Linear Passive Stationary Scattering Systems with Pontryagin State Spaces

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1 mn header will be provided by the publisher Linear Passive Stationary Scattering Systems with Pontryagin State Spaces D. Z. Arov 1, J. Rovnyak 2, and S. M. Saprikin 1 1 Department of Physics and Mathematics, Division of Mathematical Analysis, South-Ukrainian Pedagogical University, Staroportofrankovskaya 26, Odessa 652, Ukraine 2 Department of Mathematics, University of Virginia, P. O. Box 4137, Charlottesville, VA , U.S.A. Received xxx, revised xxx, accepted xxx Published online xxx Key words Scattering system, passive, conservative, transfer function, Krein-Langer factorization, Pontryagin space, minimal system, dilation, Julia operator. MSC (2) Primary: 47A48; Secondary 47A45, 47A2, 46C2, 47B5, 47N7, 93B28 Passive scattering systems having Pontryagin state spaces and their minimal conservative dilations are investigated. The transfer functions of passive scattering systems are generalized Schur functions. In the case of a simple conservative system, the right and left Kreĭn-Langer factorizations of the transfer function correspond to natural cascade syntheses of systems. A generalization of Sz.-Nagy and Foias criteria for a cascade synthesis of two simple conservative systems to be simple is obtained for systems with Pontryagin state spaces. It is shown that the state space of a simple passive system admits certain unique fundamental decompositions, which give rise to a notion of stability and a characterization of simple conservative systems whose transfer functions have unitary boundary values a.e. on the unit circle. Contents 1 Introduction 1 2 Pontryagin spaces and linear operators 3 3 Scattering systems with Pontryagin state spaces 4 4 A brief survey of known results 6 5 Schur complements and associated systems 9 6 Conservative dilations and systems with minimal losses 1 7 Cascade synthesis and Kreĭn-Langer factorizations 15 8 Application to simple conservative systems 2 9 The classes P κ and C κ 23 1 Models for C κ 26 References 29 1 Introduction In this paper we study linear stationary scattering systems x(n + 1) = Ax(n) + Bu(n), y(n) = Cx(n) + Du(n), aspect@farlep.net Corresponding author: rovnyak@virginia.edu sergey saprikin@ukr.net

2 2 Arov, Rovnyak, and Saprikin: Linear Passive Stationary Scattering Systems n =, 1, 2,..., whose states x(n) belong to a Pontryagin space X and whose inputs u(n) and outputs y(n) belong to Hilbert spaces U and Y. The transfer function of such a system is defined by Θ Σ (z) = D + zc(i za) 1 B whenever the inverse exists. The case in which the state space X is a Hilbert space is classical. In several recent papers, the classical theory, as it appears, for example, in [3], has been extended to Pontryagin state spaces. Basic properties of passive scattering systems with Pontryagin state space, including the notions of dilation and embedding of systems, are discussed in [4] and [23]. Results on Darlington representations of systems having Pontryagin state space are derived in the appendix of the English translation in [3]. In a related work, S. A. Kuzhel [21] generalizes the abstract Lax-Phillips conservative scattering scheme to Pontryagin spaces. The purpose of this paper is to continue the study of passive scattering systems having Pontryagin state spaces by investigating further properties of dilations and embeddings, minimal systems, cascade representations and invariant subspaces, and a generalization of the notion of stability. We summarize background in Sections 2 and 3. This includes properties of Pontryagin spaces, contraction operators, and Julia operators that are used in the paper. Systems (3.1) (3.3) are viewed as colligations Σ = (A, B, C, D; X, U, Y; κ), and they are classified as passive scattering systems or conservative scattering systems according as the system operator A B V Σ = C D is contractive or unitary. The definitions of controllable, observable, and simple systems are parallel to the Hilbert space case. Among several equivalent definitions of a minimal system for Hilbert state spaces, we choose one that is appropriate for Pontryagin state spaces. In Section 4 we summarize known results, mainly from [23], on passive and conservative systems and their dilations and embeddings. An important condition on indices is introduced here. The transfer function Θ Σ (z) of a passive system Σ = (A, B, C, D; X, U, Y; κ) belongs to some generalized Schur class S κ (U, Y) with κ κ. Many theorems use the hypothesis that Θ Σ (z) S κ (U, Y), that is, κ = κ. Sufficient conditions for the index condition to hold are identified. Section 5 constructs systems associated with the Schur complements D CA 1 B and A BD 1 C in the system operator. Such constructions, of course, require that the operator inverses exist. The form of a minimal conservative dilation Σ of a passive system Σ is determined in Section 6. If Σ = (A, B, C, D; X, U, Y; κ), there are many ways in which one can form a unitary operator VΣ E U Σ = : F G [ X U E ] [ X Y F ], and each such unitary operator induces a conservative dilation Σ. It is shown that Σ is a minimal conservative dilation if and only if U Σ is a Julia operator. Passive systems which admit simple conservative dilations are said to have minimal losses. The results of Section 6 determine the form of a minimal conservative dilation of a passive system, namely, they are induced by Julia operators for the system operator. In Section 7 we show that the right and left Kreĭn-Langer factorizations of the transfer function of a simple conservative system Σ correspond to natural cascade syntheses of the system. These results are applied to simple passive systems Σ. In this case, the semi-definite invariant subspaces of the main operator A, whose existence is assured by the classical theory of contraction operators on a Pontryagin space, are regular; in fact, they are unique and determine natural fundamental decompositions X = X + X and X = X + X of the state space with AX + X + and AX X. A key step in Section 7 is to show that the cascade synthesis of two particular simple conservative systems is simple. In general, such a cascade synthesis is not always simple, even with Hilbert state spaces. Section 8 is concerned with the problem to determine when the cascade synthesis of two simple conservative systems having Pontryagin state spaces is simple. We generalize a well-known analytical condition, which is related to the notion of a regular factorization of an operator-valued function, from the case of Hilbert state spaces to Pontryagin state spaces. In Section 9 we examine the notion of stability. In the case of Hilbert state spaces, we call a system stable if it is bi-stable in the sense that both of the semigroups A n and A n are stable, that is, they tend to zero strongly

