people.virginia.edupeople.virginia.edu/~jlr5m/Papers/p63.pdf · mn header will be provided by the publisher 29 Acknowledgements D. Z. Arov and S. M. Saprikin were partially supported

mn header will be provided by the publisher

Linear Passive Stationary Scattering Systemswith 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 65020, Ukraine

2 Department of Mathematics, University of Virginia, P. O. Box 400137, Charlottesville, VA 22904–4137, U.S.A.

Received xxx, revised xxx, accepted xxxPublished online xxx

Key words Scattering system, passive, conservative, transfer function, Krein-Langer factorization, Pontryaginspace, minimal system, dilation, Julia operator.MSC (2000) Primary: 47A48; Secondary 47A45, 47A20, 46C20, 47B50, 47N70, 93B28

Passive scattering systems having Pontryagin state spaces and their minimal conservative dilations are inves-tigated. The transfer functions of passive scattering systems are generalized Schur functions. In the case of asimple conservative system, the right and left Kreın-Langer factorizations of the transfer function correspond tonatural cascade syntheses of systems. A generalization of Sz.-Nagy and Foias criteria for a cascade synthesis oftwo simple conservative systems to be simple is obtained for systems with Pontryagin state spaces. It is shownthat the state space of a simple passive system admits certain unique fundamental decompositions, which giverise to a notion of stability and a characterization of simple conservative systems whose transfer functions haveunitary boundary values a.e. on the unit circle.

Copyright line will be provided by the publisher

Contents

1 Introduction 12 Pontryagin spaces and linear operators 33 Scattering systems with Pontryagin state spaces 44 A brief survey of known results 65 Schur complements and associated systems 96 Conservative dilations and systems with minimal losses 107 Cascade synthesis and Kreın-Langer factorizations 158 Application to simple conservative systems 209 The classes Pκ

00 and Cκ00 23

10 Models for Cκ00 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),

∗ e-mail: [email protected]∗∗ Corresponding author: e-mail: [email protected]∗∗∗ e-mail: sergey [email protected]


2 Arov, Rovnyak, and Saprikin: Linear Passive Stationary Scattering Systems

n = 0, 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

ΘΣ(z) = D + zC(I − zA)−1B

whenever the inverse exists. The case in which the state space X is a Hilbert space is classical. In several recentpapers, 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 dilationand embedding of systems, are discussed in [4] and [23]. Results on Darlington representations of systemshaving 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 spacesby investigating further properties of dilations and embeddings, minimal systems, cascade representations andinvariant subspaces, and a generalization of the notion of stability.

We summarize background in Sections 2 and 3. This includes properties of Pontryagin spaces, contractionoperators, 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 systemsaccording as the system operator

VΣ =[A BC D

]is contractive or unitary. The definitions of controllable, observable, and simple systems are parallel to the Hilbertspace case. Among several equivalent definitions of a minimal system for Hilbert state spaces, we choose onethat 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 isintroduced here. The transfer functionΘΣ(z) of a passive system Σ = (A,B,C,D;X ,U ,Y; κ) belongs to somegeneralized Schur class Sκ′(U ,Y) with κ′ ≤ κ. Many theorems use the hypothesis that ΘΣ(z) ∈ Sκ(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−1B and A − BD−1C in thesystem 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

UΣ =[VΣ EF G

]:

[X ⊕ UE

]→

[X ⊕ YF

],

and each such unitary operator induces a conservative dilation Σ. It is shown that Σ is a minimal conservativedilation if and only if UΣ is a Julia operator. Passive systems which admit simple conservative dilations are said tohave minimal losses. The results of Section 6 determine the form of a minimal conservative dilation of a passivesystem, namely, they are induced by Julia operators for the system operator.

In Section 7 we show that the right and left Kreın-Langer factorizations of the transfer function of a simpleconservative system Σ correspond to natural cascade syntheses of the system. These results are applied to simplepassive systems Σ. In this case, the semi-definite invariant subspaces of the main operator A, whose existence isassured by the classical theory of contraction operators on a Pontryagin space, are regular; in fact, they are uniqueand determine natural fundamental decompositions X = X+ ⊕ X− and X = X ′+ ⊕ X ′− of the state space withAX+ ⊆ X+ and AX ′− ⊆ X ′−.

A key step in Section 7 is to show that the cascade synthesis of two particular simple conservative systems issimple. In general, such a cascade synthesis is not always simple, even with Hilbert state spaces. Section 8 isconcerned with the problem to determine when the cascade synthesis of two simple conservative systems havingPontryagin state spaces is simple. We generalize a well-known analytical condition, which is related to the notionof a regular factorization of an operator-valued function, from the case of Hilbert state spaces to Pontryagin statespaces.

In Section 9 we examine the notion of stability. In the case of Hilbert state spaces, we call a system stable ifit is bi-stable in the sense that both of the semigroups An and A∗n are stable, that is, they tend to zero strongly


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 eigenvaluesof the main operator of modulus greater than one. In place of stable systems, we introduce classes Pκ

00 andCκ

00 of passive and conservative systems that are “partially stable” in a sense that depends on the fundamentaldecompositions X = X+⊕X− and X = X ′+⊕X ′− of the state space introduced in Section 7; in the definitions ofthese classes we require, roughly, that the parts of the system corresponding to the positive subspaces are stablein the usual Hilbert space sense. It is shown, for example, that a simple conservative system belongs to Cκ

00 ifand only if its transfer function has unitary boundary values a.e. on the unit circle.

The concluding Section 10 shows how the class Cκ00 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 togetherwith a linear and symmetric inner product 〈·, ·〉 = 〈·, ·〉X which admits a representation X = X+ ⊕ X−, where(X±,±〈·, ·〉X ) are Hilbert spaces and dimX− < ∞. We call such a representation a fundamental decom-position. A fundamental decomposition is in general not unique, but it determines a unique strong topology.The dimensions ind± X = dimX± are independent of the choice of fundamental decomposition and called thepositive and negative indices of X . The terms regular subspace and Hilbert subspace are used as in standardsources; the orthogonal projection operator whose range is a regular subspace M is written PM. We shall usethe 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 Mand N have negative index κ, then PMN and PNM 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 intoitself and into a Pontryagin space Y , respectively. An operator is invertible if it has an everywhere definedand 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 aredefined as for Hilbert spaces. We call A ∈ L(X ) nonnegative and write A ≥ 0 if 〈Ax, x〉X ≥ 0 for everyx ∈ X . An operator T ∈ L(X ,Y) is a contraction if I − T ∗T ≥ 0. If T ∈ L(X ,Y) is a contraction andind− 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 existHilbert spaces E ,F and operators E,F,G acting on appropriate spaces such that the operator

U =[T EF G

]:

[XE

]→

[YF

]is unitary. In this case, the conditions kerE = 0 and kerF ∗ = 0 are equivalent, and when these conditionsare 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 thereader 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 (non-positive) for all x in N . We say that N is semi-definite if it is either nonnegative or nonpositive. A semi-definitesubspace may contain nonzero vectors x such that 〈x, x〉X = 0. Notions of maximal nonnegative and maximalnonpositive 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 dimN = 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-dimensionalroot subspaces.

(ii) There exists a maximal nonpositive subspace N of X which is invariant under T and which contains all ofthe root subspaces for the eigenvalues of T in z : |z| > 1.



(iii) There exists a maximal nonnegative subspace P of X which is invariant under T and which is orthogonalto 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 theoremsof this type. Concerning the statement (i) in Theorem 2.2, see also Kuzhel′ [21, Section 4, Assertion 6], whereunitary dilations are used.

Sketch of the proof of Theorem 2.2. (i) We follow [18, Lemma 11.1]. Choose a fundamental decompositionX = X+ ⊕X− of X . Set H+ = X+, let H− be the antispace of X−, and write

T =[T11 T12

T21 T22

]:

[H+

H−

]→

[H+

H−

].

Since 〈Tx, Tx〉X ≤ 〈x, x〉X for all x in X , for any x+ ∈ X+,

‖T11x+‖2 − ‖T21x+‖2 ≤ ‖x+‖2.

Thus T ∗11T11 ≤ I+T ∗21T21 = R2, R =[I+T ∗21T21

]1/2, and so T11 = SR, where S is a contraction operator on

H+. Since T ∗21T21 has finite rank, by the spectral theorem R = I+R0 where R0 has finite rank. Hence T differsfrom the contraction operator S ⊕ 0 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-dimensionalroot 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 = 0, y ∈ M is a maximal nonnegative invariant subspace for T having therequired 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. Thedefinitions of these notions are adapted from the Hilbert state space case.

Consider a systemx(n+ 1) = Ax(n) +Bu(n),

y(n) = Cx(n) +Du(n),(3.1)

n = 1, 2, . . . , whose states x(n) belong to a Pontryagin space X of negative index κ and whose inputs u(n) andoutputs y(n) belong to Hilbert spaces U and Y . We call X the state space and U and Y the input and outputspaces. 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 VΣ ∈ L(X ⊕ U ,X ⊕ Y) is defined by

VΣ =[A BC D

]:

[XU

]→

[XY

]. (3.3)

We call A,B,C the main, input, and output operators for the system. The transfer function of the system isdefined by

ΘΣ(z) = D + zC(I − zA)−1B

whenever the inverse exists. The adjoint system is Σ∗ = (A∗, C∗, B∗, D∗;X ,Y,U ; κ). Thus VΣ∗ = V ∗Σ , andΘΣ∗(z) = ΘΣ(z)∗.



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)‖2U − ‖y(n)‖2Y (3.4)

for all initial states x(0) in X and all inputs u(n), n ≥ 0, in U . We call (3.1) a conservative scattering systemif 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 “conservativesystems” because other types of systems will not be considered. A system Σ is passive (conservative) if and onlyif the system operator VΣ is a contraction (unitary) operator. If Σ is passive or conservative, the adjoint systemΣ∗ has the same property.

For any system Σ = (A,B,C,D;X ,U ,Y; κ), set X cΣ = ∨∞0 AnB U and X o

Σ = ∨∞0 A∗nC∗Y . Then Σ issaid to be (i) controllable, (ii) observable, (iii) simple, or (iv) minimal according as

(i) X = X cΣ,

(ii) X = X oΣ,

(iii) X = X cΣ ∨ X o

Σ, or

(iv) X = X cΣ and X = X o

Σ.

It is easy to see that X cΣ∗ = X o

Σ and X oΣ∗ = X c

Σ.Remark. We have defined a minimal system as a system that is controllable and observable. This definitiondiffers from the usual definition of a minimal system with a Hilbert state space. In the case of a Hilbert statespace, 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 systemswith Pontryagin state spaces they are not the same. However, a controllable and observable system cannot be anontrivial dilation of another system, and in the special case of passive systems, which is of our main interest, theconverse statement is also true (see Corollary 4.8).

