Generalized Schur–Nevanlinna functions and their realizations

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 generalize 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 functions of self-adjoint systems with Kreĭn spaces as state spaces. A criterion 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.

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 T Σ 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 and Y are Pontryagin spaces with the same negative indices, it will be proved that for θ ∈ S κ (U, Y), the strong radial limit value θ(ζ) := lim r→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 S κ1 (U) ∩ N κ2 (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 Arlinskiȋ, Hassi and de Snoo in [4] and by Arlinskiȋ 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 ∈ S κ1 (U) ∩ N κ2 (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 ∈ S κ (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 ∈ S κ (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 see, for an instance, [8,14]. Arlinskiȋ 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 S κ1 (U)∩N κ2 (U), where U is a Pontryagin space. be chosen such that it is a generalized Nevanlinna function with the index κ; see Theorem 4.1.

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 S κ (U, Y) and N κ (U) is somewhat more complicated than in the better known Hilbert space case. For instance, when U and Y are Hilbert spaces, Kreȋn-Langer factorizations shows that a function in S κ (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 ∈ S κ (U, Y) may has any countable number of poles, see Corollary 2.3 and Example 2.7 below. However, some properties of the function θ in S κ (U, Y) or N κ (U) can be analyzed by using a suitable transformation In what follows, all notions of continuity and convergence are understood 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.
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 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 3) whose domain ρ(θ P ) consists of all the points z ∈ ρ(θ) such that θ 22 (z) is invertible. A calculation shows that 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 θ P exists, it is meromorphic on D, and if θ P is meromorphic on a set D P , then so is θ.
(iii) The identities hold whenever the corresponding functions are defined.
Proof. (i) Suppose θ P exists, i.e. θ 22 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 θ P is meromorphic on D, it is now sufficient to show that θ −1 22 is meromorphic on D, since then all the entries in (2.3) are meromorphic. To this end, note that the values of θ 22 are operators between the spaces with the same finite dimension. Therefore, θ 22 (z) can be identified as a square matrix, and θ −1 22 (z) has a representation θ −1 22 (z) = cof(θ22(z)) det(θ22(z)) , where det(θ 22 (z)) and cof(θ 22 (z)) are, respectively, the determinant and the cofactor matrix of θ 22 (z). The function det(θ 22 ) is not identically zero since θ 22 (α) is invertible. Since θ 22 is meromorphic on D, so are the functions det(θ 22 (z)) and cof(θ 22 (z)). It follows now that θ −1 22 exists and it is meromorphic on D, and so is θ P .
If θ P is meromorphic on D P , by using the same argument as above, one can show that θ P −1 22 is meromorphic on D P , and then it follows from (2.4) that θ is meromorphic on D P .
For the proof of (ii), (iii) and (iv), see [  (v) From the part (iv) it follows that θ P exists. By (2.6), it holds for the Schur kernels K θ and K θP of the form (1.1), whenever the functions are defined. Let Ω be a region such that θ and θ P both are holomorphic on Ω. Then, the values of θ 22 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 K θP restricted to Ω has the same number of negative squares than K θ restricted to Ω. Now an application of [2, Theorem 1.1.4] shows that unrestricted K θP and K θ have the same number of negative squares. Therefore, if θ ∈ S κ (U, Y) and θ −1 22 (0) exists, θ P is holomorphic at the origin and K θP has exactly κ negative squares, so θ P ∈ S κ (U , Y ). Conversely if θ P ∈ S κ (U , Y ), the function θ P and then also θ are holomorphic at the origin, K θ has exactly κ negative squares, so θ ∈ S κ (U, Y).
(vi) It follows from the assumption U = Y that U = Y , and the assumption that θ 22 is invertible for some point guarantees that θ P exists. Moreover, the function θ P is also symmetric by part (ii). From these symmetry conditions it follows that σ = τ in (2.2) and Ψ(z) = Φ(z) in (2.5). By (2.8), it then holds for the Nevanlinna kernels N θ and N θP of the form (1.3), whenever the functions are defined. Now the same argument as used in the proof of part (iii) shows that N θ and N θP have the same number of negative squares. Moreover, part (i) shows that if either θ or θ P 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 −1 22 (0) exists in parts (v) and (vii) of Proposition 2.1 is technical; it is needed because the generalized Schur function must be analytic at the origin. If θ ∈ S κ (U, Y) and θ 22 (0) is not invertible, it follows from part (iv) that θ 22 (α) is invertible for some α ∈ D. The conclusions of the part (v) of Proposition 2.1 then hold if θ P (z) is replaced by The same is true in the part (vii) of Proposition 2.1, if α ∈ (−1, 1), since then η(z) = η(z) and θ P (η(z)) = (θ P (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 .  holds as stated, if one replaces the spaces −C and C, respectively, by −C n and C n , and changes the assumptions "θ 1 not identically zero" and "θ 2 (0) = 0", respectively, by "det(θ 1 ) not identically zero" and "det(θ 2 (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 S κ (C n ) such that their values are not invertible anywhere on D.
