Conservative Realizations of Herglotz-Nevanlinna Functions

Operator Theory: Advances and Applications Volume 217 Founded in 1979 by Israel Gohberg Editors: Joseph A. Ball (Black...

Author: Yuri Arlinskii | Sergey Belyi | Eduard Tsekanovskii

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Operator Theory: Advances and Applications Volume 217 Founded in 1979 by Israel Gohberg

Editors: Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Vienna, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Vadim Adamyan (Odessa, Ukraine) Albrecht Böttcher (Chemnitz, Germany) B. Malcolm Brown (Cardiff, UK) Raul Curto (Iowa, IA, USA) Fritz Gesztesy (Columbia, MO, USA) Pavel Kurasov (Lund, Sweden) Leonid E. Lerer (Haifa, Israel) Vern Paulsen (Houston, TX, USA) Mihai Putinar (Santa Barbara, CA, USA) Leiba Rodman (Williamsburg, VA, USA) Ilya M. Spitkovsky (Williamsburg, VA, USA)

Honorary and Advisory Editorial Board: Lewis A. Coburn (Buffalo, NY, USA) Ciprian Foias (College Station, TX, USA) J.William Helton (San Diego, CA, USA) Thomas Kailath (Stanford, CA, USA) Peter Lancaster (Calgary, Canada) Peter D. Lax (New York, NY, USA) Donald Sarason (Berkeley, CA, USA) Bernd Silbermann (Chemnitz, Germany) Harold Widom (Santa Cruz, CA, USA)

Yuri Arlinskii Sergey Belyi Eduard Tsekanovskii

Conservative Realizations of Herglotz–Nevanlinna Functions

Yuri Arlinskii Department of Mathematics East Ukrainian National University Kvartal Molodizhny, 20-A 91034 Lugansk, Ukraine [email protected]

Sergey Belyi Department of Mathematics Troy University Troy, AL 36082, USA [email protected]

Eduard Tsekanovskii Department of Mathematics Niagara University, NY 14109, USA [email protected]

2010 Mathematics Subject Classification: 47A, 47B ISBN 978-3-7643-9995-5 e-ISBN 978-3-7643-9996-2 DOI 10.1007/978-3-7643-9996-2 Library of Congress Control Number: 2011930752 © Springer Basel AG 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com

To the memory of Moshe Liv˘sic, a Teacher and a great mathematician

Preface Consider the system of equations ⎧ dχ i + T χ(t) = KJψ− (t), ⎪ ⎪ ⎨ dt χ(0) = x ∈ H, ⎪ ⎪ ⎩ ψ+ = ψ− − 2iK ∗ χ(t),

(0.1)

where T is a bounded linear operator from a Hilbert space H into itself, K is a bounded linear operator from a Hilbert space E (dim E < ∞) into H, J = J ∗ = J −1 maps E into itself, and Im T = KJK ∗ . If for a given continuous in E function ψ− (t) ∈ L2[0,τ0 ] (E) we have that χ(t) ∈ H and ψ+ (t) ∈ L2[0,τ0 ] (E) satisfy the system (0.1), then the following metric conservation law holds: τ τ 2χ(τ )2 − 2χ(0)2 = (Jψ− , ψ− )E dt − (Jψ+ , ψ+ )E dt, τ ∈ [0, τ0 ]. (0.2) 0

0

Given an input vector ψ− = ϕ− eizt ∈ E, we seek solutions to the system (0.1) as an output vector ψ+ = ϕ+ eizt ∈ E and a state-space vector χ(t) = xeizt in H, (z ∈ C). Substituting the expressions for ψ± (t) and χ(t) in (0.1) allows us to cancel exponential terms and convert the system to stationary algebraic format (T − zI)x = KJϕ− , Im T = KJK ∗ , z ∈ ρ(T ), (0.3) ϕ+ = ϕ− − 2iK ∗ x; where ρ(T ) is the set of regular points of the operator T . The type of an open system in (0.3) was introduced and studied by Livˇsic [191]. A brief form of an open system (0.3) can be written as a rectangular array known in operator theory as an operator colligation [89] T K J Θ= , ImT = KJK ∗ . (0.4) H E The transfer function of the system Θ of the form (0.3)-(0.4) is given by WΘ (z) = I − 2iK ∗ (T − zI)−1 KJ,

(0.5) vii

viii

Preface

and satisﬁes, for z ∈ ρ(T ), WΘ∗ (z)JWΘ (z) ≥ J, WΘ∗ (z)JWΘ (z) = J,

(Im z > 0), (Im z = 0),

≤ J,

(Im z < 0).

WΘ∗ (z)JWΘ (z)

(0.6)

We call the function VΘ (z) = K ∗ (Re T − zI)−1 K = i[WΘ (z) + I]−1 [WΘ (z) − I]J,

(0.7)

the impedance function of the system Θ. This function VΘ (z) is a HerglotzNevanlinna function in E. The condition Im T = KJK ∗ plays a crucial role in determining the analytical properties of the functions WΘ (z) and VΘ (z). The open system Θ in (0.3)-(0.4) has the property that its transfer function WΘ (z) becomes a J-unitary operator for real z ∈ ρ(T ), i.e., [ϕ+ , ϕ+ ] = [WΘ (z)ϕ− , WΘ (z)ϕ− ], where [·, ·] = (J·, ·)E , and (·, ·)E is an inner product in E. Let us look at a simple but motivating example leading to a system of the form (0.3)-(0.4). Consider a four-terminal electrical circuit in Figure 1. Let C denote the capacity of the capacitor and let L represent the inductance of an induction coil. Given a harmonic input √ − iωt √2I − , ψ− = ϕ− e , ϕ− = 2U where I − is the current and U − is the voltage, we are trying to ﬁnd the harmonic output + I , ψ+ = ϕ+ eiωt , ϕ+ = U+ and also describe the state of the capacitor and the induction coil √ LI iωt √ χ = xe , x = . CU Here I is the current on the induction coil, U is the voltage on the capacitor, I + and U + are the output current and voltage, respectively. Using electrical circuit equations dI dU L = U − (t), = −I(t) + I − (t), C dt dt we can obtain system (0.1), separate variables, and arrive at the system