We call two systems Σ1 = (A1, B1, C1, D1;X1,U ,Y; κ) and Σ2 = (A2, B2, C2, D2;X2,U ,Y; κ) equiva-lent and write Σ1

∼= Σ2 if

A2 = WA1W−1, B2 = WB1, C2 = C1W

−1, D2 = D1.

for some unitary operator W ∈ L(X1,X2).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 embeddedin a system Σ = (A, B, C, D;X , U , Y; κ) if there exist Hilbert spaces U and Y such that U = U ⊕ U andY = Y ⊕ Y , and

VeΣ =[VΣ EF G

]:

[X ⊕ UU

]→

[X ⊕ YY

](3.5)

for some operators E,F,G. Equivalently,

VeΣ =

A B E1

C D E2

F1 F2 G

:

XUU

→

XYY

or

VeΣ =

[A B

C D

]=

A[E1 B

][CF1

] [E2 DG F2

] :

XUU

→

XYY

. (3.6)

In this case,

ΘeΣ(z) =[Θ11(z) Θ12(z)Θ21(z) Θ22(z)

], ΘΣ(z) = Θ12(z). (3.7)



If VΣ is a contraction operator (that is, Σ is a passive system) and (3.5) is a Julia operator for VΣ (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 andenlarges the state space, but without increasing its negative index. We call Σ = (A, B, C, D; X ,U ,Y; κ) adilation of Σ = (A,B,C,D;X ,U ,Y; κ) if X = D− ⊕X ⊕D+, where D+ and D− are Hilbert spaces and

VeΣ =

[A B

C D

]=

∗ 0 0∗ A 0∗ ∗ ∗

0B∗

[∗ C 0

]D

:

D−XD+

U

→

D−XD+

Y

. (3.8)

The form of the system operator is equivalent to the relations

AD+ ⊆ D+, A∗D− ⊆ D−, CD+ = 0, B∗D− = 0.

Using this form, we easily obtain ΘeΣ(z) = ΘΣ(z). A dilation Σ of Σ is called conservative or simple accordingas 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 aproper restriction of Σ if it is a restriction of Σ and Σ 6= 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 acontraction. For since Y is a Hilbert space, from the inequality

〈VΣ(x⊕ 0), VΣ(x⊕ 0)〉X⊕Y ≤ 〈x⊕ 0, x⊕ 0〉X⊕U

for any x in X , we obtain 〈Ax,Ax〉X ≤ 〈Ax,Ax〉X + 〈Cx,Cx〉Y ≤ 〈x, x〉X . Moreover, every contractionoperator A on a Pontryagin space X may be viewed as the main operator of a conservative system by consideringa Julia operator for A. We also remark that if a system Σ is embedded in any way in a system Σ, and if Σ ispassive, 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 ofa passive system Σ = (A,B,C,D;X ,U ,Y; κ) is a contraction operator. In the classical case κ = 0, X is aHilbert space and the inverse (I − zA)−1 exists at all points of the open unit disk D. In this case, the transferfunction ΘΣ(z) is defined everywhere in D and, as is well known, it is an operator-valued Schur function, thatis, it is holomorphic and bounded by one in D. When the state space X is a Pontryagin space, then according toTheorem 2.2, the inverse (I − zA)−1 exists for all but at most κ nonzero points in D. We shall usually considerthe domain of the transfer function of a passive system Σ = (A,B,C,D;X ,U ,Y; κ) to be the open unit diskwith at most κ nonzero points deleted. In some places (for example, see Section 5) we shall also consider systemsΣ with invertible main operator A; in this case, the transfer function ΘΣ(z) is holomorphic at ∞ as well, and itsrestriction to a neighborhood of ∞ plays a role.

Given Hilbert spaces U and Y , the generalized Schur class Sκ(U ,Y) is the set of functions S(z) with valuesin L(U ,Y) which are holomorphic in a neighborhood of the origin and meromorphic in D such that the kernelK(w, z) = [I−S(z)S(w)∗]/(1−zw) has κ negative squares (κ = 0, 1, 2, . . . ). This means that for any finite setof points w1, . . . , wn in the domain of holomorphy of S(z) in D, the selfadjoint operator on Y ⊕ · · ·⊕Y definedby the block Hermitian matrix

[K(wj , wk)

]n

j,k=1has at most a κ-dimensional invariant subspace in which the

operator has spectrum contained in (−∞, 0), and there exists at least one choice of points w1, . . . , wn such thatthe operator has such an invariant subspace of dimension equal to κ. By the Kreın-Langer factorization (seeSection 7), every S(z) in Sκ(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 Uκ(U ,Y) be the subclass of functions in Sκ(U ,Y) whoseboundary values are unitary a.e. For κ = 0, we write more simply S(U ,Y) for S0(U ,Y), and U(U ,Y) forU0(U ,Y).

From [1, Theorems 2.1.2, 2.3.1, and 2.1.3], we obtain:



Theorem 4.1 (i) The transfer function ΘΣ(z) of a simple conservative system Σ = (A,B,C,D;X ,U ,Y; κ)belongs to the class Sκ(U ,Y). Conversely, everyΘ(z) ∈ Sκ(U ,Y) has the formΘ(z) = ΘΣ(z) for some simpleconservative system Σ = (A,B,C,D;X ,U ,Y; κ).

(ii) Two simple conservative systems Σ1 and Σ2 are equivalent if and only if ΘΣ1(z) = ΘΣ2(z) in a neigh-borhood 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 generalizedSchur function, but for a possibly smaller index than κ. The next result, which is taken from [23, Theorems 2.2and 2.3], includes a sufficient condition for equality.

Theorem 4.2 The transfer function ΘΣ(z) of a passive system Σ = (A,B,C,D;X ,U ,Y; κ) belongs toSκ′(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 embeddingalways 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, thenΣ is simple.

P r o o f. Let Σ = (A,B,C,D;X ,U ,Y; κ) and Σ = (A, B, C, D;X , U , Y; κ). Since B U ⊇ B U andC∗Y ⊇ C∗Y ,

X ceΣ =∨∞0 AnB U ⊇∨∞0 AnB U = X cΣ,

X oeΣ =∨∞0 A∗nC∗Y ⊇∨∞0 A∗nC∗Y = X oΣ.

Therefore X ceΣ ∨ X oeΣ ⊇ X cΣ ∨ X o

Σ, 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 ΘΣ(z) ∈ Sκ(U ,Y)only under special conditions; two sufficient conditions for the inclusion ΘΣ(z) ∈ Sκ(U ,Y) are identified inTheorems 4.1 and 4.2. The property that ΘΣ(z) ∈ Sκ(U ,Y) is a hypothesis in the next two results. Theorem 4.4is 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 ΘΣ(z) belongsto Sκ(U ,Y). Then each of the spaces X c

Σ, X oΣ, and X c

Σ ∨X oΣ is regular and has negative index κ. Hence (X c

Σ)⊥,(X o

Σ)⊥, and (X cΣ ∨ 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 ΘΣ(z)belongs to Sκ(U ,Y). Then

(i) (X oΣ)⊥ is the largest Hilbert subspaceM of X which is invariant underA and contained in kerC, such that

A|M is isometric;

(ii) (X cΣ)⊥ is the largest Hilbert subspace K of X which is invariant under A∗ and contained in kerB∗, such

that A∗|K is isometric;

(iii)(X c

Σ ∨ X oΣ

)⊥is the largest Hilbert subspace L of X which is invariant under A and contained in kerC ∩

kerB∗, such that A|L is unitary.

We include the construction of a conservative dilation for any passive system from [23, Theorem 2.1] sincethis 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 aunitary operator of the form[

VΣ EF G

]:

[X ⊕ UE

]→

[X ⊕ YF

], (4.1)



or, equivalently,A B E1

C D E2

F1 F2 G

:

XUE

→

XYF

. (4.2)

This can be done in many ways; for example, we can choose (4.1) to be a Julia operator for VΣ. Set X =D− ⊕ X ⊕ D+, where D− = `2−(E) is the Hilbert space of square summable sequences . . . , e−2, e−1 withentries in E , and D+ = `2+(F) is the analogous space of sequences f0, f1, . . . with entries in F . Set

A =

V ∗− 0 0E1P−1 A 0Q0GP−1 Q0F1 V+

, B =

0B

Q0F2

,C =

[E2P−1 C 0

], D = D,

(4.3)

where

P−1 : . . . , e−3, e−2, e−1 → e−1,

Q0 : f0 → f0, 0, 0, . . . ,V− : . . . , e−3, e−2, e−1 → . . . , e−2, e−1, 0,V+ : f0, f1, f2, . . . → 0, f0, f1, . . . ,

on arbitrary elements of the appropriate spaces. The system Σ = (A, B, C, D; X ,U ,Y; κ) has the requiredproperties.

In the other direction, we can ask if a given system has restrictions with special properties. Theorem 4.7 andCorollary 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 ΘΣ(z) belongsto Sκ(U ,Y). Define subspaces

X1 = X oΣ ∩ (X c

Σ)⊥, X2 = PXoΣX c

Σ, X3 = (X oΣ)⊥.

Then X1 and X3 are Hilbert spaces, X = X1 ⊕X2 ⊕X3, and the system operator for Σ has the form

VΣ =[A BC D

]=

A11 0 0A21 A22 0A31 A32 A33

0B2

B3

[C1 C2 0

]D

:

X3

X2

X1

U

→

X3

X2

X1

Y

.Theorem 4.7 is given in [23, Theorem 3.2]. Notice that by Theorem 4.4 and Lemma 2.1, the subspace X2 in

Theorem 4.7 is regular and has negative index κ. The conclusion of Theorem 4.7 is that Σ is a dilation of thesystem

Σres,1 = (PX2A|X2, PX2B,C|X2, D;X2,U ,Y; κ),

which is called the first restriction of Σ. It is shown in [23, p. 204] that Σres,1 is minimal, and thus we obtain:

Corollary 4.8 Every passive system Σ = (A,B,C,D;X ,U ,Y; κ) with transfer functionΘΣ(z) in Sκ(U ,Y)is the dilation of a minimal passive system, namely, its first restriction.



5 Schur complements and associated systems

Consider a system Σ with system operator (3.3). Following standard matrix terminology, we call D − CA−1Bthe Schur complement of A in VΣ whenever A is invertible, and A−BD−1C the Schur complement of D in VΣ

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 isinvertible.

Theorem 5.2 Let Σ = (A,B,C,D;X ,U ,Y; κ) be a system. Assume that A is invertible and that dimX =n <∞. Define a system Σ′ = (A′, B′, C ′, D′;X ′,U ,Y; κ′), κ′ = n−κ, whose state space X ′ is the antispaceof X by setting

A′ = A−1, B′ = −A−1B,

C ′ = CA−1, D′ = D − CA−1B.(5.1)

Then

(i) Σ is conservative if and only if Σ′ is conservative;

(ii) ΘΣ(z) is analytic at infinity, ΘΣ(∞) = D′, and the identity ΘΣ′(z) = ΘΣ(1/z) holds wherever thesefunctions are defined, and in particular it holds in a neighborhood of the origin and in a neighborhood ofinfinity.

P r o o f. (i) Assume that Σ is conservative. For the purpose of computing adjoints, first write (5.1) morecorrectly as

A′ = σA−1σ−1, B′ = −σA−1B,

C ′ = CA−1σ−1, D′ = D − CA−1B,(5.2)

where σ : X → X ′ is the identity mapping. Then σ∗ = −σ−1. The identities

A′∗A′ + C ′∗C ′ = IX ′ , A′∗B′ + C ′∗D′ = 0, B′∗B′ +D′∗D′ = IU ,

A′A′∗ +B′B′∗ = IX ′ , A′C ′∗ +B′D′∗ = 0, C ′C ′∗ +D′D′∗ = IY ,

are checked by straightforward algebraic calculations. For example,

A′∗A′ + C ′∗C ′ =[− σA∗−1(−σ−1)

][σA−1σ−1

]+

[− σA∗−1C∗

][CA−1σ−1

]= σ

[A∗−1A−1 − A∗−1

(IX − A∗A

)A−1

]σ−1 = IX ′ ,

yielding the first identity. The other five identities follow similarly. Hence Σ′ is conservative. These steps arereversible.