3 mn header will be provided by the publisher 3 as n. This notion does not extend directly to Pontryagin state spaces due to the existence of eigenvalues of the main operator of modulus greater than one. In place of stable systems, we introduce classes P κ and C κ of passive and conservative systems that are partially stable in a sense that depends on the fundamental decompositions X = X + X and X = X + X of the state space introduced in Section 7; in the definitions of these classes we require, roughly, that the parts of the system corresponding to the positive subspaces are stable in the usual Hilbert space sense. It is shown, for example, that a simple conservative system belongs to C κ if and only if its transfer function has unitary boundary values a.e. on the unit circle. The concluding Section 1 shows how the class C κ can be described in terms of canonical models. 2 Pontryagin spaces and linear operators Mainly only elementary notions concerning Pontryagin spaces and operators which act on them are used here. For example, see [5, 8, 16, 18]. We recall a few basic ideas in order to fix terminology and notation. All Hilbert spaces are assumed to be separable. A Pontryagin space is a complex vector space X together with a linear and symmetric inner product, =, X which admits a representation X = X + X, where (X ±, ±, X ) are Hilbert spaces and dim X <. We call such a representation a fundamental decomposition. A fundamental decomposition is in general not unique, but it determines a unique strong topology. The dimensions ind ± X = dim X ± are independent of the choice of fundamental decomposition and called the positive and negative indices of X. The terms regular subspace and Hilbert subspace are used as in standard sources; the orthogonal projection operator whose range is a regular subspace M is written P M. We shall use the following result from [23, Lemma 3.1]. Lemma 2.1 Let M and N be regular subspaces of a Pontryagin space X which has negative index κ. If M and N have negative index κ, then P M N and P N M are regular subspaces of X having negative index κ. We write L(X ) and L(X, Y) for the spaces of continuous linear operators on a Pontryagin space X into itself and into a Pontryagin space Y, respectively. An operator is invertible if it has an everywhere defined and continuous inverse. The adjoint of an operator A L(X, Y) is the operator A L(Y, X ) such that Ax, y Y = x, A y X for all x in X and y in Y. Classes of selfadjoint, isometric, and unitary operators are defined as for Hilbert spaces. We call A L(X ) nonnegative and write A if Ax, x X for every x X. An operator T L(X, Y) is a contraction if I T T. If T L(X, Y) is a contraction and ind X = ind Y, then T L(Y, X ) is also a contraction [16, Corollary 2.5]. Let X and Y be Pontryagin spaces such that ind X = ind Y. If T L(X, Y) is a contraction, there exist Hilbert spaces E, F and operators E, F, G acting on appropriate spaces such that the operator T E X Y U = : F G E F is unitary. In this case, the conditions ker E = {} and ker F = {} are equivalent, and when these conditions are satisfied we call U a Julia operator for T. A Julia operator always exists and is essentially unique [16, Theorems 2.3 and 2.6]. Here and below, various notions of essential uniqueness appear, and we leave it to the reader to construct definitions analogous to the Hilbert space case. A subspace N of a Pontryagin space X is called nonnegative (nonpositive) if x, x X is nonnegative (nonpositive) for all x in N. We say that N is semi-definite if it is either nonnegative or nonpositive. A semi-definite subspace may contain nonzero vectors x such that x, x X =. Notions of maximal nonnegative and maximal nonpositive subspaces are defined in the usual way relative to inclusion of subspaces. If X has negative index κ, a nonpositive subspace N is maximal nonpositive if and only if dim N = κ. Theorem 2.2 Let X be a Pontryagin space of negative index κ, and let T L(X ) be a contraction operator. (i) The part of the spectrum of T in {z : z > 1} consists of isolated eigenvalues which have finite-dimensional root subspaces. (ii) There exists a maximal nonpositive subspace N of X which is invariant under T and which contains all of the root subspaces for the eigenvalues of T in {z : z > 1}.

4 4 Arov, Rovnyak, and Saprikin: Linear Passive Stationary Scattering Systems (iii) There exists a maximal nonnegative subspace P of X which is invariant under T and which is orthogonal to all of the root subspaces for the eigenvalues of T in {z : z > 1}. In particular, the part of the spectrum of T in {z : z > 1} consists of at most κ eigenvalues. See [5, Chapter 3], [15, Section 2], and [18, Section 11] for additional results and historical notes on theorems of this type. Concerning the statement (i) in Theorem 2.2, see also Kuzhel [21, Section 4, Assertion 6], where unitary dilations are used. Sketch of the proof of Theorem 2.2. (i) We follow [18, Lemma 11.1]. Choose a fundamental decomposition X = X + X of X. Set H + = X +, let H be the antispace of X, and write T11 T T = 12 H+ H+ :. T 21 T 22 H H Since T x, T x X x, x X for all x in X, for any x + X +, T 11 x + 2 T 21 x + 2 x + 2. Thus T 11T 11 I + T 21T 21 = R 2, R = [ I + T 21T 21 ] 1/2, and so T11 = SR, where S is a contraction operator on H +. Since T 21T 21 has finite rank, by the spectral theorem R = I + R where R has finite rank. Hence T differs from the contraction operator S on H + H by an operator of finite rank. Therefore by [17, Lemma 5.2], the part of the spectrum of T outside the unit circle consists of isolated eigenvalues having finite-dimensional root subspaces. (ii) See [18, Theorem 11.2]. (iii) The adjoint operator T is also a contraction, so by (ii) it has a maximal nonpositive invariant subspace M. Then P = M = {x: x, y X =, y M} is a maximal nonnegative invariant subspace for T having the required properties (see [16, Theorem 1.6]). 3 Scattering systems with Pontryagin state spaces In this paper we shall study conservative and passive scattering systems which have Pontryagin state spaces. The definitions of these notions are adapted from the Hilbert state space case. Consider a system { x(n + 1) = Ax(n) + Bu(n), (3.1) y(n) = Cx(n) + Du(n), n = 1, 2,..., whose states x(n) belong to a Pontryagin space X of negative index κ and whose inputs u(n) and outputs y(n) belong to Hilbert spaces U and Y. We call X the state space and U and Y the input and output spaces. The system (3.1) is equivalently viewed as an operator node Σ = (A, B, C, D; X, U, Y; κ). (3.2) The system operator V Σ L(X U, X Y) is defined by A B X X V Σ = :. (3.3) C D U Y We call A, B, C the main, input, and output operators for the system. The transfer function of the system is defined by Θ Σ (z) = D + zc(i za) 1 B whenever the inverse exists. The adjoint system is Σ = (A, C, B, D ; X, Y, U; κ). Thus V Σ Θ Σ (z) = Θ Σ ( z). = V Σ, and

5 mn header will be provided by the publisher 5 A system (3.1) is said to be a passive scattering system if x(n + 1), x(n + 1) X x(n), x(n) X u(n) 2 U y(n) 2 Y (3.4) for all initial states x() in X and all inputs u(n), n, in U. We call (3.1) a conservative scattering system if equality always holds in (3.4) and if the adjoint system Σ has the same property. In what follows, the term scattering will usually be omitted, and we shall speak more simply of passive systems and conservative systems because other types of systems will not be considered. A system Σ is passive (conservative) if and only if the system operator V Σ is a contraction (unitary) operator. If Σ is passive or conservative, the adjoint system Σ has the same property. For any system Σ = (A, B, C, D; X, U, Y; κ), set X c Σ = A n B U and X o Σ = A n C Y. Then Σ is said to be (i) controllable, (ii) observable, (iii) simple, or (iv) minimal according as (i) X = X c Σ, (ii) X = X o Σ, (iii) X = X c Σ X o Σ, or (iv) X = X c Σ and X = X o Σ. It is easy to see that X c Σ = X o Σ and X o Σ = X c Σ. Remark. We have defined a minimal system as a system that is controllable and observable. This definition differs from the usual definition of a minimal system with a Hilbert state space. In the case of a Hilbert state space, a system is often called minimal if it is not a nontrivial dilation (as defined below) of another system. These two definitions are equivalent in the Hilbert state space case (see [2, Proposition 3]), but for systems with Pontryagin state spaces they are not the same. However, a controllable and observable system cannot be a nontrivial dilation of another system, and in the special case of passive systems, which is of our main interest, the converse statement is also true (see Corollary 4.8). We call two systems Σ 1 = (A 1, B 1, C 1, D 1 ; X 1, U, Y; κ) and Σ 2 = (A 2, B 2, C 2, D 2 ; X 2, U, Y; κ) equivalent and write Σ 1 = Σ2 if A 2 = W A 1 W 1, B 2 = W B 1, C 2 = C 1 W 1, D 2 = D 1. for some unitary operator W L(X 1, X 2 ). We use two types of extensions of a system Σ to a larger system Σ. One does not change the state space and main operator and enlarges the input and output spaces. A system Σ = (A, B, C, D; X, U, Y; κ) is embedded in a system Σ = (A, B, C, D; X, Ũ, Ỹ; κ) if there exist Hilbert spaces Û and Ŷ such that Ũ = Û U and Ỹ = Y Ŷ, and VΣ E X U X Y V eσ = : (3.5) F G Û Ŷ for some operators E, F, G. Equivalently, A B E 1 X X V eσ = C D E 2 : U Y F 1 F 2 G Û Ŷ or [ A V eσ = C [ B ] A E1 B ] X X D = C E2 D : Û Y. (3.6) F 1 G F 2 U Ŷ In this case, Θ11 (z) Θ Θ eσ (z) = 12 (z), Θ Θ 21 (z) Θ 22 (z) Σ (z) = Θ 12 (z). (3.7)