(T − ωI)x = KJϕ− , , ω ∈ ρ(T ), ϕ+ = ϕ− − 2iK ∗x,

Preface

ix

IU

I C

L

-

I

+

U+

I+

-

Figure 1: Four-terminal circuit

where T =

0 √i LC

0 0

,

K=

√i 2L

0

0 √i 2C

,

J=

0 1

1 0

,

(0.8)

and x, ϕ± are deﬁned above. By a routine argument one obtains Im T = KJK ∗ . This open system can be re-written in the form (0.4) T K J Θ= , ImT = KJK ∗ , C2 C2 whose transfer function is of the form (0.5) and actually reads ⎛ ⎞ i 1 ωL ⎠, WΘ (ω) = I − 2iK ∗ (T − ωI)−1 KJ = ⎝ i 1 1 − ωLC ωC where T , K, and J are deﬁned in (0.8). It is easy to see that WΘ (ω) satisﬁes the conditions (0.6) with z = ω ∈ ρ(T ). When a physical system (for instance a lengthy line) has distributed parameters, the state-space operator T of the system becomes unbounded. As a result, the above mentioned system Θ does not (as an algebraic structure) have any meaning, since the imaginary part of an unbounded operator T may not be deﬁned properly because the domains of T and T ∗ may not coincide. However, some examples [191] of systems with unbounded operators show that the ranges of the channel operators K belong to some triplets of Hilbert spaces H+ ⊂ H ⊂ H− while not being a part of H. In the 1960s Livˇsic formulated a problem [191] of developing a theory of open systems and their transfer functions that would involve unbounded operators and at the same time preserve the algebraic structure existing in the case when the state-space operator of the system is bounded. The importance of a problem of using generalized functions in system theory (especially in systems with distributed parameters) was pointed out in the 1970s independently by Helton [155]. The solution to this diﬃcult problem is the main subject of the current book. The monograph also covers the research in this area for the last three

x

Preface

decades. Diﬀerent approaches to realization problems of various types of systems with continuous time and conservativity condition (or without) have been considered by Arov-Dym [56], [57], [58], [119], Ball-Staﬀans [69], Staﬀans [235], [237], Bart-Gohberg-Kaashoek-Ran [70], [71], and others. Below we provide a brief description of the results considered in the current text. • In Chapter 1 we consider some basic facts related to the theory of extensions of linear symmetric operators. In particular, we study the parameterizations of the domains of all self-adjoint extensions in dense and non-dense domain cases. This includes the von Neumann and Krasnoselki˘i decomposition and parametrization formulas. • In Chapter 2 we study the extensions of symmetric non-densely deﬁned operators in triplets of rigged Hilbert spaces. The Krasnoselski˘i formulas discussed in Chapter 1 are based on indirect decomposition, where linear manifolds such as the domain of the symmetric operator and its deﬁciency subspaces may be linearly dependent. Introduction of rigged Hilbert spaces allows us to obtain direct decomposition for the domain of the adjoint operator and parametrization of all self-adjoint extensions. This direct decomposition is written in terms of the semi-deﬁciency subspaces and is an analogue of the classical von Neumann formulas. • Chapter 3 is dedicated to the development of a new extension theory of symmetric operators in triplets of Hilbert spaces, the so-called bi-extension theory that will be put to extensive use later in the text. • Chapter 4 studies quasi-self-adjoint extensions of symmetric operators and contains the deﬁnition of the so-called (∗)-extension. The (∗)-extensions will be used later in the book in the deﬁnition of L-systems. We also present an analysis of these extensions together with their description and parametrization. • Chapter 5 contains the main concepts and ideas of the classical theory of the Liv˘sic canonical systems (operator colligations) with bounded operators. A comprehensive study of operator colligations of the form (0.4) in operator theory was developed by Brodski˘i and Liv˘sic [89], [91]. We provide a collection of known results for such a type of systems in terms of transfer functions and their linear-fractional transformations. These results include couplings of these systems and multiplication and factorization theorems for the transfer functions. • In Chapter 6, we introduce rigged canonical systems with an unbounded ˙ T ∗ ⊃ A, ˙ A˙ is a symmetric operator, operator T , (ρ(T ) = ∅), where T ⊃ A, ˙ If we consider H+ ⊂ H ⊂ H− is a rigged Hilbert space generated by A. system (0.1) and its stationary version (0.3) with an unbounded operator T such that Ran(K) ⊂ H− , we will run into substantial diﬃculties even