(ii) The invertibility of A implies that ΘΣ(z) is analytic at infinity. For any nonzero z the invertibility ofIX ′ − zA′ is equivalent to the invertibility of IX − z−1A. For any such point,

ΘΣ′(z) = D′ + zC ′(IX ′ − zA′)−1B′

= (D − CA−1B) + zCA−1(IX − zA−1)−1(−A−1B)

= D + C[−A−1 + zA−1(IX − zA−1)−1(−A−1)

]B

= D + z−1C(IX − z−1A)−1B,

and (ii) follows.



Theorem 5.3 Let Σ = (A,B,C,D;X ,U ,Y; κ) be a system. Assume that D is invertible and that dimX =n < ∞. Define a system Σ′′ = (A′′, B′′, C ′′, D′′;X ′′,Y,U ; κ′′), κ′′ = n − κ, whose state space X ′′ is theantispace of X by setting

A′′ = A−BD−1C, B′′ = BD−1,

C ′′ = −D−1C, D′′ = D−1.(5.3)

Then

(i) Σ is conservative if and only if Σ′′ is conservative;

(ii) X cΣ′′ = X c

Σ and X oΣ′′ = X o

Σ;

(iii) ΘΣ′′(z) = ΘΣ(z)−1 at all points whereΘΣ(z) andΘΣ′′(z) are defined, and in particular this identity holdsin 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∨n=0

(A′′)nB′′Y =N∨

n=0

(A−BD−1C

)nB U ⊆

N∨n=0

AnB U ,

and similarly∨N0 (A′′∗)nC ′′∗ U ⊆∨N

0 A∗nC∗ Y . Therefore X c

Σ′′ ⊆ X cΣ and X o

Σ′′ ⊆ X oΣ. 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 ΘΣ(z) and ΘΣ′′(z) are defined,

ΘΣ′′(z)ΘΣ(z) =[D′′ + zC ′′(I − zA′′)−1B′′

][D + zC(I − zA)−1B

]=

[D−1 − zD−1C

(I − z(A−BD−1C)

)−1BD−1

]·

·[D + zC(I − zA)−1B

]= I + zD−1C(I − zA)−1B

− zD−1C(I − z(A−BD−1C)

)−1BD−1D

− z2D−1C(I − z(A−BD−1C)

)−1BD−1C(I − zA)−1B

= I + zD−1C(I − z(A−BD−1C)

)−1··[(I − zA+ zBD−1C)− (I − zA)− zBD−1C

](I − zA)−1B

= I.

Similarly, ΘΣ(z)ΘΣ′′(z) = I at all points where the two functions are defined.

It should be noted that the hypothesis dimX < ∞ in Theorems 5.2 and 5.3 is only used to assure that theantispace of X has finite negative index and hence is a Pontryagin space (all of our definitions presume that thestate 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 Definition6.1 below, and we determine the form of such a dilation in terms of the construction in Theorem 4.6 (see Theorem6.4).

Definition 6.1 A conservative dilation Σ of a passive system Σ is said to be a minimal conservative dilationif there is no conservative dilation of Σ which is a proper restriction of Σ.



Theorem 6.2 Let Σ = (A, B, C, D; X ,U ,Y; κ) be a minimal conservative dilation of a passive systemΣ = (A,B,C,D;X ,U ,Y; κ). Then the operators V+ = A|D+ and V− = A∗|D− in the representationX = 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=0VnH = 0.

P r o o f. By the definition of a dilation, A, B, C, D have the formA =

A11 0 0A21 A 0A31 A32 A33

, B =

0BB3

,C =

[C1 C 0

], D = D.

(6.1)

HenceD− is invariant under A∗ andD+ is invariant under A. Since VeΣ is unitary, the operators V+ = A|D+ andV− = A∗|D− are isometric. We show that the subspace L− =

⋂∞n=0 A

∗nD− reduces to 0. In fact, since VeΣis a unitary operator from X ⊕ U onto X ⊕ Y and D− ⊇ A∗D− ⊇ A∗2D− ⊇ . . . , we have V ∗eΣL− = L− and

hence VeΣL− = L−. Now we can rewrite (6.1) according to the decomposition X = L− ⊕ L⊥− ⊕X ⊕D+:A =

A111 0 0 0

0 A112 0 00 A21 A 00 A31 A32 A33

, B =

00BB3

,C =

[0 C1|L⊥− C 0

], D = D.

(6.2)

Since Σ is a minimal conservative dilation of Σ, L− = 0. In a similar way, we obtain⋂∞

n=0 AnD+ = 0.

Theorem 6.3 Any minimal conservative dilation Σ of a passive system Σ has the form constructed in Theo-rem 4.6.

P r o o f. Let Σ = (A,B,C,D;X ,U ,Y; κ) be any passive system, and let Σ = (A, B, C, D; X ,U ,Y; κ) bea minimal conservative dilation of Σ. Write X = D−⊕X ⊕D+ for some Hilbert spaces D±, and relative to thisdecomposition let the operators A, B, C, D be given as in (6.1). We first show that

AA∗D− = D−, A∗AD+ = D+. (6.3)

Since the system operator VeΣ is unitary,

A∗31A33 = 0, A∗32A33 = 0, A∗33A33 = I,

A11A∗11 = I, A21A

∗11 = 0, A31A

∗11 = 0.

It follows that for all d− ∈ D− and d+ ∈ D+,

AA∗

d−00

=

A11 0 0A21 A 0A31 A32 A33

A∗11d−00

=

d−00

,

A∗A

00d+

=

A∗11 A∗21 A∗310 A∗ A∗320 0 A∗33

00

A33d+

=

00d+

,yielding (6.3). By Theorem 6.2 we can assume that D− = `2−(E) and D+ = `2+(F) for some Hilbert spaces Eand F and that V+ = A|D+ and V− = A∗|D− are the canonical shift operators on these spaces.



Claim 1: A21V− = 0, A31V− = 0, and C1V− = 0.

For since AA∗D− = D−,V ∗− 0 0A21 A 0A31 A32 V+

V−d−00

∈ D−, d− ∈ D−,

and hence A21 and A31 annihilate V−D−, that is, A21V− = 0 and A31V− = 0. But then for all d− ∈ D−,

VeΣV−d−

000

=

V ∗− ∗ ∗ ∗A21 ∗ ∗ ∗A31 ∗ ∗ ∗C1 ∗ ∗ ∗

V−d−

000

=

d−00

C1V−d−

.Since V− and VeΣ are isometric, ‖C1V−d−‖Y = 0. Thus C1V− = 0.

Claim 2: A∗31V+ = 0, A∗32V+ = 0, and B∗3V+ = 0.

Since A∗AD+ = D+,V− A∗21 A∗310 A∗ A∗320 0 V ∗+

00

V+d+

∈ D+, d+ ∈ D+,

and therefore A∗31V+ = 0 and A∗32V+ = 0. Then for all d+ ∈ D+,

V ∗eΣ

00

V+d+

0

=

∗ ∗ A∗31 ∗∗ ∗ A∗32 ∗∗ ∗ V ∗+ ∗∗ ∗ B∗3 ∗

00

V+d+

0

=

00d+

B∗3V+d+

.Since V+ and V ∗eΣ are isometric, ‖B∗3V+d+‖U = 0. Therefore B∗3V+ = 0.

By the two claims, A21 = E1P−1, A31 = Q0GP−1, A32 = Q0F1, B3 = Q0F2, and C1 = E2P−1 for someoperators

E1 ∈ L(E ,X ), E2 ∈ L(E ,Y), G ∈ L(E ,F), F1 ∈ L(X ,F), F2 ∈ L(U ,F).

In other words, Σ has the form (4.3). Straightforward calculations show that the unitarity of VeΣ is equivalent tothat 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 dilationΣ = (A, B, C, D; X ,U ,Y; κ) for Σ is constructed from a unitary operator

[VΣ EF G

]=

A B E1

C D E2

F1 F2 G

:

XUE

→

XYF

(6.4)

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 proofof Theorem 4.6.

Assume first that Σ is a minimal conservative dilation of Σ. We prove that (6.4) is a Julia operator by provingthe equivalent relations kerE = 0 and kerF ∗ = 0. Set kerE = E and kerF ∗ = 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. 20], (6.4) acts as a unitary operator from E onto F, and its adjoint maps F onto E; moreover (6.4) and its



adjoint coincide with G and G∗ on E and F, respectively, by the definitions of these subspaces. The assertionfollows.

Write D− = D− ⊕D−• and D+ = D+• ⊕D+, where

D− = `2−(E), D−• = `2−(E•),D+• = `2+(E•), D+ = `2+(E).

Using the definitions of the operators V+, V−, P−1, and Q0 we easily obtain that V− = V−|D− ⊕ V−|D−•,V+ = V+|D−• ⊕ V+|D−, and

E1P−1D− = 0, F ∗1Q∗0D+ = 0,

Q0GP−1D− ⊆ D+, Q0GP−1D−• ⊆ D+•,

F ∗2Q∗0D+ = 0, E2P−1D− = 0.

Hence by (4.3), relative to the decomposition X = D− ⊕D−• ⊕X ⊕D+• ⊕D+,

A =

V ∗−|D− 0 0 0 0

0 V ∗−|D−• 0 0 00 ∗ A 0 00 ∗ ∗ V+|D+• 0∗ 0 0 0 V+|D+

, B =

00B∗0

,C =

[0 ∗ C 0 0

], D = D.

It is clear now that Σ is a dilation of a system

Σ =

∗ 0 0∗ A 0∗ ∗ ∗

, 0B∗

, [∗ C 0], 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 = 0and F = 0, and thus (6.4) is a Julia operator.

Conversely, assume that (6.4) is a Julia operator. Consider any conservative dilation Σ = (A, B, C, 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), VeΣ has theform

VeΣ =

A11 0 0 0 0 0A21 A22 0 0 0 0A31 A32 A 0 0 BA41 A42 A43 A44 0 B4

A51 A52 A53 A54 A55 B5

C1 C2 C 0 0 D

. (6.6)

Since the system operator

VbΣ =

A22 0 0 0A32 A 0 BA42 A43 A44 B4

C2 C 0 D



is also unitary A21, A31, A41, A52, A53, A54, C1, B5 are zero operators. So A has the form

A =

A11 0 0 0 00 A22 0 0 00 A32 A 0 00 A42 A43 A44 0A51 0 0 0 A55

relative to the decomposition (6.5). The subspace D′+ ⊕ D′′+ is an A-invariant subspace of X X , and it iscontained in the kernel of C = PYVeΣ|X . Every vector x ∈ D′+ ⊕D′′+ has the form

. . . , e−2, e−1, 0, f0, f1, f2, . . . ∈ `2−(E)⊕X ⊕ `2+(F).