6 6 Arov, Rovnyak, and Saprikin: Linear Passive Stationary Scattering Systems If V Σ is a contraction operator (that is, Σ is a passive system) and (3.5) is a Julia operator for V Σ (see Section 2), we say that the embedding is a Julia embedding. A second type of extension of a system Σ to a larger system Σ uses the same input and output spaces and enlarges the state space, but without increasing its negative index. We call Σ = (Ã, B, C, D; X, U, Y; κ) a dilation of Σ = (A, B, C, D; X, U, Y; κ) if X = D X D +, where D + and D are Hilbert spaces and ] [Ã B A V eσ = C D = C B : D The form of the system operator is equivalent to the relations ÃD + D +, Ã D D, CD+ = {}, B D = {}. D D X D + X D +. (3.8) U Y Using this form, we easily obtain Θ eσ (z) = Θ Σ (z). A dilation Σ of Σ is called conservative or simple according as Σ is a conservative or simple system. If Σ is a dilation of Σ, we also call Σ a restriction of Σ; we say that Σ a proper restriction of Σ if it is a restriction of Σ and Σ Σ. It is not hard to see that the main operator A L(X ) of a passive system Σ = (A, B, C, D; X, U, Y; κ) is a contraction. For since Y is a Hilbert space, from the inequality V Σ (x ), V Σ (x ) X Y x, x X U for any x in X, we obtain Ax, Ax X Ax, Ax X + Cx, Cx Y x, x X. Moreover, every contraction operator A on a Pontryagin space X may be viewed as the main operator of a conservative system by considering a Julia operator for A. We also remark that if a system Σ is embedded in any way in a system Σ, and if Σ is passive, then Σ is passive too. Analogously, every restriction Σ of a passive system Σ is passive. 4 A brief survey of known results The only systems that concern us are conservative or at least passive. As noted above, the main operator A of a passive system Σ = (A, B, C, D; X, U, Y; κ) is a contraction operator. In the classical case κ =, X is a Hilbert space and the inverse (I za) 1 exists at all points of the open unit disk D. In this case, the transfer function Θ Σ (z) is defined everywhere in D and, as is well known, it is an operator-valued Schur function, that is, it is holomorphic and bounded by one in D. When the state space X is a Pontryagin space, then according to Theorem 2.2, the inverse (I za) 1 exists for all but at most κ nonzero points in D. We shall usually consider the domain of the transfer function of a passive system Σ = (A, B, C, D; X, U, Y; κ) to be the open unit disk with at most κ nonzero points deleted. In some places (for example, see Section 5) we shall also consider systems Σ with invertible main operator A; in this case, the transfer function Θ Σ (z) is holomorphic at as well, and its restriction to a neighborhood of plays a role. Given Hilbert spaces U and Y, the generalized Schur class S κ (U, Y) is the set of functions S(z) with values in L(U, Y) which are holomorphic in a neighborhood of the origin and meromorphic in D such that the kernel K(w, z) = [I S(z)S(w) ]/(1 z w) has κ negative squares (κ =, 1, 2,... ). This means that for any finite set of points w 1,..., w n in the domain of holomorphy of S(z) in D, the selfadjoint operator on Y Y defined by the block Hermitian matrix [ K(w j, w k ) ] n has at most a κ-dimensional invariant subspace in which the j,k=1 operator has spectrum contained in (, ), and there exists at least one choice of points w 1,..., w n such that the operator has such an invariant subspace of dimension equal to κ. By the Kreĭn-Langer factorization (see Section 7), every S(z) in S κ (U, Y) has strong nontangential boundary values S(ζ) a.e. on the unit circle ζ = 1, and these boundary values are contractions a.e. Let U κ (U, Y) be the subclass of functions in S κ (U, Y) whose boundary values are unitary a.e. For κ =, we write more simply S(U, Y) for S (U, Y), and U(U, Y) for U (U, Y). From [1, Theorems 2.1.2, 2.3.1, and 2.1.3], we obtain:

7 mn header will be provided by the publisher 7 Theorem 4.1 (i) The transfer function Θ Σ (z) of a simple conservative system Σ = (A, B, C, D; X, U, Y; κ) belongs to the class S κ (U, Y). Conversely, every Θ(z) S κ (U, Y) has the form Θ(z) = Θ Σ (z) for some simple conservative system Σ = (A, B, C, D; X, U, Y; κ). (ii) Two simple conservative systems Σ 1 and Σ 2 are equivalent if and only if Θ Σ1 (z) = Θ Σ2 (z) in a neighborhood of the origin. It may happen that the transfer function of a passive system Σ = (A, B, C, D; X, U, Y; κ) is a generalized Schur function, but for a possibly smaller index than κ. The next result, which is taken from [23, Theorems 2.2 and 2.3], includes a sufficient condition for equality. Theorem 4.2 The transfer function Θ Σ (z) of a passive system Σ = (A, B, C, D; X, U, Y; κ) belongs to S κ (U, Y) for some κ κ. If Σ is minimal, then κ = κ. A passive system can be embedded in a conservative system in many ways. In particular, a Julia embedding always exists and is essentially unique. Theorem 4.3 Let Σ be a passive system which is embedded in a conservative system Σ. If Σ is simple, then Σ is simple. P r o o f. Let Σ = (A, B, C, D; X, U, Y; κ) and Σ = (A, B, C, D; X, Ũ, Ỹ; κ). Since B Ũ B U and C Ỹ C Y, X c e Σ = A n B Ũ A n B U = X c Σ, X o e Σ = A n C Ỹ A n C Y = X o Σ. Therefore Xe c Σ Xe o Σ XΣ c X Σ o, and hence if Σ is simple, so is Σ. It should be emphasized that if Σ = (A, B, C, D; X, U, Y; κ) is a passive system, then Θ Σ (z) S κ (U, Y) only under special conditions; two sufficient conditions for the inclusion Θ Σ (z) S κ (U, Y) are identified in Theorems 4.1 and 4.2. The property that Θ Σ (z) S κ (U, Y) is a hypothesis in the next two results. Theorem 4.4 is from [23, Lemma 2.5], and Theorem 4.5 is from [23, Proposition 2.6]. Theorem 4.4 Let Σ = (A, B, C, D; X, U, Y; κ) be a passive system whose transfer function Θ Σ (z) belongs to S κ (U, Y). Then each of the spaces X c Σ, X o Σ, and X c Σ X o Σ is regular and has negative index κ. Hence (X c Σ ), (X o Σ ), and (X c Σ X o Σ ) are Hilbert subspaces of X. Theorem 4.5 Let Σ = (A, B, C, D; X, U, Y; κ) be a conservative system whose transfer function Θ Σ (z) belongs to S κ (U, Y). Then (i) (X o Σ ) is the largest Hilbert subspace M of X which is invariant under A and contained in ker C, such that A M is isometric; (ii) (X c Σ ) is the largest Hilbert subspace K of X which is invariant under A and contained in ker B, such that A K is isometric; (iii) ( X c Σ X o Σ) is the largest Hilbert subspace L of X which is invariant under A and contained in ker C ker B, such that A L is unitary. We include the construction of a conservative dilation for any passive system from [23, Theorem 2.1] since this construction will be needed later. Theorem 4.6 Every passive system Σ has a conservative dilation Σ. P r o o f. Let Σ = (A, B, C, D; X, U, Y; κ). Since Σ is passive, we can choose Hilbert spaces E and F and a unitary operator of the form VΣ E : F G X U E X Y, (4.1) F