Preface

xi

at the stage of deﬁning the solution of the system. This happens because, for a given input ϕ− ∈ E, the ﬁrst equation of system (0.3) does not have regular solutions x ∈ Dom(T ). In order to treat this case adequately, we need to perform a certain regularization of the system that is based on the bi-extension and (∗)-extension theory developed in Chapters 3 and 4. This regularization also allows us to determine the imaginary part Im T of the unbounded operator T . Let A, A∗ ∈ [H+ , H− ] be (∗)-extensions of T and consider the system ⎧ dχ ⎪ ⎨ i dt + Aχ(t) = KJψ− (t), (0.9) χ(0) = x ∈ H+ , ⎪ ⎩ ∗ ψ+ = ψ− − 2iK χ(t), where K ∈ [E, H− ], K ∗ ∈ [H+ , E], J is a self-adjoint and unitary operator in E, and Im A = KJK ∗ . If for a given continuous in E function ψ− (t) ∈ L2[0,τ0 ] (E) we have that a continuous in H+ and strongly diﬀerentiable in H function χ(t) ∈ H+ and a function ψ+ (t) ∈ L2[0,τ0 ] (E) satisfy the system (0.9), then the metric conservation law (0.2) also holds. In line with the approach of the bounded case, we look for stationary solutions and convert our system to the algebraic form (A − zI)x = KJϕ− , z ∈ ρ(T ), (0.10) ϕ+ = ϕ− − 2iK ∗ x, where ϕ− and ϕ+ are input and output vectors in E, respectively and vector x ∈ H+ is a vector of the state space. The system (0.10) above is called the Liv˘sic rigged, canonical system or L-system and can be written as an array A K J Θ= , z ∈ ρ(T ), (0.11) H+ ⊂ H ⊂ H− E where A is a (∗)-extension of T such that Im T = Im A = KJK ∗ ,

Ran(Im A) = Ran(K).

(0.12)

The transfer function of the system Θ has the form WΘ (z) = I − 2iK ∗ (A − zI)−1 KJ,

z ∈ ρ(T ),

and satisﬁes analytical conditions (0.6). The impedance function of the system Θ is VΘ (z) = K ∗ (Re A − zI)−1 K = i[WΘ (z) + I]−1 [WΘ (z) − I]J. Clearly, systems of the form (0.11)-(0.12) have the same algebraic structure as systems (0.3)-(0.4) with bounded operators. Therefore we can refer to the

xii

Preface canonical systems in Chapter 5 as L-systems as well. We recall that, for a holomorphic function that maps the open upper half-plane into itself, one can ﬁnd the names Herglotz [137], Nevanlinna [42], and R-functions [159] (sometimes depending on the geographical origin of authors). In this text we adopt the term Herglotz-Nevanlinna function and extend it to both scalar and operator-valued cases. An important criteria is obtained for the class of Herglotz-Nevanlinna functions in Hilbert space E (dim E < ∞) of the form 1 t V (z) = Q + Lz + − dΣ(t) t−z 1 + t2 R (dΣ(t)x,x)E where Q = Q∗ , L ≥ 0, and R < ∞ for all x ∈ E that can be 1+t2 realized as impedance functions of some scattering (J = I) L-system. The proof of this criteria relies on the constructed minimal L-system involving an operator of multiplication by the independent variable in the model functional Hilbert space. This model L-system is extensively used in the following chapters. • In Chapter 7 we introduce three distinct subclasses of the class of all realizable Herglotz-Nevanlinna operator-valued functions. Complete proofs of direct and inverse realization theorems are given for each subclass. We also provide several multiplication theorems related to this class partition. • In Chapter 8 we give the deﬁnition and describe the properties of normalized canonical L-systems. We also prove an important theorem about the constant J-unitary factor. This theorem states that if an operator-valued function W (z) is realizable as a transfer function of an L-system Θ, then for an arbitrary constant J-unitary operator B the functions W (z)B and BW (z) can be realized as transfer functions of the same type of L-system that contains the same unbounded operator T as Θ but a diﬀerent channel operator. • In Chapter 9 we present the solution to the restricted Phillips-Kato extension problem on the existence and description of all proper accretive and sectorial maximal extensions of a given densely deﬁned non-negative symmetric operator. This description (parametrization) is presented in terms of contractive extensions of a given symmetric contraction that are linear fractional transformations of the corresponding accretive operators. The established parametrization strengthens the classical Krein theorem on self-adjoint contractive extensions of symmetric contractions. On the basis of this new parametrization we establish a criterion in terms of the impedance function VΘ (z) of an L-system Θ when the state-space operator T of this system is a contraction or so-called an α-co-sectorial contraction. Also in this chapter we establish the criteria for a given Stieltjes or inverse Stieltjes function to be realized as an impedance function of some L-system Θ whose state-space operator T is maximal accretive or α-sectorial.