Since Ax ∈ D′+ ⊕ D′′+, PX Ax = 0, and therefore by the form of A in (4.3), E1e−1 = 0. Since also Cx = 0,by the form of C in (4.3), E2e−1 = 0. Therefore e−1 ∈ kerE and e−1 = 0 since (6.4) is a Julia operator.Thus e−n = 0 for every n ≥ 1, since D′+ ⊕ D′′+ is A-invariant. It follows that D′+ ⊕ D′′+ ⊆ D+. AnalogouslyD′′− ⊕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

V− = A∗|D− =[A∗11 00 A∗22

]is a simple isometry on D− = `2−(E) with the reducing subspaces D′′− and D′−. Hence there is a decompositionE = E1⊕E2 such thatD′′− = `2−(E1) andD′− = `2−(E2). Recalling that the operatorsA31 andC1 in (6.6) are zero,we see that PX AD′′− = 0 and CD′′− = 0; hence by (4.3), E1E1 = 0 and E2E1 = 0. Thus EE1 = 0.Since (6.4) is a Julia operator, E1 = 0 and hence D′′− = 0. In a similar way, D′′+ = 0. 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 conservativedilation. Below we give an example with κ = 0 of a minimal passive system which does not have a simpleconservative 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 = (A, 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 ΘeΣ(z) belongs to Sκ(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 ΘΣ(z) = ΘeΣ(z).

Theorem 6.6 A simple conservative dilation Σ = (A, 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 = (A, B, C, D; X ,U ,Y; κ), which is in turna dilation of Σ. Then X = D− ⊕X ⊕D+ and X = D− ⊕ X ⊕ D+, and relative to the latter decomposition theoperators A, B, C, D have the form

A =

A11 0 0A21 A 0A31 A32 A33

, B =

0B

B3

,C =

[C1 C 0

], D = D.

(6.7)

Since both operators

MeΣ =

[A B

C D

]and MbΣ =

[A B

C D

]Copyright line will be provided by the publisher


are unitary, A21 = 0, A31 = 0, A32 = 0, C1 = 0, B3 = 0. Then∨n≥0AnBU ⊆ X and∨n≥0A

∗nC∗Y ⊆ X ,so Σ is simple only if D− = 0 and D+ = 0.

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; 0) whose system operator

Vσ =[a bc d

]is a contraction such that abc 6= 0, |a| < 1, and I − V ∗σ Vσ and I − VσV

∗σ are invertible. Then σ is a minimal

passive system. We show that σ does not have a simple conservative dilation. Argue by contradiction, assumingthat σ has a simple conservative dilation Σ = (A, B, C, D; X ,U ,Y; 0). By Corollary 6.7, Σ is constructedfrom a Julia operator (6.4) for Vσ . Since a Julia operator is essentially unique, we can choose E = F = C2,E = (I − VσV

∗σ )1/2, F = (I − V ∗σ Vσ)1/2, and G = −V ∗σ . We construct a nonzero vector

x =

. . . , e−2, e−1x

f0, f1, . . .

(6.8)

in X = `2−(E)⊕X ⊕ `2+(F) which is orthogonal to all vectors

AnBu =

0anbu

F1an−1bu, . . . , F1bu, F2u, 0, 0, . . .

(6.9)

and

A∗nC∗y =

. . . , 0, 0, E∗2y,E∗1 cy, . . . , E∗1 an−1cyancy

0

, (6.10)

where u ∈ U , y ∈ Y , and n ≥ 0. Since dimE∗1X = dimF1X = 1, there exist unit vectors ϕ ∈ E E∗1Xand ψ ∈ F F1X . To construct x, we choose x = 1 in (6.8) and seek square summable scalars αn∞n=0 andβn∞n=0 such that the vectors e−n−1 = αnϕ and fn = βnψ meet the required orthogonality conditions. It issufficient to take u = 1 and y = 1 in (6.9) and (6.10). Without difficulty, we find that such scalars are uniquelydetermined by the orthogonality conditions and given by

αn = − anc

〈ϕ,E∗21〉, and βn = − anb

〈ψ, F21〉, n ≥ 0.

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 Kreın-Langer factorizations

The cascade synthesis of the two systems Σ1 = (A1, B1, C1, D1;X1,U ,Y1; κ1) and Σ2 = (A2, B2, C2, D2;X2,Y1,Y; κ2) is a system Σ = Σ2 Σ1 which uses the output

Σ

u −−−−−> Σ1

y1−−−−−> Σ2 −−−−−−> y



from Σ1 as input for Σ2: for all n ≥ 0,

Σ1 :

x1(n+ 1) = A1x1(n) +B1u(n),

y1(n) = C1x1(n) +D1u(n),

Σ2 :

x2(n+ 1) = A2x2(n) +B2u(n),

y(n) = C2x2(n) +D2u(n),

and hence

Σ :

[x1(n+ 1)x2(n+ 1)

]=

[A1 0B2C1 A2

] [x1(n)x2(n)

]+

[B1

B2D1

]u(n),

y(n) =[D2C1 C2

] [x1(n)x2(n)

]+D2D1u(n).

Therefore Σ = (A,B,C,D;X ,U ,Y; κ), where X = X1 ⊕X2, κ = κ1 + κ2, andA =

[A1 0B2C1 A2

], B =

[B1

B2D1

],

C =[D2C1 C2

], D = D2D1.

(7.1)

In particular, AX2 ⊆ X2. We note the following elementary properties:

(i) If Σ = Σ2 Σ1, then ΘΣ(z) = ΘΣ2(z)ΘΣ1(z).

(ii) If Σ = Σ2 Σ1, then Σ∗ = Σ∗1 Σ∗2.

(iii) Given a third system Σ3, Σ3 (Σ2 Σ1) = (Σ3 Σ2) Σ1 provided that the operations are meaningful.

(iv) If Σ = Σ2 Σ1 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 = Σ2 Σ1 where Σ1 and Σ2 are passive (conservative) systems, then Σ is a passive(conservative) system.

P r o o f. Use the identity A1 0 B1

B2C1 A2 B2D1

D2C1 C2 D2D1

=

IX1 0 00 A2 B2

0 C2 D2

A1 0 B1

0 IX2 0C1 0 D1

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 = Σ2 Σ1, where Σ2 = (A2, B2, C2, D2;X2,U ,Y; 0) is a simple conservative system with a Hilbertstate space, and Σ1 = (A1, B1, C1, D1;X1,U ,U ; κ) is a simple conservative system whose state space is aκ-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; 0) is a simple

conservative system with a Hilbert state space.



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 + ρz − w

1− wzP,

where P ∈ L(U) is a rank-one projection operator, w ∈ D, and |ρ| = 1. A right Kreın-Langer factorizationof a function S(z) in Sκ(U ,Y) is a representation

S(z) = Sr(z)Br(z)−1, (7.2)

where Sr(z) belongs to S(U ,Y) and Br(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 Sr(w)u = 0 and Br(w)u = 0for some w in D and u ∈ U , then u = 0. A left Kreın-Langer factorization of S(z) is a representation

S(z) = B`(z)−1S`(z), (7.3)

where S`(z) belongs to S(U ,Y) andB`(z) is a Blaschke-Potapov product of degree κ with values in L(Y) whichis invertible at the origin; the factorization is coprime in the sense that if S`(w)∗y = 0 andB`(w)∗y = 0 for somew in D and y ∈ Y , then y = 0. Left and right Kreın-Langer factorizations are essentially unique. Conversely, anarbitrary product (7.2) in which Sr(z) belongs to S(U ,Y) and Br(z) is a Blaschke-Potapov product of degree κwith values in L(U) which is invertible at the origin represents a function in Sκ′(U ,Y) for some κ′ ≤ κ, andκ′ = κ if the factorization is coprime. A parallel assertion holds for products (7.3). These results are due toKreın and Langer [19]; an account is given in [1, Section 4.2].

Theorem 7.3 Suppose that the function Θ(z) ∈ Sκ(U ,Y) has the right Kreın-Langer factorization Θ(z) =Θr(z)br(z)−1. Let

Σr− = (Ar

−, Br−, C

r−, D

r−;X r

−,U ,U ; κ),

Σr+ = (Ar

+, Br+, C

r+, D

r+;X r

+,U ,Y; 0),

be simple conservative systems such that ΘΣr+(z) = Θr(z) and ΘΣr

−(z) = br(z)−1. Then Σr = Σr

+ Σr− is a

simple conservative system with transfer function Θ(z).Lemma 7.4 Let Σ = Σ2 Σ1, where Σ1 = (A1, B1, C1, D1;X1,U ,Y1; κ1) and Σ2 = (A2, B2, C2, D2;

X2,Y1,Y; κ2). Then (X cΣ ∨ X o

Σ)⊥ consists of all x = x1 ⊕ x2 in X = X1 ⊕X2 such thatΘΣ2(z)C1(I − zA1)−1x1 = −C2(I − zA2)−1x2,

ΘΣ1(z)∗B∗2(I − zA∗2)

−1x2 = −B∗1(I − zA∗1)−1x1,

(7.4)

in a neighborhood of the origin. Hence if the only vectors x1 ∈ X1 and x2 ∈ X2 which satisfy (7.4) are x1 = 0and x2 = 0, then Σ = Σ2 Σ1 is simple.

Lemma 7.5 Let Θ1(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 Θ1(z) is the transfer function of a simpleconservative system Σ1 = (A1, B1, C1, D1;X1,U ,U ; κ) for which the state space X1 is the κ-dimensionalPontryagin space H(Θ1) having reproducing kernel [I −Θ1(z)Θ1(w)∗]/(1− zw), andA1 : h(z) → h(z)− h(0)

z, B1 : u→ Θ1(z)−Θ1(0)

zu,

C1 : h(z) → h(0), D1 : u→ Θ1(0)u,(7.5)

for all h(z) in H(Θ1) and all u in U . The space X1 is the antispace of a Hilbert space, and the identity⟨h1(z)−h1(0)

z,h1(z)−h1(0)

z

⟩H(Θ1)

= 〈h1(z), h1(z)〉H(Θ1)− 〈h1(0), h1(0)〉U (7.6)

holds for all elements h1(z) of the space.



Proof of Lemma 7.4. Write Σ = (A,B,C,D;X ,U ,Y; κ), where X = X1 ⊕ X2, κ = κ1 + κ2, andA,B,C,D are given by (7.1). It is easy to see that the definitions of the subspaces X c

Σ and X oΣ in Section 3

can be rewritten in the form

X cΣ =∨z∈Ω (I − zA)−1B U and X o

Σ =∨z∈Ω (I − zA∗)−1C∗Y,

where Ω is any small neighborhood of the origin. Therefore x = x1 ⊕ x2 is orthogonal to X cΣ ∨ X o

Σ if and onlyif for all z ∈ Ω, B∗(I − zA∗)−1x = 0 and C(I − zA)−1x = 0, or equivalently (by (7.1))

[D2C1 C2

] [(I − zA1)−1 0

z(I − zA2)−1B2C1(I − zA1)−1 (I − zA2)−1

] [x1

x2

]= 0,

[B∗1 D∗1B

∗2

] [(I − zA∗1)

−1 z(I − zA∗1)−1C∗1B

∗2(I − zA∗2)

−1

0 (I − zA∗2)−1

] [x1

x2

]= 0,

Expanding these identities and simplifying by means of the relations

ΘΣ2(z) = D2 + zC2(I − zA2)−1B2,

ΘΣ1(z)∗ = D∗1 + zB∗1(I − zA∗1)

−1C∗1 ,

we obtain (7.4).

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 Y1 = U , and write

Σ1 = (A1, B1, C1, D1;X1,U ,Y1; κ1) = Σr−, κ1 = κ,

Σ2 = (A2, B2, C2, D2;X2,Y1,Y; κ2) = Σr+, κ2 = 0.

