ N(U);
(iii) Kθ(w, z) = Nθ(w, z) and the kernels are nonnegative.
Proof. (i) ⇒ (ii). Denote Σ = (A,B,B∗,D;X ,U ; 0). Since Σ is a minimal
conservative self-adjoint realization of θ such that X is a Hilbert space, it
follows from Proposition 3.4 that θ ∈ S(U) ∩ N(U). Moreover, the main
operator A is self-adjoint operator in the Hilbert space X , and a similar
argument as used in the proof of Theorem 3.8 can be used to show that the
values of θ are unitary for every ζ ∈ D\{−1, 1}. Therefore θ ∈ U(U)∩N(U).
(ii) ⇒ (iii). By definition of U(U) and N(U), the kernels Kθ and Nθ are
nonnegative. If U is a Hilbert space, the other claim now follows by combining
[4, Theorem 5.1] and [5, Proposition 3.1]. Therefore assume ind−U > 0. Fix
some fundamental decomposition of U , and consider the Potapov–Ginzburg
transformation θP of θ, defined by (2.3). Since θ is symmetric, the functions
Φ and Ψ defined by (2.5) coincide. It can be assumed that θ−122 (0) exists;
otherwise, consider θP (η(z)) as in Remark 2.2. Then θP ∈ U(U ′)∩N(U ′) by
Proposition 2.1(vii), and since U ′ is a Hilbert space, it now holds KθP (w, z) =
NθP (w, z). It follows then from (2.6) and (2.8) that also Kθ(w, z) = Nθ(w, z).
(iii) ⇒ (i) Since Kθ(w, z) is nonnegative, θ ∈ S(U), and the canonical
unitary realization Σ2 defined by the operators in (3.27) is simple conservative
42 Page 22 of 29 L. Lilleberg IEOT
and the state space is a Hilbert space. Since θ = θ# and Kθ(w, z) = Nθ(w, z),
the kernel Dθ(w, z) in (3.24) reduces to
(
Kθ(w, z) Kθ(w, z)
Kθ(w, z) Kθ(w, z)
)
.
Therefore, all the functions in the space (3.25) are of the form
(
h0(z)
h0(z)
)
=
(∑n
j=1αjKθ(wj , z)uj∑n
j=1αjKθ(wj , z)uj
)
,
such that αj ∈ C, uj ∈ U and wj ∈ Ω, where Ω is the domain of holomorphy
of θ. It follows that all the L(U ⊕ U)-valued functions in the completion
D(θ) of (3.25) are of the form
(
h(z)
h(z)
)
. Then, the self-adjoint unitary similarity
mapping J = ĴD(θ) between Σ2 and Σ∗2, where Ĵ where is defined by (3.28),
is identity. That is, Σ2 is self-adjoint, and since it is simple, it is now minimal,
and the proof is complete. 
In Theorem 3.8, the condition that the space X induced by the mapping
J is a Pontryagin space with the negative index κ is equivalent to ind−J = κ,
where ind−J is with respect to the state space X1. By considering minimal
passive realizations instead of simple conservative realizations, one can obtain
a similar type of characterization when θ ∈ Sκ1(U) ∩ Nκ2(U).
Denote EX (x) = 〈x, x〉X for the vector x in an inner product space
X . For θ ∈ Sκ(U ,Y), where U and Y are Pontryagin spaces with the same
negative index, the realization Σ of θ is called κ-admissible, if the nega-
tive index of the state space of Σ is κ. A κ-admissible passive realization
Σ = (A,B,C,D;X ,U ,Y;κ) of θ ∈ Sκ(U ,Y) is called optimal if for any κ-
admissible passive realization Σ0 = (A0, B0, C0,D;X0,U ,Y;κ) of θ it holds
EX
(
N∑
n=0
AnBun
)
≤ EX0
(
N∑
n=0
A0
nB0un
)
,
for any N ∈ N0 and {un}Nn=0 ⊂ U . Moreover, an observable passive realization
Σ = (A,B,C,D;X ,U ,Y;κ) of θ ∈ Sκ(U ,Y) is called ∗-optimal if for any
observable κ-admissible passive realization Σ0 = (A0, B0, C0,D;X0,U ,Y;κ)
of θ it holds
EX
(
N∑
n=0
AnBun
)
≥ EX0
(
N∑
n=0
A0
nB0un
)
,
for any N ∈ N0 and {un}Nn=0 ⊂ U . The requirement of the observability in the
definition of ∗-optimality is essential to avoid trivialities, see [9, Proposition
3.5 and example on page 144]. Moreover, the requirement that the considered
realizations are κ-admissible is also essential, see [32, Example 3.1].