Preface

xiii

• Chapter 10 is dedicated to an important application: it provides the descriptions of all accretive and sectorial boundary value problems Th , where Th is a Schr¨odinger operator in L2 [a, +∞) with a complex boundary parameter h. We also provide a complete description of all L-systems with Schr¨odinger operator Th . Moreover, we describe the class of scalar Stieltjes (inverse Stieltjes) like functions that can be realized as impedance functions of L-systems with Schr¨odinger operator Th . It is shown that the Schr¨odinger operator Th of an L-system is accretive if and only if the impedance function of this L-system is either a Stieltjes or inverse Stieltjes function. We derive the formulas that restore an L-system uniquely from a given Stieltjes (inverse Stieltjes) like function to become the impedance function of this L-system. These formulas allow us to solve the inverse problem and ﬁnd the exact value of the parameter h in the deﬁnition of Th , as well as a real parameter μ that appears in the construction of the elements of the realizing L-system. An elaborate investigation of these formulas shows the dynamics of the restored parameters h and μ in terms of the changing constant term from the integral representation of the realizable function. We also point out an important connection between the impedance functions of L-systems with Schr¨odinger operator Th and the Krein-von Neumann and Friedrichs extensions of a minimal non-negative Schr¨odinger operator. • In Chapter 11 we consider a new type of solutions of Nevanlinna-Pick interpolation problems for the class of scalar Herglotz-Nevanlinna functions. These are explicit system solutions that are impedance functions of some L-systems with bounded operators. The conditions for the existence and uniqueness of solutions are presented in terms of interpolation data. We also ﬁnd new properties of the classical Pick matrices. The exact formula for the angle of sectoriality of the corresponding state-space operator in the explicit system solution is derived. The criterion for this operator to be accretive, but not α-sectorial for any angle α ∈ (0, π/2), is obtained in terms of interpolation data and classic Pick matrices. We ﬁnd conditions on interpolation data under which the explicit system solution of a scalar Nevanlinna-Pick interpolation problem is generated by the dissipative L-system whose state-space operator is a non-self-adjoint, prime dissipative Jacobi matrix with a rank-one imaginary part. In order to obtain these results, we establish a new model for prime, bounded, dissipative operators with rank-one imaginary part and show that a semi-inﬁnite (ﬁnite) bounded Jacobi matrix is a new model. In addition, an inverse spectral problem for ﬁnite non-self-adjoint Jacobi matrices with rankone imaginary part is solved. It is shown that any ﬁnite sequence of non-real numbers in the open upper half-plane is the set of eigenvalues (counting multiplicities) of some dissipative non-self-adjoint Jacobi matrix with rank-one imaginary part. The algorithm of reconstruction of the unique Jacobi matrix from its non-real eigenvalues is presented. • In Chapter 12 we consider non-canonical rigged systems and show that the metric conservation law holds for them as well. Moreover, it is easily seen

xiv

Preface that, in the special case when a non-canonical system becomes canonical, the metric conservation law for the non-canonical system matches its canonical version. Later on in this chapter we utilize non-canonical systems to present the solution of the general realization problem for an arbitrary HerglotzNevanlinna function. In particular it is shown that an arbitrary HerglotzNevanlinna operator-valued function can be realized as the transfer function of the corresponding non-canonical impedance system or NCI-system. The conditions on an arbitrary Herglotz-Nevanlinna function to be an impedance function of a non-canonical L-system (or NCL-system) are also provided.

Over the last several decades many books and papers have been dedicated to the analysis of inﬁnite-dimensional systems and realization problems for diﬀerent function classes. The literature on this subject is too extensive to be discussed exhaustively but we refer in this matter to [1]–[274] and the literature therein. In this text we propose a comprehensive analysis of the above mentioned L-systems with, generally speaking, unbounded operators that satisfy the metric conservation law. We also treat realization problems for Herglotz-Nevanlinna functions and their various subclasses when members of these subclass are realized as impedance functions of L-systems. This type of realizations is called conservative. The detailed study provided relies on a new method involving extension theory of linear operators with the exit into rigged triplets of Hilbert spaces. In particular, it is possible to set a one-to-one correspondence between the impedance of L-systems and related (∗)-extensions of unbounded operators. The theory of singular systems developed in the current monograph leads to several useful and important applications including systems with non-self-adjoint Schr¨odinger operator, nonself-adjoint Jacobi matrices, and system interpolation. In summary, we hope that this book contains new developments and will be of value and interest to researchers in the ﬁeld of operator theory, spectral analysis of diﬀerential operators, and system theory. We also think that this text may be used to teach a graduate level special topics course on this subject.

Acknowledgements We are very grateful to Harm Bart and Olof Staﬀans for their reviews and valuable comments. We are also indebted to Vladimir Peller and Peter Kuchment for their helpful suggestions. Special thanks goes to our co-authors and colleagues Vladimir Derkach, Seppo Hassi, Fritz Gesztesy, Konstantin Makarov, Mark Malamud, and Henk de Snoo for their valuable input and fruitful discussions. Lugansk, Troy, Niagara Falls, October 2010

Yury Arlinskii Sergey Belyi Eduard Tsekanovskii

Contents Preface

vii

1 Extensions of Symmetric Operators 1.1 Deﬁciency indices of symmetric operators . . . . . . . . . . . 1.2 The ﬁrst von Neumann formula in the dense case . . . . . . . 1.3 Parametrization of symmetric and self-adjoint extensions. The second von Neumann formula . . . . . . . . . . . . . . . 1.4 The Cayley transform . . . . . . . . . . . . . . . . . . . . . . 1.5 Non-densely deﬁned symmetric operators and semi-deﬁciency subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Symmetric extensions of a non-densely deﬁned symmetric operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Indirect decomposition and the Krasnoselski˘i formulas . . . .

. . . . . .

1 1 4

. . . . . .

5 8

. . .

10

. . . . . .

14 18

2 Geometry of Rigged Hilbert Spaces 2.1 The Riesz-Berezansky operator . . . . . . . . . . . . . . . . . . 2.2 Construction of the operator generated rigging . . . . . . . . . 2.3 Direct decomposition and analogue of the ﬁrst von Neumann’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Regular and singular symmetric operators . . . . . . . . . . . . 2.5 Closed symmetric extensions . . . . . . . . . . . . . . . . . . .

. . . .

23 23 27

. . . . . .

28 34 35

. . . .

. . . .

45 45 52 54 61

4 Quasi-self-adjoint Extensions 4.1 Quasi-self-adjoint extensions of symmetric operators . . . . . . . . 4.2 Quasi-self-adjoint bi-extension . . . . . . . . . . . . . . . . . . . . . 4.3 The (∗)-extensions and uniqueness theorems . . . . . . . . . . . . .