By Theorem 7.1, Σr = Σr+ Σr

− = Σ2 Σ1 is conservative. It has transfer functionΘΣr (z) = ΘΣ2(z)ΘΣ1(z) =Θr(z)br(z)−1 = Θ(z) at all points in D where the functions are defined.

The main problem is to show that Σr is simple, and for this we use Lemma 7.4. Let x1 ∈ X1 and x2 ∈ X2

satisfy (7.4). Since all simple conservative realizations are equivalent by Theorem 4.1, without loss of generalitywe can assume that Σ1 is given as in Lemma 7.5 with Θ1(z) = br(z)−1. Thus X1 = H(b−1

r ) is the antispace ofa κ-dimensional Hilbert space, and for every h(z) in the space,

C1(I − wA1)−1 : h(z) → h(w) (7.7)

for allw in a neighborhood of the origin [1, p. 89]. We remark also that since κ2 = 0,A2 is a contraction operatoron the Hilbert space X2 and therefore (I − zA2)−1 exists for z in the unit disk D.

We first show that x1 = 0. Argue by contradiction, assuming x1 6= 0. LetM be the subspace of elements h(z)in H(b−1

r ) such that Θr(z)h(z) ∈ Hol(D), that is, apart from removable singularities Θr(z)h(z) is holomorphicon D. Then x1(z) belongs to M by (7.7) and the first relation in (7.4), and so M 6= 0. The subspace M isinvariant under A1, since if Θr(z)h(z) ∈ Hol(D), then

Θr(z)h(z)− h(0)

z=Θr(z)h(z)−Θr(z)h(0)

z∈ Hol(D).

Since M is finite dimensional, A1|M has an eigenvalue β1. Thus M contains an element of the form

h1(z) =u

1− β1z, ‖u‖U = 1.

The identity (7.6), together with the fact that X1 is the antispace of a Hilbert space, implies that |β1| > 1. Setα1 = 1/β1. By the definition of M, Θr(z)h1(z) ∈ Hol(D), and this is only possible if Θr(α1)u = 0. Writing

b1(z) = I − P1 +z − α1

1− α1zP1, P1 = 〈·, u〉Uu,



we therefore have Θr(z)b1(z)−1 ∈ S(U ,U). An argument in [1, p. 142] shows that

br(z)−1 = b1(z)−1S1(z),

where S1(z) belongs to Sκ−1(U ,U). But then [1, Theorem 4.1.1]

Θ(z) =[Θr(z)b1(z)−1

]S1(z) ∈ Sκ′(U ,Y), κ′ ≤ κ − 1,

which contradicts our assumption that Θ(z) ∈ Sκ(U ,Y). Therefore M = 0 and hence x1 = 0.Returning to (7.4), we see that C2(I − zA2)−1x2 = 0 and B∗2(I − zA∗2)

−1x2 = 0 in a neighborhood of theorigin. Since Σ2 is simple, it follows that x2 = 0. Hence by Lemma 7.4, Σ is simple, as was to be shown.

A parallel result holds for left Kreın-Langer factorizations.

Theorem 7.6 Suppose that the function Θ(z) ∈ Sκ(U ,Y) has the left Kreın-Langer factorization Θ(z) =b`(z)−1Θ`(z). Let

Σ`+ = (A`

+, B`+, C

`+, D

`+;X `

+,U ,Y; 0),

Σ`− = (A`

−, B`−, C

`−, D

`−;X `

−,Y,Y; κ),

be simple conservative systems such that ΘΣ`+(z) = Θ`(z) and ΘΣ`

−(z) = b`(z)−1. Then Σ` = Σ`

− Σ`+ 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) ∈ Sκ(Y,U) (for example,see [1, p. 68]), and we can define a right Kreın-Langer factorization of Θ(z) by setting

Θ(z) = Θr(z)br(z)−1, Θr(z) = Θ`(z), br(z) = b`(z).

Then Σ` ∗+ and Σ` ∗

− are simple conservative systems with transfer functions

ΘΣ` ∗+

(z) = ΘΣ`+(z) = Θ`(z) = Θr(z),

ΘΣ` ∗−

(z) = ΘΣ`−(z) = b`(z)−1 = br(z)−1.

By Theorem 7.3, Σ` ∗+ Σ`∗

− is a simple conservative system such that ΘΣ` ∗+ Σ` ∗

−(z) = Θr(z)br(z)−1 = Θ(z).

It follows that Σ` = Σ`− Σ`

+ is a simple conservative system whose transfer function is given by ΘΣ(z) =Θ(z).

Every contraction operator on a Pontryagin space admits semi-definite invariant subspaces (see Section 2). Weobtain 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 systemthe statement of Theorem 7.7 then follows from this fact and an embedding into a simple conservative system asin our proof below. The authors learned of this connection from the referee’s report, and they thank the refereefor his remark.

Lemma 7.8 The main operator of a simple passive system Σ = (A,B,C,D;X ,U ,Y; κ) has no eigenvalueof modulus one.



Proof of Lemma 7.8. By embedding Σ in a simple conservative system which has the same state space andmain operator (see Theorem 4.3), without loss of generality we can assume that Σ is conservative, that is, thesystem operator VΣ is unitary.

Suppose x ∈ X , |λ| = 1, and Ax = λx. Then

VΣ

[x0

]=

[A BC D

] [x0

]=

[λxCx

].

Since VΣ is unitary and |λ| = 1,

〈x, x〉X = |λ|2〈x, x〉X + ‖Cx‖2Y = 〈x, x〉X + ‖Cx‖2Y , (7.8)

and hence Cx = 0. Therefore CAnx = λnCx = 0, n ≥ 0. Since VΣ is unitary,[x0

]= V ∗Σ

[λx0

].

This yields A∗x = λx and B∗x = 0, and therefore B∗A∗nx = λnB∗x = 0, n ≥ 0. Hence since Σ is simple,x = 0.

Proof of Theorem 7.7. As in the proof of Lemma 7.8, we can assume that Σ is a simple conservative system.For the existence part of (i), let Θ(z) = ΘΣ(z) ∈ Sκ(U ,Y) have right Kreın-Langer factorization Θ(z) =

Θr(z)br(z)−1. By Theorem 7.3, we can assume that Σ = Σr+ Σr

− in the notation of that result. In particular,X = X r

− ⊕ X r+, and we can choose X+ = X r

+ and X− = X r−. The existence part of (ii) is handled in the same

way, but using Theorem 7.6 in place of Theorem 7.3.We prove uniqueness in (ii). By Theorem 7.6 we can assume that Σ = Σ`

− Σ`+, where Σ`

+ and Σ`− are simple

conservative systems such that

ΘΣ`+(z) = Θ`(z), ΘΣ`

−(z) = b`(z)−1,

for some left Kreın-Langer factorization Θ(z) = b`(z)−1Θ`(z) of the function Θ(z) = ΘΣ(z) ∈ Sκ(U ,Y).Then in the notation of Theorem 7.6,

A =[A`

+ 0∗ A`

−

]relative to the decomposition X = X `

+ ⊕X `−.

Consider any fundamental decomposition X = X ′+ ⊕ X ′− as in (ii). We show that X ′− ⊆ X `−, and hence

X ′− = X `− because the two spaces have the same finite dimension. It is sufficient to show that any nonzero root

vector of A|X ′− belongs to X `−. Let x ∈ X ′−, x 6= 0, and assume that (A−λI)n x = 0 for some complex number

λ and positive integer n. By Lemma 7.8, |λ| 6= 1; since A|X ′− is a contraction operator on the antispace of aHilbert space, |λ| > 1. Write write x = x+ ⊕ x−, where x± ∈ X `

±. Then[(A`

+ − λI)n 0∗ (A`

− − λI)n

] [x+

x−

]=

[00

].

It follows that (A+ − λI)nx+ = 0. Since A`+ is a contraction operator on a Hilbert space and |λ| > 1, x+ = 0.

Hence x = x− ∈ X `−, as was to be shown.

Uniqueness in (i) can be proved similarly or by applying what has been shown to the adjoint system Σ∗.

8 Application to simple conservative systems

If a simple system Σ is represented as the cascade synthesis Σ = Σ2 Σ1 of two systems Σ1 and Σ2, it is wellknown that Σ1 and Σ2 are simple (for example, see [1, Theorem 1.2.1]). The converse is false in general, evenfor conservative systems and Hilbert state spaces. In the case of Hilbert state spaces, the condition for the cascade



synthesis Σ = Σ2 Σ1 of two simple conservative systems Σ1 and Σ2 to be simple is usually expressed in termsof the factorization ΘΣ(z) = ΘΣ2(z)ΘΣ1(z) of the corresponding transfer functions: in standard terminology,the factorization is said to be regular if Σ is simple. This notion originates in the study of invariant subspaces.Invariant subspaces and regular factorizations are studied from the point of view of a functional model in Sz.-Nagy and Foias [24, Chapter VII]. In the latter work, necessary and sufficient conditions for a factorization to beregular are derived from an analysis of the unitary dilation of a contraction operator; another account is given inthe survey of Ball and Cohen [7]. The theory of regular factorizations was put into the framework of operatornodes, again in the case of Hilbert state spaces, by M. S. Brodskiı [10], based in part on previous works by M. S.Brodskiı and V. M. Brodskiı [11] and V. M. Brodskiı and P. A. Svarcman [12]. Theorem 8.1 below generalizes[10, Theorem 8.2] to Pontryagin state spaces. We do not, however, go so far as to allow the input space Uand output space Y to be Pontryagin spaces, even with κ = 0. The latter situation occurs in the model theoryof noncontractions (for example, see Kuzhel [20], Ball [6], and McEnnis [22]); Clark [13] defines a notion ofregular factorization in such a situation, but this notion does not overlap with our account except in the classicalcase of Hilbert state, input, and output spaces.

Recall that every function S(z) ∈ Sκ(U ,Y) has strong boundary values S(ζ) a.e. on the circle |ζ| = 1, andthese boundary values are contraction operators. Set

∆S(ζ) = [I − S(ζ)∗S(ζ)]1/2, ∆∗S(ζ) = [I − S(ζ)S(ζ)∗]1/2,

a.e. for |ζ| = 1. Consider a cascade synthesis Σ = Σ2 Σ1 of two simple conservative systems having transferfunctions Θ1(z) = ΘΣ1(z) and Θ2(z) = ΘΣ2(z). Define an operator

V : ∆Θ2Θ1L2(U) → ∆Θ2L

2(Y1)⊕∆Θ1L2(U)

by setting

V : ∆Θ2Θ1(ζ)u(ζ) → ∆Θ2(ζ)Θ1(ζ)u(ζ)⊕∆Θ1(ζ)u(ζ)

for every u(ζ) in L2(U). We easily check that V is isometric. For fixed ζ on the circle |ζ| = 1 with the exceptionof a set of measure zero, define

Vζ : ∆Θ2Θ1(ζ)U → ∆Θ2(ζ)Y1 ⊕∆Θ1(ζ)U

by setting

Vζ : ∆Θ2Θ1(ζ)u→ ∆Θ2(ζ)Θ1(ζ)u⊕∆Θ1(ζ)u

for every u ∈ U . Then Vζ is isometric a.e. for |ζ| = 1.Theorem 8.1 Let Σ = Σ2 Σ1, where Σ1 = (A1, B1, C1, D1;X1,U ,Y1; κ1) and Σ2 = (A2, B2, C2, D2;

X2,Y1,Y; κ2) are simple conservative systems with transfer functions Θ1(z) = ΘΣ1(z) and Θ2(z) = ΘΣ2(z).Then Σ is simple if and only if

(i) Θ2(z)Θ1(z) ∈ Sκ1+κ2(U ,Y), and

(ii) one of the following conditions is satisfied

(a) ∆Θ2(ζ)Y1 ∩∆∗Θ1(ζ)Y1 = 0 a.e. for |ζ| = 1;

(b) the equality

Θ1(ζ)∗∆Θ2(ζ)y1(ζ) + ∆Θ1(ζ)u(ζ) = 0

with y1(ζ) ∈ ∆Θ2L2(Y1) and u(ζ) ∈ ∆Θ1L

2(U) holds a.e. for |ζ| = 1 only for the zero elements ofthe spaces;

(c) V is unitary;

(d) Vζ is unitary a.e. for |ζ| = 1.