Let θ ∈ Sκ1(U) be symmetric. It follows from [34, Theorem 5.3] and
[32, Theorem 3.5] that there exists a ∗-optimal minimal passive realization
Σ = (A,B,C,D;X ,U ;κ) of θ. Since θ = θ#, it follows from [34, Theorem 5.2]
and [32, Theorem 3.5] that the dual system Σ∗ = (A∗, C∗, B∗,D;X ,U ;κ) of
IEOT Generalized Schur–Nevanlinna functions Page 23 of 29 42
Σ is optimal minimal passive. Define
Z
(
N∑
n=0
AnBun
)
=
N∑
n=0
A∗nC∗un
for the vectors of the form
∑N
n=0 A
nBun. Since Σ and Σ∗ are minimal and
Σ∗ is optimal, the linear relation Z is densely defined, contractive, and it has
a dense range in X . It follows from [2, Theorem 1.4.2] that the closure of
Z, which is still denoted as Z, is an everywhere defined bounded contractive
linear operator in X . By proceeding as in the proof of [31, Theorem 2.5], one
deduces that Z is injective, it has a dense range, and it holds
ZA = A∗Z, C = B∗Z and ZB = C∗. (3.29)
That is, Z is an everywhere defined weak similarity. Moreover, it holds
Z∗
(
N∑
n=0
AnBun
)
=
N∑
n=0
A∗nZ∗Bun =
N∑
n=0
A∗nC∗un = Z
(
M∑
n=0
AnBun
)
.
Since Σ is minimal, it follows now that Z : X → X is self-adjoint. That
is, Z is bounded injective self-adjoint operator. Moreover, an optimal (∗-
optimal) minimal passive realization of θ ∈ Sκ(U ,Y) is unique up to unitary
similarity [32, Theorem 3.5]. Therefore, the mapping Z is unique up to unitary
equivalence, and the properties of Z in Theorem 3.10 below do not depend
of the choice of a ∗-optimal minimal passive realization of θ.
Theorem 3.10. Let θ ∈ Sκ1(U) be symmetric and let Σ=(A,B,C,D;X ,U ;κ1)
be a ∗-optimal minimal passive realization of θ. Then θ ∈ Sκ1(U) ∩ Nκ2(U)
if and only if ind−Z = κ2, where Z is the self-adjoint contraction in X with
the properties (3.29).
Proof. Since Z is self-adjoint, bounded and injective with a dense range, it has
a Bogna´r–Kra´mli factorization of the form Z = VZV ∗Z , where VZ : DZ → X ,
is bounded, DZ is a Kre˘ın space with ind−DZ = ind−Z, and VZ and V ∗Z
have zero kernels and dense ranges; see [23, Theorem 2.1]. By using the
realization Σ, the properties (3.29) of the self-adjoint mapping Z imply that
Z(I−zA)−1 = (I−zA∗)−1Z whenever (I−zA)−1 and (I−zA∗) exist. Hence
the Nevanlinna kernel of θ can be represented as
Nθ(w, z)=C(I − zA)−1(I − w¯A)−1B=B∗Z(I − zA)−1(I − w¯A)−1B
=B∗(I−zA∗)−1Z(I−w¯A)−1B=B∗(I−zA∗)−1VZV ∗Z (I−w¯A)−1B;
cf. (3.23) on page 20. This yields the identity
(〈Nθ(wj ,wi)fj ,fi〉U
)n
i,j=1
=
(〈
Z(I−w¯jA)−1Bfj ,(I−w¯iA)−1Bfi
〉
X
)n
i,j=1
=
(〈
V ∗Z (I−w¯jA)−1Bfj , V ∗Z (I−w¯iA)−1Bfi
〉
DZ
)n
i,j=1
(3.30)
where n ∈ N, {fj}nj=1 ⊂ U , and {wj}nj=1 ⊂ Ω for some sufficiently small
symmetric neighbourhood Ω of the origin, for the kernel Nθ. Moreover, since
Σ is minimal, the space span{ran (I − zA)−1B : z ∈ Ω} := S is dense in X .