69 69 81 90

3 Bi-extensions of Closed Symmetric Operators 3.1 Bi-extensions . . . . . . . . . . . . . . . . . . . . 3.2 Bi-extensions of O-operators . . . . . . . . . . . . 3.3 Self-adjoint and t-self-adjoint bi-extensions . . . 3.4 The case of a densely deﬁned symmetric operator

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

xv

xvi

Contents 4.4 4.5

The (∗)-extensions in the densely-deﬁned case . . . . . . . . . . . 104 Resolvents of quasi-self-adjoint extensions . . . . . . . . . . . . . . 110

5 The 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Livˇsic Canonical Systems with Bounded Operators The Livˇsic canonical system and the Brodski˘i theorem Minimal canonical systems . . . . . . . . . . . . . . . . Couplings of canonical systems . . . . . . . . . . . . . Transfer functions of canonical systems . . . . . . . . . Class ΩJ and its realization . . . . . . . . . . . . . . . Finite-dimensional state-space case . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

119 119 122 124 129 135 139 143

6 The 6.1 6.2 6.3 6.4 6.5 6.6

Herglotz-Nevanlinna Functions and Rigged Canonical Systems The Herglotz-Nevanlinna functions and their representations . Extended resolvents and resolution of identity . . . . . . . . . . Deﬁnition of an L-system . . . . . . . . . . . . . . . . . . . . . Realizable Herglotz-Nevanlinna operator-functions. Class N (R) Realization of the class N (R) . . . . . . . . . . . . . . . . . . . Minimal realization and the theorem on bi-unitary equivalence

. . . . . .

. . . . . .

147 147 150 161 170 176 194

. . . . . .

. . . . . .

205 205 211 214 224 228 233

7 Classes of realizable Herglotz-Nevanlinna functions 7.1 Sub-classes of the class N (R) and their realizations . . 7.2 Class Ω(R, J). The Potapov-Ginzburg Transformation 7.3 Multiplication Theorems for Ω(R, J) classes . . . . . . 7.4 Boundary triplets and self-adjoint bi-extensions . . . . 7.5 The Krein-Langer Q-functions and their realizations . 7.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . .

. . . . . . .

. . . . . .

. . . . . . .

. . . . . .

. . . . . . .

. . . . . .

. . . . . .

8 Normalized L-Systems 8.1 Auxiliary canonical system . . . . . . . . . . . . . . . . . . . . 8.2 Constant J-unitary factor . . . . . . . . . . . . . . . . . . . . . 8.3 The Donoghue transform and impedance functions of scattering L-systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Normalized (∗)-extensions and normalized L-systems . . . . . . 8.5 Realizations of eizl and eil/z as transfer functions of L-systems

239 . . 239 . . 245

9 Canonical L-systems with Contractive and Accretive Operators 9.1 Contractive extensions and their block-matrix forms . . . . . . 9.2 Quasi-self-adjoint contractive extensions of symmetric contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 The Weyl-Titchmarsh functions of quasi-self-adjoint contractive extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Canonical L-systems with contractive state-space operators . .

261 . . 261

. . 250 . . 252 . . 256

. . 269 . . 281 . . 286

Contents 9.5 9.6 9.7 9.8 9.9

The restricted Phillips-Kato extension problem . . Bi-extensions of non-negative symmetric operators Accretive bi-extensions . . . . . . . . . . . . . . . . Realization of Stieltjes functions . . . . . . . . . . Realization of inverse Stieltjes functions . . . . . .

xvii . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

292 308 311 321 334

10 L-systems with Schr¨odinger operator 10.1 (∗)-extensions of ordinary diﬀerential operators . . . . . . . . . . 10.2 Canonical L-systems with Schr¨ odinger operator . . . . . . . . . . 10.3 Accretive and sectorial boundary problems for a Schr¨odinger operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Functional model for symmetric operator with deﬁciency indices (1,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Accretive (∗)-extensions of a Schr¨odinger operator . . . . . . . . 10.6 Stieltjes functions and L-systems with accretive Schr¨odinger operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Inverse Stieltjes functions and systems with Schr¨ odinger operator 10.8 Stieltjes-like functions and inverse spectral problems for systems with Schr¨ odinger operator . . . . . . . . . . . . . . . . . 10.9 Inverse Stieltjes-like functions and inverse spectral problems for systems with Schr¨ odinger operator . . . . . . . . . . . . . . . . .

341 . 341 . 344

11 Non-self-adjoint Jacobi Matrices and System Interpolation 11.1 Systems with Jacobi matrices . . . . . . . . . . . . . . . . . . . . 11.2 The Stone theorem and its generalizations . . . . . . . . . . . . . 11.3 Inverse spectral problems for ﬁnite dissipative Jacobi matrices . . 11.4 Reconstruction of a dissipative Jacobi matrix from its triangular form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 System Interpolation and Sectorial Operators . . . . . . . . . . . 11.6 The Liv˘sic interpolation systems in the Pick form . . . . . . . . 11.7 The Nevanlinna-Pick rational interpolation with distinct poles . . 11.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

413 . 414 . 417 . 424 . . . . .

427 431 436 446 451

12 Non-canonical Systems 12.1 F -systems: deﬁnition and basic properties . . . . . . . . . 12.2 Multiplication theorems for F -systems . . . . . . . . . . . 12.3 Realizations in the case of a compactly supported measure 12.4 Deﬁnitions of NCI-systems and NCL-systems . . . . . . . 12.5 NCI realizations of Herglotz-Nevanlinna functions . . . . . 12.6 Realization by NCL-systems . . . . . . . . . . . . . . . . 12.7 Minimal NCL-realization . . . . . . . . . . . . . . . . . . . 12.8 Examples and non-canonical system interpolation . . . . .