In this case, all of the conditions (a)–(d) in (ii) hold.In the case of Hilbert state spaces, that is, when κ1 = κ2 = 0, condition (i) in Theorem 8.1 is automatically

satisfied. This case of Theorem 8.1 is known from [10, Theorem 8.2]. For reference purposes, we state the resultin the following form.

Lemma 8.2 Let Σ+ = Σ2+ Σ1+, where Σ1+ and Σ2+ are simple conservative systems with transferfunctions Θ1r(z) and Θ2`(z) and input and output spaces U ,Y1 and Y1,Y , respectively. Define operators V+

and Vζ+ for Σ+ = Σ2+ Σ1+ in the same way as V and Vζ are defined above for Σ = Σ2 Σ1. Then thefollowing statements are equivalent:

(a) ∆Θ2`(ζ)Y1 ∩∆∗Θ1r (ζ)Y1 = 0 a.e. for |ζ| = 1;

(b) the equality

Θ1r(ζ)∗∆Θ2`(ζ)y1(ζ) + ∆Θ1r (ζ)u(ζ) = 0

with y1(ζ) ∈ ∆Θ2`L2(Y1) and u(ζ) ∈ ∆Θ1rL

2(U) holds a.e. for |ζ| = 1 only for the zero elements of thespaces;

(c) V+ is unitary;

(d) Vζ+ is unitary a.e. for |ζ| = 1;

(e) Σ+ is simple.

Lemma 8.3 Let Σ = Σ2 Σ1, where Σ1 and Σ2 are simple conservative systems whose input and outputspaces are the same Hilbert space U . If the transfer functions of Σ1 and Σ2 have the form b1(z)−1 and b2(z)−1,respectively, where b1(z) and b2(z) are finite Blaschke products which are invertible at the origin, then Σ is asimple conservative system whose transfer function is b2(z)−1b1(z)−1.

Proof of Lemma 8.3. Everything is clear from Section 7 except that Σ is simple. To see this, construct thecomplementary systems Σ′′, Σ′′1 , and Σ′′2 as in Theorem 5.3. The state spaces of Σ1 and Σ2 are antispacesof Hilbert spaces by, for example, [1, Theorem A3, p. 197], and therefore the state spaces of Σ′′1 , Σ′′2 , andΣ′′ = Σ′′1 Σ′′2 are Hilbert spaces. By Theorem 5.3(iii), the transfer functions of Σ′′1 and Σ′′2 are b1(z) and b2(z),and these have unitary boundary values on the unit circle. Condition (a) in Lemma 8.2 is trivially satisfied for thecascade synthesis Σ′′ = Σ′′1 Σ′′2 , and therefore Σ′′ is simple. Hence by Theorem 5.3(ii), Σ is simple.

Lemma 8.4 Let Σ = Σ2 Σ1, where Σ1 = (A1, B1, C1, D1;X1,U ,Y1; κ1) and Σ2 = (A2, B2, C2, D2;X2,Y1,Y; κ2) are simple conservative systems with transfer functions Θ1(z) = ΘΣ1(z) and Θ2(z) = ΘΣ2(z).Let

Θ1(z) = Θ1r(z)b1r(z)−1 and Θ2(z) = b2`(z)−1Θ2`(z) (8.1)

be right and left Kreın-Langer factorizations, respectively. Choose choose simple conservative systems Σ1+ andΣ2+ such that Θ1r(z) = ΘΣ1+(z) and Θ2`(z) = ΘΣ2+(z). Then Σ is simple if and only if

(i) Θ2(z)Θ1(z) ∈ Sκ1+κ2(U ,Y), and

(ii) Σ2+ Σ1+ is simple.

Proof of Lemma 8.4. Choose simple conservative systems Σ2− and Σ1− such that b1r(z)−1 = ΘΣ1−(z) andb2`(z)−1 = ΘΣ2−(z). Since a simple conservative realization is essentially unique by Theorem 4.1, by Theorems7.3 and 7.6 without loss of generality we can assume that Σ1 = Σ1+ Σ1− and Σ2 = Σ2− Σ2+.

Assume that Σ is simple. Since Σ = Σ2− (Σ2+ Σ1+) Σ1−, it follows that Σ2+ Σ1+ is simple, thatis, (ii) holds. Since ΘΣ(z) = Θ2(z)Θ1(z) where Σ is a simple conservative system with state space X1 ⊕ X2,Θ2(z)Θ1(z) = ΘΣ(z) ∈ Sκ1+κ2(U ,Y), and thus (i) holds.

Conversely, assume that (i) and (ii) hold. To see that Σ = Σ2− Σ2+ Σ1+ Σ1− is simple, we firstargue that Σ = Σ2− (Σ2+ Σ1+) is simple. By (ii) and Theorem 7.6 it is sufficient to show that Θ(z) =



b2`(z)−1[Θ2`(z)Θ1r(z)] is a left Kreın-Langer factorization of Θ(z) = ΘbΣ(z). In fact, since b2`(z) has degreeκ2, Θ(z) ∈ Sκ′(U ,Y) where κ′ ≤ κ2. Since b1r(z) has degree κ1, it follows that

Θ2(z)Θ1(z) = Θ(z)b1r(z)−1 ∈ Sκ′′(U ,Y),

where κ′′ ≤ κ′+κ1 ≤ κ2 +κ1. By (i), κ′′ = κ1 +κ2, and so κ′ = κ2. Thus Θ(z) = b2`(z)−1[Θ2`(z)Θ1r(z)]is a left Kreın-Langer factorization, and hence Σ = Σ2− (Σ2+ Σ1+) is simple by Theorem 7.6.

To see that Σ = Σ Σ1− is simple, first choose a right Kreın-Langer factorization Θ(z) = Θr(z)br(z)−1. ByTheorem 7.3, we may assume that Σ = Σ+ Σ−, where Σ+ and Σ− are simple conservative systems havingtransfer functions Θr(z) and br(z)−1. Thus Σ = Σ+ (Σ− Σ1−). In view of Lemma 8.3, to prove that Σ issimple, it is sufficient to show that

Θ2(z)Θ1(z) = Θr(z)[b1r(z)br(z)]−1

is a right Kreın-Langer factorization. This is clear because by (i), Θ2(z)Θ1(z) ∈ Sκ1+κ2(U ,Y) and b1r(z)br(z)is a Blaschke product of degree κ1 + κ2. Hence Σ is simple by Theorem 7.3.

Proof of Theorem 8.1. We use the Kreın-Langer factorizations (8.1) and notation of Lemma 8.4. By thatlemma, Σ is simple if and only if Θ2(z)Θ1(z) ∈ Sκ1+κ2(U ,Y) and Σ2+ Σ1+ is simple. Thus to complete theproof, it is sufficient to show that the four conditions (a)–(d) in Theorem 8.1(ii) are equivalent to their counterparts(a)–(d) in Lemma 8.2. For this we use the relations

∆Θ2(z) = ∆Θ2`(ζ), ∆∗Θ1(z) = ∆∗Θ1r

(ζ), (8.2)

∆Θ1(ζ) = b1r(ζ)∆Θ1r(ζ)b1r(ζ)−1, (8.3)

which hold a.e. for |ζ| = 1.(a) The equivalence of the conditions (a) is immediate from (8.2).(b) Assume that condition (b) in Lemma 8.2 holds. Suppose y1(ζ) ∈ ∆Θ2L

2(Y1), u(ζ) ∈ ∆Θ1L2(U), and

Θ1(ζ)∗∆Θ2(ζ)y1(ζ) + ∆Θ1(ζ)u(ζ) = 0 a.e. Then

b1r(ζ)−1[Θ1(ζ)∗∆Θ2(ζ)y1(ζ) + ∆Θ1(ζ)u(ζ)

]= 0 a.e.,

and with the aid of (8.2) and (8.3) we obtain

Θ1r(ζ)∗∆Θ2`(ζ)y1(ζ) + ∆Θ1r (ζ)u(ζ) = 0 a.e.

where y1(ζ) ∈ ∆Θ2L2(Y1) = ∆Θ2`

L2(Y1) and

u(ζ) = b1r(ζ)−1u(ζ) ∈ b1r(ζ)−1(∆Θ1L

2(U))

= ∆Θ1rL2(U).

Thus y1(ζ) = 0 a.e. and u(ζ) = 0 a.e., and hence condition (b) in Theorem 8.1 is satisfied. These steps arereversible, and so the conditions (b) are equivalent.

(c) In a routine way, we verify that an element y1(ζ)⊕ u(ζ) of ∆Θ2L2(Y1)⊕∆Θ1L

2(U) is orthogonal to therange of V if and only if the element y1(ζ) ⊕ b1r(ζ)−1u(ζ) of ∆Θ2`

L2(Y1) ⊕∆Θ1rL2(U) is orthogonal to the

range of V+, which yields the equivalence of the conditions (c).(d) The equivalence of the conditions (d) is a pointwise version of the preceding argument.Since we assume the Hilbert space form of the theorem as stated in Lemma 8.2, the result follows.

9 The classes Pκ00 and Cκ

00

A system Σ is said to be bi-stable if the main operator A satisfies An s→ 0 and A∗n s→ 0. The class of contractivebi-stable operators in the Hilbert space case is denoted C00. When state spaces are Pontryagin spaces, it is notpossible to define stability in exactly the same way because in this case there are eigenvalues λ such that |λ| > 1.We introduce a corresponding notion which says, roughly, that states that can be stable are stable.



Definition 9.1 Let Σ = (A,B,C,D;X ,U ,Y; κ) be a simple passive system, and let X = X+ ⊕ X− be theunique fundamental decomposition in Theorem 7.7(i) such that AX+ ⊆ X+.

(i) We say that Σ belongs to the class Pκ00 if A|X+ ∈ C00.

(ii) We say that Σ belongs to the class Cκ00 if Σ is conservative Σ ∈ Pκ

00.

Theorem 9.2 A simple conservative system Σ = (A,B,C,D;X ,U ,Y; κ) belongs to Cκ00 if and only if

ΘΣ(z) ∈ Uκ(U ,Y).