8 8 Arov, Rovnyak, and Saprikin: Linear Passive Stationary Scattering Systems or, equivalently, A B E 1 C D E 2 : F 1 F 2 G X U E X Y. (4.2) F This can be done in many ways; for example, we can choose (4.1) to be a Julia operator for V Σ. Set X = D X D +, where D = l 2 (E) is the Hilbert space of square summable sequences {..., e 2, e 1 } with entries in E, and D + = l 2 +(F) is the analogous space of sequences {f, f 1,... } with entries in F. Set where à = V E 1 P 1 A Q GP 1 Q F 1 V + C = [ E 2 P 1 C ], D = D,, P 1 : {..., e 3, e 2, e 1 } e 1, Q : f {f,,,... }, V : {..., e 3, e 2, e 1 } {..., e 2, e 1, }, V + : {f, f 1, f 2,... } {, f, f 1,... }, B = B, Q F 2 on arbitrary elements of the appropriate spaces. The system Σ = (Ã, B, C, D; X, U, Y; κ) has the required properties. (4.3) In the other direction, we can ask if a given system has restrictions with special properties. Theorem 4.7 and Corollary 4.8 also use the condition of equality κ = κ in Theorem 4.2. Theorem 4.7 Let Σ = (A, B, C, D; X, U, Y; κ) be a passive system whose transfer function Θ Σ (z) belongs to S κ (U, Y). Define subspaces X 1 = X o Σ (X c Σ), X 2 = P X o Σ X c Σ, X 3 = (X o Σ). Then X 1 and X 3 are Hilbert spaces, X = X 1 X 2 X 3, and the system operator for Σ has the form A 11 X 3 X 3 A B A V Σ = = 21 A 22 B 2 C D A 31 A 32 A 33 B 3 [ C1 C 2 ] : X 2 X 1 X 2 X 1. D U Y Theorem 4.7 is given in [23, Theorem 3.2]. Notice that by Theorem 4.4 and Lemma 2.1, the subspace X 2 in Theorem 4.7 is regular and has negative index κ. The conclusion of Theorem 4.7 is that Σ is a dilation of the system Σ res,1 = (P X2 A X 2, P X2 B, C X 2, D; X 2, U, Y; κ), which is called the first restriction of Σ. It is shown in [23, p. 24] that Σ res,1 is minimal, and thus we obtain: Corollary 4.8 Every passive system Σ = (A, B, C, D; X, U, Y; κ) with transfer function Θ Σ (z) in S κ (U, Y) is the dilation of a minimal passive system, namely, its first restriction.

9 mn header will be provided by the publisher 9 5 Schur complements and associated systems Consider a system Σ with system operator (3.3). Following standard matrix terminology, we call D CA 1 B the Schur complement of A in V Σ whenever A is invertible, and A BD 1 C the Schur complement of D in V Σ whenever D is invertible. In this section, we construct systems Σ and Σ associated with these operators. Theorem 5.1 Let Σ = (A, B, C, D; X, U, Y; κ) be a conservative system. Then A is invertible if and only if D is invertible. P r o o f. Since Σ is conservative, A A = I C C, AA = I BB, D D = I B B, DD = I CC. The invertibility of A is thus equivalent to the invertibility of both I C C and I BB, which by [1, (1.3.15)] is equivalent to the invertibility of both I CC and I B B. Hence A is invertible if and only if D is invertible. Theorem 5.2 Let Σ = (A, B, C, D; X, U, Y; κ) be a system. Assume that A is invertible and that dim X = n <. Define a system Σ = (A, B, C, D ; X, U, Y; κ ), κ = n κ, whose state space X is the antispace of X by setting { A = A 1, B = A 1 B, Then C = CA 1, D = D CA 1 B. (i) Σ is conservative if and only if Σ is conservative; (5.1) (ii) Θ Σ (z) is analytic at infinity, Θ Σ ( ) = D, and the identity Θ Σ (z) = Θ Σ (1/z) holds wherever these functions are defined, and in particular it holds in a neighborhood of the origin and in a neighborhood of infinity. P r o o f. (i) Assume that Σ is conservative. For the purpose of computing adjoints, first write (5.1) more correctly as { A = σa 1 σ 1, B = σa 1 B, C = CA 1 σ 1, D = D CA 1 B, (5.2) where σ : X X is the identity mapping. Then σ = σ 1. The identities A A + C C = I X, A B + C D =, B B + D D = I U, A A + B B = I X, A C + B D =, C C + D D = I Y, are checked by straightforward algebraic calculations. For example, A A + C C = [ σa 1 ( σ 1 ) ][ σa 1 σ 1] + [ σa 1 C ][ CA 1 σ 1] = σ [ A 1 A 1 A 1( I X A A ) A 1] σ 1 = I X, yielding the first identity. The other five identities follow similarly. Hence Σ is conservative. These steps are reversible. (ii) The invertibility of A implies that Θ Σ (z) is analytic at infinity. For any nonzero z the invertibility of I X za is equivalent to the invertibility of I X z 1 A. For any such point, and (ii) follows. Θ Σ (z) = D + zc (I X za ) 1 B = (D CA 1 B) + zca 1 (I X za 1 ) 1 ( A 1 B) = D + C [ A 1 + za 1 (I X za 1 ) 1 ( A 1 ) ] B = D + z 1 C(I X z 1 A) 1 B,