. . . . . . . .

453 455 460 464 470 474 480 487 490

Notes and Comments

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. 349 . 357 . 361 . 364 . 370 . 378 . 396

497

xviii

Contents

Bibliography

505

Index

525

Chapter 1

Extensions of Symmetric Operators In this chapter we deal with extensions of densely and non-densely deﬁned symmetric operators. The parametrization of the domains of all self-adjoint extensions in both dense and non-dense cases is given in terms of von Neumann’s and Krasnoselski˘i’s formulas, respectively. The so-called admissible unitary operators serve as parameters in Krasnoselski˘i’s formulas.

1.1 Deﬁciency indices of symmetric operators Let H be a complex separable Hilbert space with inner product ( · , · ) and the identity operator I. A linear operator A˙ in a Hilbert space H is called symmetric if ˙ y) = (x, Ay), ˙ ˙ (Ax, ∀x, y ∈ Dom(A). ˙ is dense in H and A˙ = A˙ ∗ , then the operator A˙ is If in this case also Dom(A) ˙ xn → x, self-adjoint. An operator A˙ is closed if the relations {xn } ⊂ Dom(A), ˙ ˙ ˙ Axn → y imply that x ∈ Dom(A) and Ax = y. ˙ is dense in H. A Let A˙ be a closed linear operator whose domain Dom(A) −1 ˙ ˙ number λ ∈ C is called a regular point of the operator A if the operator (A−λI) exists, is bounded, and is deﬁned on the entire space H. The set of all regular points ˙ We call a complex of the operator A˙ forms a resolvent set and is denoted by ρ(A). ˙ number λ a point of regular type of the operator A if there exists k = k(λ) > 0 ˙ such that, for all x ∈ Dom(A), (A˙ − λI)x ≥ kx. It follows from this deﬁnition that λ is a point of regular type of the operator A˙ if ˙ of all points of regular and only if (A˙ − λI)−1 exists and is bounded. The set π(A) Y. Arlinskii et al., Conservative Realizations of Herglotz–Nevanlinna Functions, Operator Theory: Advances and Applications, DOI 10.1007/978-3-7643-9996-2_1, © Springer Basel AG 2011

1

2

Chapter 1. Extensions of Symmetric Operators

˙ It is easy to see that π(A) ˙ type is called the ﬁeld of regularity of the operator A. ˙ ˙ ˙ ˙ is an open set and ρ(A) ⊂ π(A). By σ(A) = C \ ρ(A) we denote the spectrum of ˙ its point spectrum. A complex number λ belongs to the operator A˙ and by σp (A) ˙ If A˙ is a symmetric operator the point spectrum of A˙ if λ is an eigenvalue of A. ˙ and z = x + iy, (y = 0), then for any f ∈ Dom(A), (A˙ − zI)f 2 = (A˙ − xI)f 2 + y 2 f 2 ≥ y 2 f 2 , which implies that both the upper and the lower half-planes are connected components of the ﬁeld of regularity of any symmetric operator. For a closed symmetric operator A˙ we consider the set ˙ Mλ = (A˙ − λI)Dom(A). Then the subspace Nλ = H Mλ¯ ˙ Obviously, Mλ is closed and is called the deﬁciency subspace of the operator A. ˙ makes a subspace in H for λ ∈ π(A). ˙ Then the Lemma 1.1.1. Let A˙ be a closed symmetric operator and λ0 ∈ π(A). dimension of the deﬁciency subspaces Nλ is the same for each λ in some neighborhood of λ0 . Proof. Let H1 and H2 be subspaces of a Hilbert space H and dim H2 > dim H1 . Let also G ⊂ H2 be the set of all vectors in H2 that are orthogonal to H1 . We are going to show that G = {0}. Indeed, let P1 be an orthoprojection operator in H onto H1 and let T = P1 H2 . Suppose that G = {0}. We will show that in this case Ker (T ) = {0}. Indeed, let x ∈ Ker (T ), i.e., x ∈ H2 and T x = P1 x = 0. The latter means that x ⊥ H1 and hence x ∈ G = {0} implying x = 0. Thus, Ker (T ) = {0}. Taking into account the invertibility of T we get dim H2 = dim T H2 ≤ dim H1 and arrive at a contradiction. Therefore, G = {0}. ¯ 0 is ˙ then λ Now we get back to the proof of the lemma. Since λ0 ∈ π(A) ˙ also a point of regular type of the operator A, and hence, there exists a number ¯ 0 ) > 0 such that k0 = k(λ ¯0 I)g ≥ k0 g, (A˙ − λ

˙ ∀g ∈ Dom(A).

(1.1)

˙ is an open set. Hence π(A) ˙ contains some We have already mentioned that π(A) ¯ 0 . Without loss of generality we can assume that ε ≤ k0 . Let ε-neighborhood of λ |λ − λ0 | < ε ≤ k0 . Assume that dim Nλ > dim Nλ0 .

(1.2)

Then, according to the above, the set of vectors G ⊂ Nλ orthogonal to Nλ0 is nontrivial, and thus there exists an element h ∈ Nλ (h = 0) such that h ⊥ Nλ0 and

1.1. Deﬁciency indices of symmetric operators

3

¯ 0 I)g, (g = 0). But h = f + (λ ¯ −λ ¯ 0 )g, therefore h ∈ Mλ¯ 0 . Consequently, h = (A˙ − λ ¯ where f = (A − λI)g ∈ Mλ¯ , and hence f ⊥ h, h ∈ Nλ . Then |λ − λ0 |2 g2 = h − f 2 = h2 + f 2,

(1.3)

where by (1.1) h ≥ k0 g,

f ≥ kg,

(k = k(λ) > 0).