When U and Y are Hilbert spaces, the class S κ (U, Y) has characterizations which do not involve the Schur kernel (1.1). For a proof of the following lemma, combine [22,Proposition 7.11]   When U and Y are finite dimensional anti-Hilbert spaces with the same negative index, i.e. U = Y = −C n , the results of Lemma 2.5 have counterparts; 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 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 S κ (−C n ) if and only if θ has exactly κ zeros in D, counting multiplicities, and where γ(θ(z)) is taken with respect to the usual norm of L(C n ). Proof. The values of θ can be considered as the operators in L(−C n ). Then, the Potapov-Ginzburg transformation θ P of θ is θ −1 . Suppose θ ∈ S κ (−C n ). 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 ∈ S κ (C n ) by Proposition 2.1. It follows from Lemma 2.5 that θ −1 has exactly κ poles in D, counting multiplicities, and it holds 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 ∈ S κ (C n ), and then by Proposition 2.1 that θ ∈ S κ (−C n ). 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 S κ (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 ∈ S κ (U, Y) has infinite number of zeros and poles, as Example 2.7 below shows. However, a function θ ∈ S κ (U, Y) still has some properties similar to (2.11) or (2.12). Indeed, the radial limit values of θ ∈ S κ (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 b 1 and b 2 be scalar infinite Blaschke products such that . It easily follows from Lemma 2.5 that θ P ∈ S(C 2 ), 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 index.
(i) If θ ∈ S κ (U, Y), then strong radial limit values lim r→1 − θ(rζ) exist for a.e. ζ ∈ T, and the limit values are contractive with respect to the indefinite inner products of U and Y. Assume then that the negative index of U and Y is not zero. By Proposition 2.1, the Potapov-Ginzburg transform θ P of θ exists. It can be assumed that θ 22 (0) invertible; if not, one only need to consider θ P (η(z)), where η is as in Remark 2.2. By Proposition 2.1, θ P ∈ S κ (U , Y ), and since U and Y are Hilbert spaces, θ P is meromorphic on D, has strong contractive radial limit values almost everywhere on T, and the same holds for the entries θ P11 , θ P12 , θ P21 and θ P22 in (2.3). By Lemma 2.5, θ P has exactly κ poles in D, counting multiplicities, and therefore θ P22 has no more than κ poles in D. It now follows again from Lemma 2.
Then, θ P22 has the Kreȋn-Langer factorization of the form . The values of θ P22 are operators between the spaces with same finite dimension, and they can be identified with square matrices. Moreover, the values of θ P22 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 θ 0 are invertible on ρ(θ) \ Ξ. In particular, the function det(θ 0 ), is not identically zero. These facts combined with (2.13) , (2.14) where cof means the cofactor matrix. The function θ P22 is meromorphic on D and has strong contractive radial limit values a.e. on T, so clearly cof(θ P22 ) 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 θ 0 are contractive everywhere on D, and therefore det(θ 0 ) is bounded holomorphic function in D. This implies that radial limit values of det(θ 0 ) exist, and since det(θ 0 ) is not identically zero, the radial limit values also differ from zero a.e. on T. It now follows from (2.14) that θ −1 P22 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 θ P 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 −1 22 = σ −1 θ P 22 τ and the strong radial limit values of θ P 22 exist a.e. on T, the strong radial limit values of θ −1 22 also exist a.e. on T. Especially, the strong radial limit values of θ 22 are invertible a.e. on T. An easy calculation then shows that the strong radial 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 ∈ S κ (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 ∈ S κ (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 U κ (U, Y) is defined to be the subclass of the generalized bi-J -inner functions in S κ (U, Y). The class U κ (U, U) is written as U κ (U). For a symmetric function, it is evident that if the radial values are isometric or co-isometric a.e., they are also unitary.

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 = (T Σ ; 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 = (T Σ ; 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 T Σ . 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 θ Σ * (z) = θ Σ # (z) holds for the transfer function θ Σ * of the dual system Σ * .