10 1 Arov, Rovnyak, and Saprikin: Linear Passive Stationary Scattering Systems Theorem 5.3 Let Σ = (A, B, C, D; X, U, Y; κ) be a system. Assume that D is invertible and that dim X = n <. Define a system Σ = (A, B, C, D ; X, Y, U; κ ), κ = n κ, whose state space X is the antispace of X by setting { A = A BD 1 C, B = BD 1, (5.3) C = D 1 C, D = D 1. Then (i) Σ is conservative if and only if Σ is conservative; (ii) X c Σ = X c Σ and X o Σ = X o Σ ; (iii) Θ Σ (z) = Θ Σ (z) 1 at all points where Θ Σ (z) and Θ Σ (z) are defined, and in particular this identity holds in a neighborhood of the origin. P r o o f. (i) This is verified by calculations as in the proof of Theorem 5.2. (ii) For any positive integer N, N (A ) n B Y = n= N ( A BD 1 C ) n N B U A n B U, n= and similarly N (A ) n C U N A n C Y. Therefore XΣ c X c Σ and X Σ o X o Σ. The reverse inclusions follow since the same construction applied to Σ yields the system Σ. This yields (ii). (iii) For all z for which both functions Θ Σ (z) and Θ Σ (z) are defined, Θ Σ (z)θ Σ (z) = [ D + zc (I za ) 1 B ][ D + zc(i za) 1 B ] n= = [ D 1 zd 1 C ( I z(a BD 1 C) ) 1 BD 1 ] [D + zc(i za) 1 B ] = I + zd 1 C(I za) 1 B zd 1 C ( I z(a BD 1 C) ) 1 BD 1 D z 2 D 1 C ( I z(a BD 1 C) ) 1 BD 1 C(I za) 1 B = I + zd 1 C ( I z(a BD 1 C) ) 1 = I. [(I za + zbd 1 C) (I za) zbd 1 C ] (I za) 1 B Similarly, Θ Σ (z)θ Σ (z) = I at all points where the two functions are defined. It should be noted that the hypothesis dim X < in Theorems 5.2 and 5.3 is only used to assure that the antispace of X has finite negative index and hence is a Pontryagin space (all of our definitions presume that the state space is a Pontryagin space). The algebraic manipulations in no way make any use of this assumption, however. 6 Conservative dilations and systems with minimal losses In this section we show that every passive system has a minimal conservative dilation in the sense of Definition 6.1 below, and we determine the form of such a dilation in terms of the construction in Theorem 4.6 (see Theorem 6.4). Definition 6.1 A conservative dilation Σ of a passive system Σ is said to be a minimal conservative dilation if there is no conservative dilation of Σ which is a proper restriction of Σ.

11 mn header will be provided by the publisher 11 Theorem 6.2 Let Σ = (Ã, B, C, D; X, U, Y; κ) be a minimal conservative dilation of a passive system Σ = (A, B, C, D; X, U, Y; κ). Then the operators V + = à D + and V = à D in the representation X = D X D + are simple isometries. By a simple isometry on a Hilbert space H we mean an isometry V L(H) such that n=v n H = {}. P r o o f. By the definition of a dilation, Ã, B, C, D have the form à = A 11 A 21 A, B = B, A 31 A 32 A 33 B 3 C = [ C 1 C ], D = D. (6.1) Hence D is invariant under à and D + is invariant under Ã. Since V e Σ is unitary, the operators V + = à D + and V = à D are isometric. We show that the subspace L = n= à n D reduces to {}. In fact, since V eσ is a unitary operator from X U onto X Y and D à D à 2 D..., we have V e Σ L = L and hence V eσ L = L. Now we can rewrite (6.1) according to the decomposition X = L L X D + : à = A 111 A 112 A 21 A A 31 A 32 A 33 C = [ C 1 L C ], D = D., B = B, Since Σ is a minimal conservative dilation of Σ, L = {}. In a similar way, we obtain n= Ãn D + = {}. Theorem 6.3 Any minimal conservative dilation Σ of a passive system Σ has the form constructed in Theorem 4.6. P r o o f. Let Σ = (A, B, C, D; X, U, Y; κ) be any passive system, and let Σ = (Ã, B, C, D; X, U, Y; κ) be a minimal conservative dilation of Σ. Write X = D X D + for some Hilbert spaces D ±, and relative to this decomposition let the operators Ã, B, C, D be given as in (6.1). We first show that B 3 Ãà D = D, à ÃD + = D +. (6.3) Since the system operator V eσ is unitary, A 31A 33 =, A 32A 33 =, A 33A 33 = I, A 11 A 11 = I, A 21 A 11 =, A 31 A 11 =. It follows that for all d D and d + D +, A 11 d Ãà = A 21 A A 31 A 32 A 33 à à = d + A 11 A 21 A 31 A A 32 A 33 A 11d A 33 d + = = d,, d + yielding (6.3). By Theorem 6.2 we can assume that D = l 2 (E) and D + = l 2 +(F) for some Hilbert spaces E and F and that V + = à D + and V = à D are the canonical shift operators on these spaces. (6.2)

12 12 Arov, Rovnyak, and Saprikin: Linear Passive Stationary Scattering Systems Claim 1: A 21 V =, A 31 V =, and C 1 V =. For since Ãà D = D, V A 21 A V d D, d D, A 31 A 32 V + and hence A 21 and A 31 annihilate V D, that is, A 21 V = and A 31 V =. But then for all d D, V d V V d d V eσ = A 21 A 31 =. C 1 C 1 V d Since V and V eσ are isometric, C 1 V d Y =. Thus C 1 V =. Claim 2: A 31V + =, A 32V + =, and B 3V + =. Since à ÃD + = D +, V A 21 A 31 A A 32 D +, d + D +, V+ V + d + and therefore A 31V + = and A 32V + =. Then for all d + D +, A 31 Ve Σ V + d + = A 32 V+ V + d + = d +. B3 B3V + d + Since V + and V e Σ are isometric, B 3V + d + U =. Therefore B 3V + =. By the two claims, A 21 = E 1 P 1, A 31 = Q GP 1, A 32 = Q F 1, B 3 = Q F 2, and C 1 = E 2 P 1 for some operators E 1 L(E, X ), E 2 L(E, Y), G L(E, F), F 1 L(X, F), F 2 L(U, F). In other words, Σ has the form (4.3). Straightforward calculations show that the unitarity of V eσ is equivalent to that of (4.2). Therefore the dilation Σ has the form constructed in Theorem 4.6. Theorem 6.4 Let Σ = (A, B, C, D; X, U, Y; κ) be a passive system. Assume that a conservative dilation Σ = (Ã, B, C, D; X, U, Y; κ) for Σ is constructed from a unitary operator A B E VΣ E 1 X X = C D E F G 2 : U Y (6.4) F 1 F 2 G E F as in Theorem 4.6. Then Σ is a minimal conservative dilation of Σ if and only if (6.4) is a Julia operator. P r o o f. In both the necessity and sufficiency parts of the theorem, we use the same notation as in the proof of Theorem 4.6. Assume first that Σ is a minimal conservative dilation of Σ. We prove that (6.4) is a Julia operator by proving the equivalent relations ker E = {} and ker F = {}. Set ker E = E and ker F = F, and write E = E E, F = F F. Then the restriction of G to E maps E isometrically onto F, and GE F. For as in [1, p. 2], (6.4) acts as a unitary operator from E onto F, and its adjoint maps F onto E ; moreover (6.4) and its