(1.4)

Substituting (1.4) into (1.3) and canceling g2 we get |λ − λ0 |2 ≥ k02 + k 2 , which contradicts |λ − λ0 | < k0 . Thus (1.2) yields a contradiction. Similarly we conclude that the opposite to (1.3) inequality does not take place. Therefore dim Nλ = dim Nλ0 ,

(|λ − λ0 | < ε).

Theorem 1.1.2. The dimension of a deﬁciency subspace Nλ of a closed symmetric operator A˙ is the same for each λ in the open upper (lower) half-plane. Proof. It is easy to see that all non-real points are points of regular type of a ˙ Let λ and μ be two arbitrary ﬁxed points from upper symmetric operator A. (lower) half-plane. We connect them with a line segment. For each point of this segment there is a neighborhood such that dimensions of deﬁciency subspaces of the operator A˙ are the same for every λ from this neighborhood. Thus we get an open cover of a line segment that contains a ﬁnite subcover. As a result we conclude that dimensions of Nλ and Nμ are the same. Let us note that neither Lemma 1.1.1 nor Theorem 1.1.2 assume that operator A˙ is densely deﬁned. For a closed symmetric operator A˙ with the deﬁciency subspaces Nλ the numbers n+ = dim Ni , n− = dim N−i ˙ They are also written are called deﬁciency indices (numbers) of the operator A. in the form of an ordered pair (n+ , n− ). It follows from Theorem 1.1.2 that n+ , Imλ > 0, dim Nλ = n− , Imλ < 0. We note that deﬁciency indices of a symmetric operator are not necessarily equal and can be ﬁnite or inﬁnite. Theorem 1.1.3. If a symmetric operator A˙ has a real point of a regular type, then its deﬁciency indices are equal. ˙ and λ0 ∈ R, then in some neighborhood the dimensions of Proof. If λ0 ∈ π(A) deﬁciency subspaces of A˙ are the same. Then by Theorem 1.1.2, n+ = n− .

4

Chapter 1. Extensions of Symmetric Operators

1.2 The ﬁrst von Neumann formula in the dense case In this section we assume that our symmetric operator A˙ has a dense domain, i.e., ˙ = H. In what follows, for a linear set B in a Hilbert space H by B we Dom(A) denote the closure of B in H. As we have already mentioned above for a closed symmetric operator A˙ we have ˙ g) = (f, Ag), ˙ (Af,

˙ ∀f, g ∈ Dom(A).

˙ then g ∈ Dom(A˙ ∗ ) and A˙ ∗ g = Ag. ˙ Thus A˙ ∗ is This implies that if g ∈ Dom(A), ˙ an extension of A. Let us consider a deﬁciency subspace Nλ of the operator A˙ and ¯ ˙ we have ((A˙ − λI)g, ϕ ∈ Nλ . For any g ∈ Dom(A) ϕ) = 0 or ˙ ϕ) = (g, λ ϕ), (Ag,

˙ ∀g ∈ Dom(A).

(1.5)

That means ϕ ∈ Dom(A˙ ∗ ) and A˙ ∗ ϕ = λϕ,

ϕ ∈ Nλ

Evidently, the converse statement is also true, i.e., A˙ ∗ f = λf implies f ∈ Nλ . Thus the deﬁciency subspace of the operator A˙ is a kernel of the operator A˙ ∗ − λI, or Nλ = Ker (A˙ ∗ − λI). (1.6) Let us agree to call the linear manifolds L1 , L2 , . . ., Ln (n < ∞) linearly independent if the equation x1 + x2 + · · · + xn = 0,

xk ∈ Lk ,

k = 1, 2, . . . , n,

implies that x1 = x2 = · · · = xn = 0. It is easy to see that two linear manifolds L and M are linearly independent if and only if L ∩ M = {0}. The following result is known as the ﬁrst von Neumann’s formula. Theorem 1.2.1. Let A˙ be an arbitrary closed symmetric operator and Im λ = 0. ˙ Nλ , and Nλ¯ are linearly independent. MoreThen the linear manifolds Dom(A), over, ˙ Nλ Nλ¯ . Dom(A˙ ∗ ) = Dom(A) (1.7) ˙ ϕ ∈ Nλ , and ψ ∈ N ¯ . Suppose that Proof. Let g ∈ Dom(A), λ

g + ϕ + ψ = 0.

(1.8)

˙ Applying A˙ ∗ to both sides of this equation and taking into account that A˙ ∗ g = Ag, ¯ we get A˙ ∗ ϕ = λϕ, and A˙ ∗ ψ = λψ, ¯ = 0. ˙ + λϕ + λψ Ag

(1.9)

¯ and subtracting the result from (1.9) yields Multiplying both sides of (1.8) by λ ¯ ¯ = 0. (A˙ − λI)g + (λ − λ)ϕ