Let y ∈ DZ such that 〈y, V ∗Zx〉DZ = 0 for all x ∈ S. Then, 〈y, V ∗Zx〉DZ =
42 Page 24 of 29 L. Lilleberg IEOT
〈VZy, x〉DZ = 0, which implies VZy = 0 and then y = 0, since S is a dense
set and VZ has only the trivial kernel. That is, V ∗ZS is a dense set in DZ .
⇐: Suppose that ind−Z = κ2. Then ind−DZ = κ2, and it follows from
the identity (3.30) that Nθ has at most κ2 negative squares. Since V ∗ZS is a
dense set in DZ , [2, Lemma 1.1.1] shows that there exists a finite sequence
{V ∗Z (I−w′iA)−1Bf ′i}ni=1 ⊂ DZ of vectors such that the linear span of {V ∗Z (I−
w′iA)
−1Bf ′i}ni=1 contains a κ2-dimensional anti-Hilbert subspace of DZ . Then
by the identity (3.30) the kernel Nθ has at least κ2 negative squares. It
has been showed that Nθ has exactly κ2 negative squares, and therefore
θ ∈ Nκ2(U).
⇒: Assume θ ∈ Nκ2(U). Then the identity (3.30) shows that ind−Z is
at least κ2. Since V ∗ZS is a dense set, a similar argument as in the proof of
Theorem 3.8 can be used to conclude that ind−Z = κ2. 
4. Dilations and subclasses of generalized Schur–Nevanlinna
functions
A dilation of the function θ ∈ Sκ(U ,Y) is any function Θ holomorphic at the
origin, which is of the form
Θ(z) =
(
θ(z) θ2(z)
θ3(z) θ4(z)
)
(4.1)
and has values in L(U ⊕ U ′,Y ⊕ Y ′), where U ′ and Y ′ are Hilbert spaces. In
the case where U ′ or Y ′ is a zero space, the corresponding row or column
in (4.1) will be left out. This definition is a straightforward generalization of
the definition of a dilation of θ ∈ S(U ,Y), where U and Y are Hilbert spaces,
as represented in [14]. A function θ ∈ Sκ(U ,Y) has a generalized bi-J -inner
dilation, if there exists a dilation Θ of θ such that Θ ∈ Uκ(U ⊕ U ′,Y ⊕ Y ′).
The case when U and Y are Hilbert spaces and κ = 0 corresponds to the
ordinary bi-inner dilation. It is known from [14] that θ ∈ S(U ,Y), where U
and Y are Hilbert spaces, has a bi-inner dilation if and only if there exists ϕ ∈
S(U ,Y ′) and ψ ∈ S(Y,U ′), where such that IU − θ∗(ζ)θ(z) = ϕ∗(ζ)ϕ(ζ) and
IY − θ(ζ)θ∗(ζ) = ψ(ζ)ψ∗(ζ) for a.e. ζ ∈ T. Moreover, every θ ∈ S(U)∩ N(U)
has a bi-inner dilation Θ, which can be chosen such that Θ ∈ U (U ⊕ U ′) ∩
N (U ⊕ U ′) [5, pp. 4)].