13 mn header will be provided by the publisher 13 adjoint coincide with G and G on E and F, respectively, by the definitions of these subspaces. The assertion follows. Write D = D D and D + = D + D +, where D = l 2 (E ), D = l 2 (E ), D + = l 2 +(E ), D + = l 2 +(E ). Using the definitions of the operators V +, V, P 1, and Q we easily obtain that V = V D V D, V + = V + D V + D, and E 1 P 1 D = {}, F 1 Q D + = {}, Q GP 1 D D +, Q GP 1 D D +, F 2 Q D + = {}, E 2 P 1 D = {}. Hence by (4.3), relative to the decomposition X = D D X D + D +, V D V D Ã = A V + D +, B = B, V + D + C = [ C ], D = D. It is clear now that Σ is a dilation of a system Σ = A, B, [ C ], D; D X D +, U, Y; κ, which is a conservative dilation of Σ. Since Σ is a minimal conservative dilation of Σ, it follows that E = {} and F = {}, and thus (6.4) is a Julia operator. Conversely, assume that (6.4) is a Julia operator. Consider any conservative dilation Σ = (Â, B, Ĉ, D; X, U, Y; κ) of Σ which is a restriction of Σ. We show that Σ = Σ. By the definition of dilations and restrictions, X = D X D +, X = D X D +, where D ± and D ± are Hilbert spaces. We show that the resulting decomposition X = ( D D ) ( X D + D + ). (6.5) coincides with the decomposition X = D X D + in the proof of Theorem 4.6. Relative to (6.5), V eσ has the form A 11 A 21 A 22 V eσ = A 31 A 32 A B A 41 A 42 A 43 A 44 B 4. (6.6) A 51 A 52 A 53 A 54 A 55 B 5 C 1 C 2 C D Since the system operator A 22 V bσ = A 32 A B A 42 A 43 A 44 B 4 C 2 C D

14 14 Arov, Rovnyak, and Saprikin: Linear Passive Stationary Scattering Systems is also unitary A 21, A 31, A 41, A 52, A 53, A 54, C 1, B 5 are zero operators. So à has the form A 11 A 22 à = A 32 A A 42 A 43 A 44 A 51 A 55 relative to the decomposition (6.5). The subspace D + D + is an Ã-invariant subspace of X X, and it is contained in the kernel of C = P Y V eσ X. Every vector x D + D + has the form {..., e 2, e 1,, f, f 1, f 2,...} l 2 (E) X l 2 +(F). Since à x D + D +, P X à x =, and therefore by the form of à in (4.3), E 1e 1 =. Since also C x =, by the form of C in (4.3), E 2 e 1 =. Therefore e 1 ker E and e 1 = since (6.4) is a Julia operator. Thus e n = for every n 1, since D + D + is Ã-invariant. It follows that D + D + D +. Analogously D D D. Since D D D + D + = D D +, we obtain D + D + = D + and D D = D. By the construction of a unitary dilation in Theorem 4.6, the operator A V = à D = 11 A 22 is a simple isometry on D = l 2 (E) with the reducing subspaces D and D. Hence there is a decomposition E = E 1 E 2 such that D = l 2 (E 1 ) and D = l 2 (E 2 ). Recalling that the operators A 31 and C 1 in (6.6) are zero, we see that P X ÃD = {} and CD = {}; hence by (4.3), E 1 E 1 = {} and E 2 E 1 = {}. Thus EE 1 = {}. Since (6.4) is a Julia operator, E 1 = {} and hence D = {}. In a similar way, D + = {}. Therefore Σ = Σ, and we have shown that Σ is a minimal conservative dilation of Σ. Every passive system has a conservative dilation, but not every passive system has a simple conservative dilation. Below we give an example with κ = of a minimal passive system which does not have a simple conservative dilation. Definition 6.5 We say that a passive system Σ has minimal losses if it has a simple conservative dilation Σ. If a simple conservative dilation Σ = (Ã, B, C, D; X, U, Y; κ) of a passive system Σ = (A, B, C, D; X, U, Y; κ) exists, then its transfer function Θ eσ (z) belongs to S κ (U, Y) by Theorem 4.1(i). By Theorem 4.1(ii), Σ is essentially unique because any two such dilations have the same transfer function Θ Σ (z) = Θ eσ (z). Theorem 6.6 A simple conservative dilation Σ = (Ã, B, C, D; X, U, Y; κ) of a given passive system Σ = (A, B, C, D; X, U, Y; κ) is a minimal conservative dilation. P r o o f. Assume that Σ is a dilation of a conservative system Σ = (Â, B, Ĉ, D; X, U, Y; κ), which is in turn a dilation of Σ. Then X = D X D + and X = D X D +, and relative to the latter decomposition the operators Ã, B, C, D have the form  11 à =  21   31  32  33 [ C = Ĉ1 Ĉ, B = ], D = D. B, B 3 (6.7) Since both operators ] [à B M eσ = C D ] [ B and M bσ = Ĉ D

15 mn header will be provided by the publisher 15 are unitary,  21 =,  31 =,  32 =, Ĉ1 =, B 3 =. Then n à n BU X and n à n C Y X, so Σ is simple only if D = {} and D + = {}. Theorems 6.3, 6.4 and 6.6 yield the following corollary. Corollary 6.7 Let Σ be an arbitrary passive system with minimal losses. Then a simple conservative dilation Σ of Σ has the form of the dilation constructed in Theorem 4.6 with Julia operator (4.1). Example of a minimal passive system without minimal losses. Let X = U = Y = C in the Euclidean metric. Choose a system σ = (a, b, c, d; X, U, Y; ) whose system operator a b V σ = c d is a contraction such that abc, a < 1, and I Vσ V σ and I V σ Vσ are invertible. Then σ is a minimal passive system. We show that σ does not have a simple conservative dilation. Argue by contradiction, assuming that σ has a simple conservative dilation Σ = (Ã, B, C, D; X, U, Y; ). By Corollary 6.7, Σ is constructed from a Julia operator (6.4) for V σ. Since a Julia operator is essentially unique, we can choose E = F = C 2, E = (I V σ Vσ ) 1/2, F = (I Vσ V σ ) 1/2, and G = Vσ. We construct a nonzero vector {..., e 2, e 1 } x = x (6.8) {f, f 1,... } in X = l 2 (E) X l 2 +(F) which is orthogonal to all vectors à n Bu = a n bu (6.9) {F 1 a n 1 bu,..., F 1 bu, F 2 u,,,... } and {...,,, E2y, E à n 1 cy,..., E1ā n 1 cy} C y = ā n cy, (6.1) where u U, y Y, and n. Since dim E 1X = dim F 1 X = 1, there exist unit vectors ϕ E E 1X and ψ F F 1 X. To construct x, we choose x = 1 in (6.8) and seek square summable scalars {α n } n= and {β n } n= such that the vectors e n 1 = α n ϕ and f n = β n ψ meet the required orthogonality conditions. It is sufficient to take u = 1 and y = 1 in (6.9) and (6.1). Without difficulty, we find that such scalars are uniquely determined by the orthogonality conditions and given by α n = an c ϕ, E2 1, and β n = ān b ψ, F 2 1, n. The denominators here do not vanish under our assumptions, and the sequences are square summable because a < 1. Therefore Σ is not simple, and hence σ is without minimal losses. 7 Cascade synthesis and Kreĭn-Langer factorizations The cascade synthesis of the two systems Σ 1 = (A 1, B 1, C 1, D 1 ; X 1, U, Y 1 ; κ 1 ) and Σ 2 = (A 2, B 2, C 2, D 2 ; X 2, Y 1, Y; κ 2 ) is a system Σ = Σ 2 Σ 1 which uses the output u > Σ 1 y 1 > Σ 2 > y Σ