The following subspaces 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 whenever z ∈ Ω, then also 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 Pontryagin spaces such that U and Y have the same negative index, the transfer function of the passive system Σ = (T Σ ; X , U; κ) is a generalized Schur function, with the index not larger that the negative index of the state space [32,Proposition 2.4]. Conversely, every θ ∈ S κ (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) [ 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 Σ 1 and Σ 2 above are said to be weakly similar if D 1 = D 2 and there exists an injective closed densely defined possible unbounded linear operator Z : X 1 → X 2 with the dense range such that For a generalized Nevanlinna function θ ∈ N κ (U) in the special case where U is a Hilbert space, the realization of θ usually means a representation of the form such that X is a Pontryagin space, Γ ∈ L(U, X ), H is a self-adjoint linear relation in X and z 0 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 ∈ N κ (U), and it holds In that case, the realization is unique up to unitary equivalence. In general, a function θ ∈ N κ (U) is not necessary holomorphic at the origin, and therefore it cannot be realized in the form (1.6). However, a selfadjoint system with a Pontryagin state space always induces some generalized Nevanlinna function. 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) −1 B 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 Therefore, it follows from [2, Lemma 1.1.1'] that the number of negative eigenvalues of the Gram matrix of the form . It now follows that the Nevanlinna kernel N θ 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 Kreȋn space operator theory will be needed to prove this.
Let X be a Kreȋn 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 Hx j , 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.
and linear operators D A ∈ L(D A , X 1 ) and D A * ∈ L(D A * , X 2 ) with zero kernels and a linear operator L ∈ L(D A , D A * ) such that it holds The operator U A in Theorem 3.2 is called as a Julia operator of A, the operators D A and D A * are called, respectively, defect operators of A and A * , and the spaces D A and D A * 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 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.  are defect operators of T and T * , respectively, then the identities 14) (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 where U and Y are Hilbert spaces. The transfer function of the embedded system is where θ Σ is the transfer function of the original system.
where Ω is a sufficiently small symmetric neighbourhood of the origin . Moreover, if U is a Hilbert space, then κ 1 = κ 2 .
Proof. It follows from Proposition 3.1 that θ ∈ N κ2 (U). Moreover, since Σ is passive, θ is also a generalized Schur function with the negative index κ 1 , 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 κ 1 ≤ 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 selfadjoint system, A = A * and C = B * . Then the same argument as in the proof of Proposition 3.1 shows that κ 2 = κ 2 . 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 D T * , and it follows that the kernel (1 − zw) −1 (ψ(z)ψ * (w)) has no negative square; for details, see the proof of [ T with the properties described therein. Since T = T * is contractive, the domain D T of D T is a Hilbert space. Moreover, (3.12) shows that it holds That is, the operator T D T is co-isometric. Therefore the system Σ with the system operator is a co-isometric embedding of Σ. The transfer function of Σ is given by 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 Since N θ has κ 2 negative squares, so has K θ , and therefore θ is a generalized Schur function with the index κ 2 . 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.
The identity above yields that N θ and Nθ have the same number of negative squares, and thereforeθ ∈ N κ1 (U). Since θ is holomorphic at the origin, it has the Neumann series of the form θ(z) = Ω is a sufficiently small symmetric neighbourhood of the origin. Therefore lim z→0 z −1 (θ(z) − θ(0)) = θ 1 , and also lim z→∞ This implies y lim y→∞ θ (iy)f, f U < ∞ for every f ∈ U. Therefore,θ has the realization of the form (3.9) which reduces toθ where B ∈ L(U, X ) and A is a self-adjoint operator in a Pontryagin space X with the negative index κ 2 [28], [33, pp. 348-349]. But then θ can be realized as where D = θ(0) = θ * (0). That is, Σ = ( A, B, B * , D; X , U; κ 2 ) 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 − z A) −1 B : z ∈ Ω} := S, where Ω is some sufficiently small symmetric neighbourhood of the origin, is κ 2 , the negative index of X . Then, the closure X c of S must be a regular subspace of X . Therefore, X = X c ⊕ X c ⊥ , and X c ⊥ is a Hilbert subspace of X . Since Σ is self-adjoint system, the spaces X c , X o and X s coincide. These facts and (3.1) imply that the system operator T of Σ can be represented as 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 ∈ S κ1 (U) ∩ N κ2 (U), where U is Pontryagin space, it follows from Proposition 3.4 that a self-adjoint minimal realization Σ = (T Σ ; X , U; κ) of θ can be passive only if κ 1 ≤ κ 2 = κ, and in the case where U is a Hilbert space, only if κ 1 = κ 2 = κ.