P r o o f. The case κ = 0 follows from [24, Proposition 3.5, p. 257]. In the general case, by Theorem 7.3 wecan assume that Σ = Σ2 Σ1, where Σ1 and Σ2 are simple conservative systems such that ΘΣ2(z) = Θr(z) andΘΣ1(z) = br(z)−1 for a Kreın-Langer factorization ΘΣ(z) = Θr(z)br(z)−1. Then A|X+ is the main operatorof Σ2; by the case κ = 0, A|X+ ∈ C00 if and only if Θr(z) ∈ Uκ(U ,Y). Since br(z)−1 has unitary boundaryvalues, Θr(z) ∈ Uκ(U ,Y) if and only if ΘΣ(z) ∈ Uκ(U ,Y), and the result follows.

Theorem 9.3 Let Σ = (A,B,C,D;X ,U ,Y; κ) be a simple passive system, and let X = X+ ⊕ X− andX = X ′+⊕X ′− be the unique fundamental decompositions in Theorem 7.7 such thatAX+ ⊆ X+ andAX ′− ⊆ X ′−.Then the conditions

A|X+ ∈ C00 and A∗|X ′+ ∈ C00

are equivalent. Hence if Σ belongs to Pκ00 or Cκ

00, so does Σ∗.

P r o o f. Since we can embed any simple passive system into a simple conservative system having the samestate space and main operator by Theorem 4.3, we can assume that Σ is conservative. By Theorem 9.2, A|X+ ∈C00 if and only ifΘΣ(z) ∈ Uκ(U ,Y), or equivalentlyΘΣ∗(z) = ΘΣ(z)∗ ∈ Uκ(Y,U). By Theorem 9.2 appliedto Σ∗, the last condition is equivalent to A∗|X ′+ ∈ C00.

Theorem 9.4 Let Σ = (A,B,C,D;X ,U ,Y; κ) be a simple passive system with transfer function ΘΣ(z) ∈Uκ(U ,Y). Then Σ is conservative.

In the case κ = 0, this result is Theorem 1 in [3]. It follows from Theorem 10.2 that the system Σ inTheorem 9.4 is minimal.

P r o o f. Consider a Julia embedding of Σ into Σ = (A, B, C, D;X , U , Y; κ). Write the system operator andtransfer function for Σ in the forms (3.5), (3.6), and (3.7). By (3.6),

ΘeΣ(z) =[E2 DG F2

]+ z

[CF1

](I − zA)−1

[E1 B

]=

[E2 + zC(I − zA)−1E1 D + zC(I − zA)−1BG+ zF1(I − zA)−1E1 F2 + z + F1(I − zA)−1B

],

and so

Θ11(z) = E2 + zC(I − zA)−1E1, Θ12(z) = D + zC(I − zA)−1B,

Θ21(z) = G+ zF1(I − zA)−1E1, Θ22(z) = F2 + zF1(I − zA)−1B.

Since the values of Θ(ζ) are contractive a.e. on the unit circle,[IbU 00 IU

]−

[Θ11(ζ)∗ Θ21(ζ)∗

Θ12(ζ)∗ Θ22(ζ)∗

] [Θ11(ζ) Θ12(ζ)Θ21(ζ) Θ22(ζ)

]≥ 0,

[IY 00 IbY

]−

[Θ11(ζ) Θ12(ζ)Θ21(ζ) Θ22(ζ)

] [Θ11(ζ)∗ Θ21(ζ)∗

Θ12(ζ)∗ Θ22(ζ)∗

]≥ 0,



and hence

IU −Θ12(ζ)∗Θ12(ζ)−Θ22(ζ)∗Θ22(ζ) ≥ 0,

IY −Θ11(ζ)Θ11(ζ)∗ −Θ12(ζ)Θ12(ζ)∗ ≥ 0,

a.e. Since Θ12(z) = ΘΣ(z) ∈ Uκ(U ,Y) by assumption, it follows that Θ11(ζ) = 0 and Θ22(ζ)∗ = 0 a.e. for|ζ| = 1 and Θ11(z) ≡ 0 and Θ22(z) ≡ 0 on D. Hence

E2 = 0, CAkE1 = 0, k ≥ 0, (9.1)

F2 = 0, F1AkB = 0, k ≥ 0. (9.2)

In particular, E1U ⊥ X oΣ and F ∗1 Y ⊥ X c

Σ. We shall show that E1U ⊥ X cΣ and F ∗1 Y ⊥ X o

Σ and hence, becauseΣ is simple, E1 = 0 and F ∗1 = 0. Granting this, we obtain U ⊆ kerE1 ∩ kerE2 = kerE = 0 andY ⊆ kerF ∗1 ∩ kerF ∗2 = kerF ∗ = 0, and therefore Σ = Σ is conservative.

Thus to complete the proof, it remains to show that

B∗A∗kE1 = 0 and CAkF ∗1 = 0, k ≥ 0. (9.3)

From the relationsA∗ C∗ F ∗1B∗ D∗ F ∗2E∗1 E∗2 G∗

A B E1

C D E2

F1 F2 G

=

IX 0 00 IU 00 0 IbU

, (9.4)

A B E1

C D E2

F1 F2 G

A∗ C∗ F ∗1B∗ D∗ F ∗2E∗1 E∗2 G∗

=

IX 0 00 IY 00 0 IbY

, (9.5)

we obtain

B∗E1 +D∗E2 + F ∗2G = 0, CF ∗1 +DF ∗2 + E2G∗ = 0.

Since E2 = 0 and F2 = 0, (9.3) holds with k = 0. By (9.4) and (9.5) we also have

A∗E1 + C∗E2 + F ∗1G = 0, AF ∗1 +BF ∗2 + E1G∗ = 0,

which gives A∗E1 = −F ∗1G and AF ∗1 = −E1G∗. Consequently, for k ≥ 1, from (9.1) and (9.2), we have

B∗A∗kE1 = B∗(A∗)k−1(A∗E1) = −B∗(A∗)k−1F ∗1G = −(F1Ak−1B)∗G = 0,

and

CAkF ∗1 = CAk−1(AF ∗1 ) = −CAk−1E1G∗ = 0.

We have verified (9.3) for all k ≥ 0, and the result follows.

Corollary 9.5 The class Cκ00 coincides with the set of all simple passive systems Σ in Pκ

00 such that ΘΣ(z) ∈Uκ(U ,Y).

P r o o f. This follows from Theorem 9.2 and Theorem 9.4.



10 Models for Cκ00

In this section we briefly describe two models that further illustrate Kreın-Langer factorizations and the cascadesynthesis of corresponding systems.

We first consider simple conservative systems in the class Cκ00 and use a special case of the canonical model

of de Branges and Rovnyak [9], as presented in an indefinite form in [1]. By [1, Theorem 2.2.1] every Θ(z) ∈Sκ(U ,Y) is the transfer function of an observable system ΣΘ = (T, F,G,H;H(Θ),U ,Y; κ), where H(Θ) isthe Pontryagin space with reproducing kernel [I −Θ(z)Θ(w)∗]/(1− zw), and

T : h(z) → h(z)− h(0)z

, F : u→ Θ(z)−Θ(0)z

u,

G : h(z) → h(0), H : u→ Θ(0)u,

for all h(z) in H(Θ) and u in U . The system operator VΣΘis coisometric, and thus ΣΘ is passive.

Theorem 10.1 (i) If Θ(z) ∈ Uκ(U ,Y) and Θ(z) = Θr(z)br(z)−1 = b`(z)−1Θ`(z) are right and leftKreın-Langer factorizations, then ΣΘ

∼= ΣΘr Σb−1r

and ΣΘ∼= Σb−1

` ΣΘ`

.(ii) IfΘ(z) ∈ Uκ(U ,Y), ΣΘ is a simple conservative system in Cκ

00. Conversely, if Σ is a simple conservativesystem in Cκ

00, then Σ ∼= ΣΘ where Θ(z) = ΘΣ(z).The relation ∼= of equivalence of systems is defined in Section 3.

P r o o f. (i) If Θ(z) ∈ Uκ(U ,Y), we deduce that Σb−1r,Σb−1

`,ΣΘr

,ΣΘ`are simple conservative systems (see

Lemma 7.5, [24, Proposition 3.5, p. 257] and [3, Proposition 2]). Write

Σb−1r

= (A1, B1, C1, D1;H(b−1r ),U ,Y; κ),

ΣΘr = (A2, B2, C2, D2;H(Θr),U ,Y; κ).

We exhibit an equivalence between the systems

ΣΘr Σb−1r

= (A,B,C,D;H(b−1r )⊕H(Θr),U ,Y; κ),

ΣΘ = (T, F,G,H;H(Θ),U ,Y; κ).

By [1, Theorem 4.2.3(4)], H(Θ) = H(Θr) ⊕ ΘrH(b−1r ). This decomposition allows us to define an operator

W : H(b−1r )⊕H(Θr) → H(Θ) by

W :[u(z)f(z)

]→ f(z) +Θr(z)u(z)

for all u(z) in H(b−1r ) and f(z) in H(Θr). Straightforward verifications show that T = WAW−1, F = WB,

G = CW−1, and H = D. Hence ΣΘ∼= ΣΘr Σb−1

r. We obtain ΣΘ

∼= Σb−1` ΣΘ`

by considering adjoints.(ii) If Θ(z) ∈ Uκ(U ,Y), ΣΘ is conservative by (i) and Theorem 7.1. Since ΣΘ is observable, it is simple.

Thus ΣΘ belongs to Cκ00 by Theorem 9.2. Conversely, if Σ is any simple conservative system in Cκ

00 andΘ(z) = ΘΣ(z), then Σ and ΣΘ are simple conservative systems having the same transfer function, and hencethey are equivalent by Theorem 4.1.

Theorem 10.2 Every simple conservative system Σ in Cκ00 is minimal.

A different criterion for a simple conservative system to be minimal is given in [14, Proposition 3.5].

P r o o f. By Theorem 10.1, we can take Σ = ΣΘ where Θ(z) ∈ Uκ(U ,Y). Set Θ(z) = Θ(z)∗. ThenΣ∗ = Σ eΘ is a simple conservative system in Cκ

00 by Theorem 9.2. We know that Σ = ΣΘ is observable. Weprove that Σ is controllable using a criterion for controllability given in Theorem 3.4.1 of [1]. Define an operatorΛ: H(Θ) → H(Θ) first on a fundamental set by

Λ:I −Θ(z)Θ(w)∗

1− zwy → Θ(z)− Θ(w)

z − wy



for all w in the domain of holomorphy of Θ(z) in the unit disk and all y in Y , and then by linearity and continuityon all of H(Θ). By Theorem 3.4.1 of [1], Σ = ΣΘ is controllable if and only if Λ has zero kernel. In fact, Λ isan isometry. For since the system operator for Σ∗ = Σ eΘ is unitary, Theorem 3.3.5 of [1] gives⟨

Θ(z)− Θ(α)z − α

y1,Θ(z)− Θ(β)

z − βy2

⟩H( eΘ)

=⟨I −Θ(β)Θ(α)∗

1− βαy1, y2

⟩Y

=

⟨I −Θ(z)Θ(α)∗

1− zαy1,

I −Θ(z)Θ(β)∗

1− zβy2

⟩H(Θ)

for all α, β in the domain of holomorphy ofΘ(z) in the unit disk and all y1, y2 ∈ Y . The kernel of an everywheredefined and continuous isometry on a Pontryagin space is zero (see [16, Corollary 1.9]). Thus Σ = ΣΘ is bothobservable and controllable and hence minimal.