As mentioned on page 12, any two minimal passive realizations of θ ∈
Sκ(U ,Y) are only weakly similar in general. However, for some general-
ized Schur functions, any two minimal passive κ-admissible realizations are
unitary similar. This happens, for an example, if the boundary values of
θ ∈ Sκ(U ,Y) are (co-)isometric a.e. on T, or when the generalized right or
left defect function is identically zero; see [32, pp. 25]. Let SUκ (U ,Y) be the
class that consist of those functions θ ∈ Sκ(U ,Y) with the property that any
two minimal passive κ-admissible realizations of θ are unitarily similar. A
criterion for θ to be in SU (U ,Y), where U and Y are Hilbert spaces, were ob-
tained by Arov and Nudelman in [10,11]. This result was generalized for the
IEOT Generalized Schur–Nevanlinna functions Page 25 of 29 42
class Sκ(U ,Y), where U and Y are Pontryagin spaces with the same negative
indices, by the author in [32, pp. 25].
Define RSκ(U) to be the class of those functions in Sκ(U)∩Nκ(U) which
have κ-admissible passive self-adjoint realizations. If U is a Hilbert space and
κ = 0, the class RS0(U) coincides with S(U) ∩ N(U) [5].
Theorem 4.1. Every θ ∈ SUκ1(U)∩Nκ2(U), where U is a Pontryagin space, has
a generalized bi-J -inner dilation Θ ∈ Uκ1 (U ⊕ U ′), and every θ ∈ RSκ(U)
has a generalized bi-J -inner dilation Θ ∈ Uκ (U ⊕ U ′) ∩ Nκ (U ⊕ U ′).
Proof. Suppose θ ∈ SUκ1(U) ∩ Nκ2(U). Consider a ∗-optimal minimal real-
ization Σ = (A,B,C,D;X ,U ;κ1) of θ and the self-adjoint contraction Z
with the properties (3.29). Since θ ∈ SUκ1 and Σ and Σ∗ are both mini-
mal passive κ1-admissible realizations of θ, the system Σ is unitarily sim-
ilar with Σ∗. It follows then easily that Z is actually unitary in X , and
therefore Z = Z∗ = Z−1 i.e. Then, since Z in boundedly invertible, the
space XZ , which coincide with X as a vector space but which is endowed
with the inner product 〈x, y〉XZ = 〈Zx, y〉X , is a Kre˘ın space [15, 6.13 on
page 40]. The same arguments as used in the proof of Proposition 3.7 shows
that A and A∗ are self-adjoint with respect to the inner product of XZ , and
ΣZ := (A,B,C,D;XZ ,U) is a self-adjoint realization of θ. Since Σ is minimal
and θ ∈ Nκ2(U), similar arguments as in the proof of Theorem 3.8 can be
used to conlude that ΣZ is minimal and XZ is a Pontryagin space with the
negative index κ2. Then, the non-real spectra of A and A∗ consist only of
finitely many points, i,e., (I − ζA)−1 and (I − ζA∗)−1 are defined for all but
finitely many ζ ∈ T.
Denote the system operator of Σ as T. By Theorem 3.2, the system
operator T has a Julia operator of the form
(
T DT ∗
D∗T −L∗
)
:
(X ⊕ U
DT ∗
)
→
(X ⊕ U
DT
)
, (4.2)
where DT ∗ and DT are Hilbert spaces, DT ∗D∗T ∗ = I − TT ∗ and DTD∗T =
I − T ∗T such that DT and DT ∗ have zero kernels. Then, one can form the
Julia embedding Σ˜ of the system Σ; recall the embeddings introduced after
Lemma 3.3. That is, the corresponding system operator TΣ˜ of the embedding
Σ˜ is a Julia operator of T, and it is of the form
TΣ˜ =
⎛
⎝
A
(
B DT ∗,1
)
(
C
D∗T,1
) (
D DT ∗,2
D∗T,2 −L∗
)
⎞
⎠ :
⎛
⎝
X( U
DT ∗
)
⎞
⎠ →
⎛
⎝
X( U
DT
)
⎞
⎠ ,
where DT ∗ =
(
DT ∗,1
DT ∗,2
)
and DT =
(
DT,1
DT,2
)
. The system Σ˜ is conservative,
and since Σ is minimal, the inclusions ranB ⊂ ran (B DT ∗,1
)
and ranC∗ ⊂
ran
(
C∗ DT,1
)
imply that Σ˜ is also minimal. The transfer function of the Julia
embedding is
Θ(z)=
(
D + zC(I − zA)−1B DT ∗,2 + zC(I − zA)−1DT ∗,1
D∗T,2 + zD
∗
T,1(I − zA)−1B −L∗ + zD∗T,1(I − zA)−1DT ∗,1
)
=
(
θ(z) ψ(z)
ϕ(z) χ(z)
)
.