The conditions of Theorem 3.5 that U is a Hilbert space there and θ ∈ N κ2 (U) can be relaxed slightly; with a cost of weakened conclusions. realization of θ. Therefore Σ 1 and Σ * 1 are unitarily similar, that is, there exists a unitary mapping J : X 1 → X 1 such that, see (3.7), The letter J is used, because the operator J is now also self-adjoint in X 1 . Indeed, let N ∈ N 0 . Easy calculations show that it holds (3.21) and similarly Since Σ 1 is simple, it follows from (3.21) and (3.22) that J and J * coincide on a dense lineal of X 1 , and then by continuity, everywhere. That is, J is unitary and self-adjoint. Now introduce the inner product space X , which coincides with X 1 as a vector space but which is endowed with the inner product x, y X = Jx, y X1 . Then X is a Kreȋn space. Moreover, it holds This implies that A is self-adjoint in the Kreȋn 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 = (T Σ ; X , U) with the Kreȋn space X was constructed from a simple conservative realization Σ 1 = (T Σ1 ; X 1 , U; κ). If X is a Pontryagin space with the negative index κ , it follows from Proposition 3.1 that the transfer function θ ∈ S κ (U) belongs also in the class N κ (U), where κ ≤ κ . One might conjecture that this happens for every θ ∈ S κ (U) ∩ N κ (U). However, Theorem 3.8 below shows that this is not true; it happens only when θ ∈ U κ (U) ∩ N κ (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. properties (3.20). Then, since 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 Σ 1 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 for all but finitely many ζ ∈ T, which shows that θ = θ # ∈ U κ1 (U). Choose some fundamental decomposition of U, and consider the Potapov-Ginzburg transformation θ P as in (2. ] that all simple conservative realizations of θ are minimal. Therefore Σ 1 is minimal, which implies that Σ is also minimal, since the norms of spaces X 1 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 ∈ N κ2 (U). ⇒: Let θ ∈ U κ1 (U) ∩ N κ2 (U). By using the Potapov-Ginzburg transformation similarly as above, it can be deduced that Σ 1 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 Then the matrix of the form where Ω is a sufficiently small symmetric neighbourhood of the origin, is a Gram matrix. Since θ ∈ N κ2 (U), the kernel N θ has κ 2 negative squares. These facts combined with [2, Lemma 1.1. 1'] imply that there exists a finite sequence contains a κ 2 -dimensional anti-Hilbert subspace of X . Therefore, ind − X is at least κ 2 . Suppose that ind − X > κ 2 . Then there exists a finite sequence {x i } κ i=1 ⊂ X of linearly independent negative vectors such that κ > κ 2 . Since Σ is minimal, span{ran (I − 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 ∈ S κ (U, Y) is unitarily similar with the system Σ 1 , and any simple conservative realization is unitarily similar with Σ 2 .
Suppose next the simple conservative realization of the symmetric function θ ∈ S κ1 (U) in Theorem 3.8 is chosen to be the canonical unitary realization. Then it can be derived from [3,Theorem 3.6] that the self-adjoint unitary similarity J is of the form J = J D(θ) , where In addition, if also θ ∈ U κ1 (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 Σ 1 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 (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) K θ (w, z) = N θ (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).
(iii) ⇒ (i) Since K θ (w, z) is nonnegative, θ ∈ S(U), and the canonical unitary realization Σ 2 defined by the operators in (3.27) is simple conservative Therefore, all the functions in the space (3.25) are of the form such that α j ∈ C, u j ∈ U and w j ∈ 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 = J D(θ) between Σ 2 and Σ * 2 , where J where is defined by (3.28), is identity. That is, Σ 2 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 X 1 . By considering minimal passive realizations instead of simple conservative realizations, one can obtain a similar type of characterization when θ ∈ S κ1 (U) ∩ N κ2 (U).
Denote E X (x) = x, x X for the vector x in an inner product space X . For θ ∈ S κ (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 negative index of the state space of Σ is κ.
for any N ∈ N 0 and {u n } N n=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 [ A * n C * u n for the vectors of the form N n=0 A n Bu n . 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 That is, Z is an everywhere defined weak similarity. Moreover, it holds 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 ∈ S κ (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. where n ∈ N, {f j } n j=1 ⊂ U, and {w j } n j=1 ⊂ Ω for some sufficiently small symmetric neighbourhood Ω of the origin, for the kernel N θ . Moreover, since Σ is minimal, the space span{ran (I − zA) −1 B : z ∈ Ω} := S is dense in X . Let y ∈ D Z such that y, V * Z x DZ = 0 for all x ∈ S. Then, y, where D T * = 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 ∈ RS κ (U), and let Σ = ( A, B, C, D; X , U; κ) be a κ-admissible passive self-adjoint realization of θ. Since Σ is a κ-admissible and passive, the space X s is a regular subspace with the negative index κ, and ( X s ) ⊥ is a Hilbert space [32,Proposition 2.7]. This implies that the system operator T Σ can be represented as in (3.19). It then easily follows that the restriction Σ = (A, B, C, D; X s , U; κ) of Σ 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 U T of T of the form (4.2) where D T * and D T are Hilbert spaces. Since T is self-adjoint, it follows from [24, Theorem 5 and pp. 88] that U T can be chosen such that D T * = D T := U and U T ∈ L (X ⊕ U ⊕ U ) is self-adjoint; cf. (3.13). Now construct a Julia embedding Σ of Σ similarly as above, by using U T , 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 ∈ S κ ∈ (U ⊕ U ), and by Proposition 3.1 also Θ ∈ N κ (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 ∈ U κ (U ⊕ U ) , and the proof is complete.