We next consider the more general case of functions Θ(z) ∈ Sκ(U ,Y) and the model of Sz.-Nagy and Foias[24] and M.S. Brodskiı [10]. Write L2(U) and H2(U) for the usual Lebesgue and Hardy spaces of functions onthe unit circle |ζ| = 1 with values in a Hilbert space U . Define ∆S(ζ) for any function S(z) ∈ Sκ(U ,Y) as inSection 8.

In the case κ = 0, given a function Θ(z) ∈ S(U ,Y), the model of Sz.-Nagy and Foias produces a systemΣ= (

A,

B,

C,

D;

X (Θ),U ,Y; 0) with a Hilbert state space

X (Θ) =

(H2(Y)⊕∆ΘL2(U)

)

[Θ(ζ)

∆Θ(ζ)

]H2(U) (10.1)

and operators defined byA :

[y(ζ)u(ζ)

]→ ζ

[y(ζ)− y(0)

u(ζ)

],

B : u0 → ζ

[(Θ(ζ)−Θ(0))u0

∆Θ(ζ)u0

],

C :

[y(ζ)u(ζ)

]→ y(0),

D : u0 → Θ(0)u0,

(10.2)

for all y(ζ)⊕ u(ζ) inX (Θ) and all u0 in U . The system

Σ is simple and conservative, and has transfer function

Θ(z). In the special case of inner functions, ∆Θ(ζ) = 0 a.e., and therefore we have the simpler formulas

X (Θ) = H2(Y)Θ(ζ)H2(U) (10.3)

and A : y(ζ) → ζ (y(ζ)− y(0)) ,

B : u0 → ζ (Θ(ζ)−Θ(0))u0,

C : y(ζ) → y(0),

D : u0 → Θ(0)u0.

(10.4)

for all y(ζ) inX (Θ) and all u in U . We formulate an analogous model for a simple conservative system with

transfer function Θ(z) ∈ Sκ(U ,Y).

Theorem 10.3 Let Θ(z) ∈ Sκ(U ,Y) and let Θ(z) = Θr(z)br(z)−1 be a right Kreın-Langer factorization.Define a system

Σ= (

A,

B,

C,

D;

X (Θ),U ,Y; κ),

X (Θ) =

X (Θr)⊕

(−

X (br)

),



where

A :

y(ζ)u1(ζ)u2(ζ)

→ ζ

y(ζ)− y(0)−(Θr(ζ)br(0)−1 −Θ(0)

)u2(ζ)

u1(ζ)−∆Θr(ζ)br(0)−1u2(0)

u2(ζ)− u2(0)

,B : u0 → ζ

(Θr(ζ)br(0)−1 −Θ(0)

)u0

∆Θr(ζ)br(0)−1u0(

br(ζ)br(0)−1 − I)u0

,C :

y(ζ)u1(ζ)u2(ζ)

→ y(0)−Θ(0)u2(0),

D : u0 → Θ(0)u0,

(10.5)

for all y(ζ) ⊕ u1(ζ) ⊕ u2(ζ) inX (Θ) and all u0 in U . Then

Σ is a simple conservative system with transfer

function Θ(z). Moreover, the identities

(I − zA)−1

B u0 =

1eit − z

Θr(ζ)−Θr(z)∆Θr

(ζ)br(ζ)− br(z)

br(z)−1u0 (10.6)

(I − wA∗)−1

C∗ y0 =

11− weit

I −Θr(ζ)Θ∗r(w)−∆Θr

(ζ)Θ∗r(w)b∗−1r (w)− br(ζ)Θ∗r(w)

y0. (10.7)

hold for all u0 ∈ U and all y0 ∈ Y .

P r o o f. Construct a simple conservative system Σ1 with transfer function Θr(z) ∈ S(U ,Y) using (10.1)and (10.2). A simple conservative system Σ2 with inner transfer function br(z) ∈ U(U ,U) can be constructedusing (10.3) and (10.4). Starting with Σ2 and using (5.3), we can construct a system Σ′′2 which, by Theorem 5.3,is also simple and conservative and satisfies ΘΣ2(z) = b−1

r (z). By Theorem 7.3,

Σ= Σ1 Σ′′2

is a simple conservative system. Straightforward calculations verify (10.5), (10.6), and (10.7).

If Θ(z) ∈ Uκ(U ,Y), (10.5) –(10.7) have the simpler form

A :

[y(ζ)u2(ζ)

]→ ζ

[y(ζ)− y(0)−

(Θr(ζ)br(0)−1 −Θ(0)

)u2(ζ)

u2(ζ)− u2(0)

],

B : u0 → ζ

[(Θr(ζ)br(0)−1 −Θ(0)

)u0(

br(ζ)br(0)−1 − I)u0

],

C :

[y(ζ)u2(ζ)

]→ y(0)−Θ(0)u2(0),

D : u0 → Θ(0)u0,

and

(I − zA)−1

B u0 =

1ζ − z

[Θr(ζ)−Θr(z)br(ζ)− br(z)

]b−1r (z)u0,

(I − wA∗)−1

C∗ y0 =

11− wζ

[I −Θr(ζ)Θ∗r(w)

b∗−1r (w)− br(ζ)Θ∗r(w)

]y0.



Acknowledgements D. Z. Arov and S. M. Saprikin were partially supported by the joint grant UM1-2567-00 03 from theU. S. Civilian Research & Development Foundation (CRDF) and Ukrainian Government, and they thank the University ofVirginia for hospitality during visits when this work was carried out. J. Rovnyak was supported by the National ScienceFoundation Grant DMS-0100437.

References[1] D. Alpay, A. Dijksma, J. Rovnyak, and H. S. V. de Snoo, Schur functions, operator colligations, and reproducing kernel

Pontryagin spaces, Oper. Theory Adv. Appl., vol. 96, Birkhauser, Basel, 1997.[2] D. Z. Arov, Passive linear steady-state dynamical systems, Sibirsk. Mat. Zh. 20 (1979), no. 2, 211–228, 457, English

transl. Siberian Math. J. 20 (1979), no. 2, 149–162.[3] , Stable dissipative linear stationary dynamical scattering systems, J. Operator Theory 2 (1979), no. 1, 95–126,

English transl. with an appendix by D. Z. Arov and J. Rovnyak, Interpolation theory, systems theory, and related topics,Oper. Theory Adv. Appl. 134, Birkhauser, Basel, 2002, pp. 99–136.

[4] D. Z. Arov and S. M. Saprikin, Maximal solutions for the embedding problem for a generalized Schur function andoptimal dissipative scattering systems with Pontryagin state spaces, Methods Funct. Anal. Topology 7 (2001), no. 4,69–80.

[5] T. Ya. Azizov and I. S. Iokhvidov, Linear operators in spaces with an indefinite metric, “Nauka”, Moscow, 1986, Englishtransl., John Wiley & Sons Ltd., Chichester, 1989.

[6] J. A. Ball, Models for noncontractions, J. Math. Anal. Appl. 52 (1975), no. 2, 235–254.[7] J. A. Ball and N. Cohen, de Branges-Rovnyak operator models and systems theory: a survey, Topics in matrix and

operator theory (Rotterdam, 1989), Oper. Theory Adv. Appl., vol. 50, Birkhauser, Basel, 1991, pp. 93–136.[8] J. Bognar, Indefinite inner product spaces, Springer-Verlag, New York, 1974, Ergebnisse der Mathematik und ihrer

Grenzgebiete, Band 78.[9] L. de Branges and J. Rovnyak, Canonical models in quantum scattering theory, Perturbation Theory and its Applications

in Quantum Mechanics (Proc. Adv. Sem. Math. Res. Center, U.S. Army, Theoret. Chem. Inst., Univ. of Wisconsin,Madison, Wis., 1965), Wiley, New York, 1966, pp. 295–392.

[10] M. S. Brodskiı, Unitary operator colligations and their characteristic functions, Uspekhi Mat. Nauk 33 (1978),no. 4(202), 141–168, 256, English transl. Russian Math. Surveys 33 (1978), no. 4, 159-191.

[11] V. M. Brodskiı and M. S. Brodskiı, Factorization of the characteristic function and invariant subspaces of a contractionoperator, Funkcional. Anal. i Prilozen. 8 (1974), no. 2, 63–64, English transl. Funct. Anal. Appl. 8 (1974), 145–147.

[12] V. M. Brodskiı and Ja. S. Svarcman, Invariant subspaces of a contraction, and factorization of the characteristic func-tion, Teor. Funkciı Funkcional. Anal. i Prilozen. 22 (1975), 15–35, 160.

[13] D. N. Clark, On functional models for non-contraction operators, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom.Phys. 23 (1975), no. 2, 159–163.

[14] B. Curgus, A. Dijksma, H. Langer, and H. S. V. de Snoo, Characteristic functions of unitary colligations and of boundedoperators in Kreın spaces, The Gohberg anniversary collection, Vol. II (Calgary, AB, 1988), Oper. Theory Adv. Appl.,vol. 41, Birkhauser, Basel, 1989, pp. 125–152.

[15] A. Dijksma, H. Langer, and H. S. V. de Snoo, Characteristic functions of unitary operator colligations in πκ-spaces,Operator theory and systems (Amsterdam, 1985), Oper. Theory Adv. Appl., vol. 19, Birkhauser, Basel, 1986, pp. 125–194.

[16] M. A. Dritschel and J. Rovnyak, Operators on indefinite inner product spaces, Lectures on operator theory and itsapplications (Waterloo, ON, 1994), Amer. Math. Soc., Providence, RI, 1996, pp. 141–232.

[17] I. C. Gohberg and M. G. Kreın, Introduction to the theory of linear nonselfadjoint operators, Izdat. “Nauka”, Moscow,1965, English transl. American Mathematical Society, Providence, R.I., 1969.

[18] I. S. Iokhvidov, M. G. Kreın, and H. Langer, Introduction to the spectral theory of operators in spaces with an indefinitemetric, Akademie-Verlag, Berlin, 1982.

[19] M. G. Kreın and H. Langer, Uber die verallgemeinerten Resolventen und die charakteristische Funktion einesisometrischen Operators im Raume Πκ, Hilbert space operators and operator algebras (Proc. Internat. Conf., Tihany,1970), North-Holland, Amsterdam, 1972, pp. 353–399. Colloq. Math. Soc. Janos Bolyai, 5.

[20] A. V. Kuzhel′, A characteristic operator function of an arbitrary bounded operator, Dopovıdı Akad. Nauk Ukraın. RSRSer. A 1968 (1968), 233–236.

[21] S. A. Kuzhel′, The abstract Lax-Phillips scattering scheme in Pontryagin spaces, Mat. Sb. 187 (1996), no. 10, 87–108,English transl. Sb. Math. 187 (1996), no. 10, 1503–1523.

[22] B. W. McEnnis, Models for operators with bounded characteristic function, Acta Sci. Math. (Szeged) 43 (1981), no. 1-2,71–90.

[23] S. M. Saprikin, The theory of linear discrete time-invariant dissipative scattering systems with πκ state spaces, Zap.Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 282 (2001), Issled. po Linein. Oper. i Teor. Funkts. 29,192–215, 281, English transl. J. Math. Sci. (N. Y.) 120 (2004), no. 5, 1752–1765.

[24] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam,1970.


Documents

people.virginia.edupeople.virginia.edu/~jlr5m/Papers/p63.pdf · mn header will be provided by the publisher 29 Acknowledgements D. Z. Arov and S. M. Saprikin were partially supported