42 Page 26 of 29 L. Lilleberg IEOT
The funtion Θ is a dilation of θ, and since it is the transfer function of the
minimal conservative system, it is a generalized Schur function with the index
κ1 [2, Theorem 2.1.2]. Since Σ˜ is conservative, it follows from Lemma 3.3 that
I − Θ(z)Θ∗(w) = (1 − zw)
(
C
D∗T,1
)
(I − zA)−1(I − wA∗)−1 (C∗ DT,1
)
I − Θ∗(w)Θ(z) = (1 − zw)
(
B∗
D∗T ∗,1
)
(I − wA∗)−1(I − zA)−1 (B DT ∗,1
)
,
and therefore that I − Θ(ζ)Θ∗(ζ) = 0 and I − Θ∗(ζ)Θ(ζ) = 0 for all but
finitely many ζ ∈ T. That is, the radial limit values of Θ are unitary for all
but finitely many ζ ∈ T, and therefore Θ is a unitary dilation of θ.
Assume then that θ ∈ RSκ(U), and let Σ̂ = (Â, B̂, Ĉ,D; X̂ ,U ;κ) be a
κ-admissible passive self-adjoint realization of θ. Since Σ̂ is a κ-admissible
and passive, the space X̂ s is a regular subspace with the negative index κ,
and (X̂ s)⊥ is a Hilbert space [32, Proposition 2.7]. This implies that the
system operator TΣ̂ can be represented as in (3.19). It then easily follows
that the restriction Σ = (A,B,C,D; X̂ s,U ;κ) of Σ̂ to the simple subspace
X s is a minimal passive self-adjoint κ-admissible realization of θ. Denote the
system operator of Σ as T. By Theorem 3.2 there exists a Julia operator
UT of T of the form (4.2) where DT ∗ and DT are Hilbert spaces. Since T
is self-adjoint, it follows from [24, Theorem 5 and pp. 88] that UT can be
chosen such that DT ∗ = DT := U ′ and UT ∈ L (X ⊕ U ⊕ U ′) is self-adjoint;
cf. (3.13). Now construct a Julia embedding Σ˜ of Σ similarly as above, by
using UT , and denote the transfer function of Σ˜ as Θ. Then Θ is a dilation of
θ and Σ˜ is minimal self-adjoint conservative. Therefore Θ ∈ Sκ ∈ (U ⊕ U ′),
and by Proposition 3.1 also Θ ∈ Nκ (U ⊕ U ′). Since A = A∗ is a self-adjoint
operator in a Pontryagin space with the negative index κ, a similar argument
as above shows that the values of Θ are unitary for all but finitely many
ζ ∈ T. Therefore also Θ ∈ Uκ (U ⊕ U ′) , and the proof is complete. 
Acknowledgements
I wish to thank the Vilho, Yrjo¨ and Kalle Va¨isa¨la¨ Foundation of the Finnish
Academy of Science and Letters for a received grant. I also wish to thank
Seppo Hassi for helpful discussions while preparing this paper.
Funding Open access funding provided by University of Vaasa (UVA).
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Lassi Lilleberg(B)
Department of Mathematics and Statistics
University of Vaasa
P.O. Box 70065101 Vaasa
Finland
e-mail: lassi.lilleberg@uva.fi
Received: May 28, 2020.
Revised: September 10, 2020.