16 16 Arov, Rovnyak, and Saprikin: Linear Passive Stationary Scattering Systems from Σ 1 as input for Σ 2 : for all n, Σ 1 : { x1 (n + 1) = A 1 x 1 (n) + B 1 u(n), y 1 (n) = C 1 x 1 (n) + D 1 u(n), and hence Σ 2 : Σ : { x2 (n + 1) = A 2 x 2 (n) + B 2 u(n), y(n) = C 2 x 2 (n) + D 2 u(n), [ x1 (n + 1) A1 x1 (n) = x 2 (n + 1) B 2 C 1 A 2 x 2 (n) y(n) = [ D 2 C 1 C 2 ] [ x 1 (n) x 2 (n) ] B1 + u(n), B 2 D 1 ] + D 2 D 1 u(n). Therefore Σ = (A, B, C, D; X, U, Y; κ), where X = X 1 X 2, κ = κ 1 + κ 2, and A1 B1 A =, B =, B 2 C 1 A 2 B 2 D 1 C = D 2 C 1 C 2, D = D2 D 1. (7.1) In particular, AX 2 X 2. We note the following elementary properties: (i) If Σ = Σ 2 Σ 1, then Θ Σ (z) = Θ Σ2 (z)θ Σ1 (z). (ii) If Σ = Σ 2 Σ 1, then Σ = Σ 1 Σ 2. (iii) Given a third system Σ 3, Σ 3 (Σ 2 Σ 1 ) = (Σ 3 Σ 2 ) Σ 1 provided that the operations are meaningful. (iv) If Σ = Σ 2 Σ 1 and Σ, Σ 1, and Σ 2 are defined as in Theorem 5.3, then Σ = Σ 1 Σ 2. Theorem 7.1 If Σ = Σ 2 Σ 1 where Σ 1 and Σ 2 are passive (conservative) systems, then Σ is a passive (conservative) system. P r o o f. Use the identity A 1 B 1 I X1 A 1 B 1 B 2 C 1 A 2 B 2 D 1 = A 2 B 2 I X2 D 2 C 1 C 2 D 2 D 1 C 2 D 2 C 1 D 1 and the fact that the product of two contraction (unitary) operators is a contraction (unitary) operator. We shall prove: Theorem 7.2 If Σ = (A, B, C, D; X, U, Y; κ) is a simple conservative system, then (i) Σ = Σ 2 Σ 1, where Σ 2 = (A 2, B 2, C 2, D 2 ; X 2, U, Y; ) is a simple conservative system with a Hilbert state space, and Σ 1 = (A 1, B 1, C 1, D 1 ; X 1, U, U; κ) is a simple conservative system whose state space is a κ-dimensional antispace of a Hilbert space; (ii) Σ = Σ 2 Σ 1, where Σ 2 = (A 2, B 2, C 2, D 2; X 2, Y, Y; κ) is a simple conservative system whose state space is a κ-dimensional antispace of a Hilbert space, and Σ 1 = (A 1, B 1, C 1, D 1; X 1, U, Y; ) is a simple conservative system with a Hilbert state space.

17 mn header will be provided by the publisher 17 This result will be established in more precise form in Theorems 7.3 and 7.6. Let U and Y be Hilbert spaces. A Blaschke-Potapov product of degree κ with values in L(U) is a product of κ factors of the form I P + ρ z w 1 wz P, where P L(U) is a rank-one projection operator, w D, and ρ = 1. A right Kreĭn-Langer factorization of a function S(z) in S κ (U, Y) is a representation S(z) = S r (z)b r (z) 1, (7.2) where S r (z) belongs to S(U, Y) and B r (z) is a Blaschke-Potapov product of degree κ with values in L(U) which is invertible at the origin; the factorization is coprime in the sense that if S r (w)u = and B r (w)u = for some w in D and u U, then u =. A left Kreĭn-Langer factorization of S(z) is a representation S(z) = B l (z) 1 S l (z), (7.3) where S l (z) belongs to S(U, Y) and B l (z) is a Blaschke-Potapov product of degree κ with values in L(Y) which is invertible at the origin; the factorization is coprime in the sense that if S l (w) y = and B l (w) y = for some w in D and y Y, then y =. Left and right Kreĭn-Langer factorizations are essentially unique. Conversely, an arbitrary product (7.2) in which S r (z) belongs to S(U, Y) and B r (z) is a Blaschke-Potapov product of degree κ with values in L(U) which is invertible at the origin represents a function in S κ (U, Y) for some κ κ, and κ = κ if the factorization is coprime. A parallel assertion holds for products (7.3). These results are due to Kreĭn and Langer [19]; an account is given in [1, Section 4.2]. Theorem 7.3 Suppose that the function Θ(z) S κ (U, Y) has the right Kreĭn-Langer factorization Θ(z) = Θ r (z)b r (z) 1. Let Σ r = (A r, B r, C r, D r ; X r, U, U; κ), Σ r + = (A r +, B r +, C r +, D r +; X r +, U, Y; ), be simple conservative systems such that Θ Σ r + (z) = Θ r (z) and Θ Σ r (z) = b r (z) 1. Then Σ r = Σ r + Σ r is a simple conservative system with transfer function Θ(z). Lemma 7.4 Let Σ = Σ 2 Σ 1, where Σ 1 = (A 1, B 1, C 1, D 1 ; X 1, U, Y 1 ; κ 1 ) and Σ 2 = (A 2, B 2, C 2, D 2 ; X 2, Y 1, Y; κ 2 ). Then (XΣ c X Σ o) consists of all x = x 1 x 2 in X = X 1 X 2 such that { ΘΣ2 (z)c 1 (I za 1 ) 1 x 1 = C 2 (I za 2 ) 1 x 2, (7.4) Θ Σ1 ( z) B2(I za 2) 1 x 2 = B1(I za 1) 1 x 1, in a neighborhood of the origin. Hence if the only vectors x 1 X 1 and x 2 X 2 which satisfy (7.4) are x 1 = and x 2 =, then Σ = Σ 2 Σ 1 is simple. Lemma 7.5 Let Θ 1 (z) = b(z) 1, where b(z) is a Blaschke product of degree κ which has values in L(U) for some Hilbert space U and which is invertible at the origin. Then Θ 1 (z) is the transfer function of a simple conservative system Σ 1 = (A 1, B 1, C 1, D 1 ; X 1, U, U; κ) for which the state space X 1 is the κ-dimensional Pontryagin space H(Θ 1 ) having reproducing kernel [I Θ 1 (z)θ 1 (w) ]/(1 z w), and h(z) h() A 1 : h(z), B 1 : u Θ 1(z) Θ 1 () u, z z (7.5) C 1 : h(z) h(), D 1 : u Θ 1 ()u, for all h(z) in H(Θ 1 ) and all u in U. The space X 1 is the antispace of a Hilbert space, and the identity h1 (z) h 1 (), h 1(z) h 1 () = h 1 (z), h 1 (z) z z h H(Θ1) 1(), h 1 () U (7.6) holds for all elements h 1 (z) of the space. H(Θ 1)

18 18 Arov, Rovnyak, and Saprikin: Linear Passive Stationary Scattering Systems Proof of Lemma 7.4. Write Σ = (A, B, C, D; X, U, Y; κ), where X = X 1 X 2, κ = κ 1 + κ 2, and A, B, C, D are given by (7.1). It is easy to see that the definitions of the subspaces XΣ c and X Σ o in Section 3 can be rewritten in the form X c Σ = z Ω (I za) 1 B U and X o Σ = z Ω (I za ) 1 C Y, where Ω is any small neighborhood of the origin. Therefore x = x 1 x 2 is orthogonal to XΣ c X Σ o if and only if for all z Ω, B (I za ) 1 x = and C(I za) 1 x =, or equivalently (by (7.1)) (I za 1 ) 1 [x1 ] D2 C 1 C 2 =, z(i za 2 ) 1 B 2 C 1 (I za 1 ) 1 (I za 2 ) 1 x 2 [ B 1 D1B2] (I za 1) 1 z(i za 1) 1 C 1 B2(I za 2) 1 [x1 ] =, (I za 2) 1 x 2 Expanding these identities and simplifying by means of the relations we obtain (7.4). Θ Σ2 (z) = D 2 + zc 2 (I za 2 ) 1 B 2, Θ Σ1 ( z) = D 1 + zb 1(I za 1) 1 C 1, Proof of Lemma 7.5. This follows from [1, Theorem A3] and the last statement in [1, Theorem 3.2.5]. Proof of Theorem 7.3. Set Y 1 = U, and write Σ 1 = (A 1, B 1, C 1, D 1 ; X 1, U, Y 1 ; κ 1 ) = Σ r, κ 1 = κ, Σ 2 = (A 2, B 2, C 2, D 2 ; X 2, Y 1, Y; κ 2 ) = Σ r +, κ 2 =. By Theorem 7.1, Σ r = Σ r + Σ r = Σ 2 Σ 1 is conservative. It has transfer function Θ Σ r(z) = Θ Σ2 (z)θ Σ1 (z) = Θ r (z)b r (z) 1 = Θ(z) at all points in D where the functions are defined. The main problem is to show that Σ r is simple, and for this we use Lemma 7.4. Let x 1 X 1 and x 2 X 2 satisfy (7.4). Since all simple conservative realizations are equivalent by Theorem 4.1, without loss of generality we can assume that Σ 1 is given as in Lemma 7.5 with Θ 1 (z) = b r (z) 1. Thus X 1 = H(b 1 r ) is the antispace of a κ-dimensional Hilbert space, and for every h(z) in the space, C 1 (I wa 1 ) 1 : h(z) h(w) (7.7) for all w in a neighborhood of the origin [1, p. 89]. We remark also that since κ 2 =, A 2 is a contraction operator on the Hilbert space X 2 and therefore (I za 2 ) 1 exists for z in the unit disk D. We first show that x 1 =. Argue by contradiction, assuming x 1. Let M be the subspace of elements h(z) in H(b 1 r ) such that Θ r (z)h(z) Hol(D), that is, apart from removable singularities Θ r (z)h(z) is holomorphic on D. Then x 1 (z) belongs to M by (7.7) and the first relation in (7.4), and so M = {}. The subspace M is invariant under A 1, since if Θ r (z)h(z) Hol(D), then Θ r (z) h(z) h() z = Θ r(z)h(z) Θ r (z)h() z Hol(D). Since M is finite dimensional, A 1 M has an eigenvalue β 1. Thus M contains an element of the form h 1 (z) = u 1 β 1 z, u U = 1. The identity (7.6), together with the fact that X 1 is the antispace of a Hilbert space, implies that β 1 > 1. Set α 1 = 1/β 1. By the definition of M, Θ r (z)h 1 (z) Hol(D), and this is only possible if Θ r (α 1 )u =. Writing b 1 (z) = I P 1 + z α 1 1 ᾱ 1 z P 1, P 1 =, u U u,

19 mn header will be provided by the publisher 19 we therefore have Θ r (z)b 1 (z) 1 S(U, U). An argument in [1, p. 142] shows that b r (z) 1 = b 1 (z) 1 S 1 (z), where S 1 (z) belongs to S κ 1 (U, U). But then [1, Theorem 4.1.1] Θ(z) = [ Θ r (z)b 1 (z) 1] S 1 (z) S κ (U, Y), κ κ 1, which contradicts our assumption that Θ(z) S κ (U, Y). Therefore M = {} and hence x 1 =. Returning to (7.4), we see that C 2 (I za 2 ) 1 x 2 = and B 2(I za 2) 1 x 2 = in a neighborhood of the origin. Since Σ 2 is simple, it follows that x 2 =. Hence by Lemma 7.4, Σ is simple, as was to be shown. A parallel result holds for left Kreĭn-Langer factorizations. Theorem 7.6 Suppose that the function Θ(z) S κ (U, Y) has the left Kreĭn-Langer factorization Θ(z) = b l (z) 1 Θ l (z). Let Σ l + = (A l +, B l +, C l +, D l +; X l +, U, Y; ), Σ l = (A l, B l, C l, D l ; X l, Y, Y; κ), be simple conservative systems such that Θ Σ l + (z) = Θ l (z) and Θ Σ l (z) = b l (z) 1. Then Σ l = Σ l Σ l + is a simple conservative system with transfer function Θ(z). P r o o f. Write F (z) = F ( z) for any operator-valued function F (z). Then Θ(z) S κ (Y, U) (for example, see [1, p. 68]), and we can define a right Kreĭn-Langer factorization of Θ(z) by setting Θ(z) = Θ r (z)b r (z) 1, Θ r (z) = Θ l (z), b r (z) = b l (z). Then Σ l + and Σ l are simple conservative systems with transfer functions Θ Σ l + (z) = Θ Σ l + (z) = Θ l (z) = Θ r (z), Θ Σ l (z) = Θ Σ l (z) = b l (z) 1 = b r (z) 1. By Theorem 7.3, Σ l + Σ l is a simple conservative system such that Θ Σ l + Σl (z) = Θ r(z)b r (z) 1 = Θ(z). It follows that Σ l = Σ l Σ l + is a simple conservative system whose transfer function is given by Θ Σ (z) = Θ(z). Every contraction operator on a Pontryagin space admits semi-definite invariant subspaces (see Section 2). We obtain a stronger conclusion when the contraction operator is the main operator of a simple passive system. Theorem 7.7 Let Σ = (A, B, C, D; X, U, Y; κ) be any simple passive system. (i) The state space X has a unique fundamental decomposition X = X + X such that AX + X +. (ii) The state space X has a unique fundamental decomposition X = X + X such that AX X. Remark. The statement of Theorem 7.7 for a simple conservative system is a consequence of [18, Lemma 11.5, p. 82], as described in Corollary 2.3 and the discussion in Section 4 of the paper [15]. For a simple passive system the statement of Theorem 7.7 then follows from this fact and an embedding into a simple conservative system as in our proof below. The authors learned of this connection from the referee s report, and they thank the referee for his remark. Lemma 7.8 The main operator of a simple passive system Σ = (A, B, C, D; X, U, Y; κ) has no eigenvalue of modulus one.