1.3. The second von Neumann formula

5

¯ ¯ Consequently, (A˙ − λI)g = 0 and ϕ = 0 because the vectors (A˙ − λI)g ∈ Mλ¯ and ˙ ϕ ∈ Nλ are orthogonal. But λ ∈ π(A), (Im λ = 0) and so g = 0. Hence (1.8) can only take place if all the terms in the left-hand side are zeros. This proves linear ˙ Nλ , and N ¯ . independence of Dom(A), λ ˙ Nλ Nλ¯ . Since all Now let us consider the linear manifold L = Dom(A) ∗ ˙ ˙ the sets Dom(A), Nλ , and Nλ¯ are contained in Dom(A ), we have L ⊂ Dom(A˙ ∗ ). Then all we have to show is that Dom(A˙ ∗ ) ⊂ L. ¯ Let f ∈ Dom(A˙ ∗ ) and f1 = (A˙ ∗ − λI)f . The Hilbert space H can be written as H = Mλ¯ ⊕ Nλ . Then f1 = g1 + ϕ1 , where g1 ∈ Mλ¯ , ϕ1 ∈ Nλ . Therefore, ¯ ¯ g1 = (A˙ − λI)g = (A˙ ∗ − λI)g, ˙ Consider the vector where g ∈ Dom(A). ϕ=

1 ¯ ϕ1 . λ−λ

Following (1.5) we get ¯ = (A˙ ∗ − λI)ϕ. ¯ ϕ1 = (λ − λ)ϕ Using this we can rewrite f1 = g1 + ϕ1 in the form ¯ ¯ ¯ (A˙ ∗ − λI)f = (A˙ ∗ − λI)g + (A˙ ∗ − λI)ϕ, or

¯ (A˙ ∗ − λI)(f − g − ϕ) = 0.

The latter means that the vector ψ = f − g − ϕ ∈ Nλ¯ , and thus, f = g + ϕ + ψ ∈ L. This proves Dom(A˙ ∗ ) ⊂ L and (1.7). Corollary 1.2.2. A closed symmetric operator is self-adjoint if and only if both its deﬁciency indices equal zero. ˙ = Dom(A˙ ∗ ) and by (1.7), Nλ = Nλ¯ = {0} Proof. If A˙ is self-adjoint, then Dom(A) ˙ = Dom(A˙ ∗ ) implying n+ = n− = 0. Conversely, if n+ = n− = 0, then Dom(A) and A˙ is self-adjoint.

1.3 Parametrization of symmetric and self-adjoint extensions. The second von Neumann formula Let A be a closed symmetric or self-adjoint extension of a densely deﬁned symmet˙ Since A˙ ⊂ A and A ⊂ A∗ , we have that A∗ is also an extension of ric operator A. ˙ i.e., A˙ ⊂ A∗ . But then A∗ ⊂ A˙ ∗ and A ⊂ A˙ ∗ . Thus for any symmetric extension A, A of a symmetric operator A˙ we have A˙ ⊂ A ⊂ A˙ ∗ ,

A˙ ⊂ A∗ ⊂ A˙ ∗ .

(1.10)

6

Chapter 1. Extensions of Symmetric Operators

Consequently, for any f1 , f2 ∈ Dom(A), f1 = g1 + ϕ1 + ψ1 ,

f2 = g2 + ϕ2 + ψ2 ,

(1.11)

˙ where g1 , g2 ∈ Dom(A), ϕ1 , ϕ2 ∈ Nλ , ψ1 , ψ2 ∈ Nλ¯ . Also, because A ⊂ A˙ ∗ we have ¯ 1 , Af2 = Ag ¯ 2. ˙ 1 + λϕ1 + λψ ˙ 2 + λϕ2 + λψ Af1 = Ag Using the relations ¯ 1 , ϕ2 ), ˙ 1 , ϕ2 ) = λ(g (Ag ˙ (ϕ1 , Ag2 ) = λ(ϕ1 , g2 ), we get

˙ 1 , ψ2 ) = λ(g1 , ψ2 ), (Ag ¯ 1 , g2 ), ˙ 2 ) = λ(ψ (ψ1 , Ag

¯ − λ) (ψ1 , ψ2 ) − (ϕ1 , ϕ2 ) . 0 = (Af1 , f2 ) − (f1 , Af2 ) = (λ

Therefore, if f1 , f2 ∈ Dom(A), where both f1 and f2 are deﬁned by (1.11), then (ψ1 , ψ2 ) = (ϕ1 , ϕ2 ). In particular, if f1 = f2 , then we have ψ1 = ϕ1 .

(1.12)

Recall that an operator U acting from a Hilbert space H1 into a Hilbert space H2 is called an isometric operator or an isometry if for any x, y ∈ Dom(U ) we have that (U x, U y) = (x, y) or, equivalently, U x = x for all x ∈ Dom(U ). In particular, if Dom(U ) = H1 and Ran(U ) = H2 , the isometric operator U is called unitary. Theorem 1.3.1 (von Neumann). Let A˙ be a closed symmetric operator and let Nλ and Nλ¯ (Im λ = 0) be any pair of its deﬁciency subspaces. Let also A be a symmet˙ Then there exists an isometric operator U from Dom(U ) ⊂ Nλ ric extension of A. into Nλ¯ such that the following representation for f ∈ Dom(A), ϕ ∈ Dom(U ), and Af takes place: f = g + ϕ − U ϕ, (1.13) ¯ ϕ. ˙ + λϕ − λU Af = Ag Conversely, if U is an isometric operator from Dom(U ) ⊂ Nλ into Nλ¯ , then the ˙ operator A deﬁned by (1.13) is a symmetric extension of the operator A. Proof. Since any element f ∈ Dom(A) is uniquely represented in the form ˙ ϕ ∈ Nλ , ψ ∈ Nλ¯ , f = g + ϕ + ψ, g ∈ Dom(A), (1.14) we can consider operator Pλ : Dom(A) → Nλ deﬁned by the rule Pλ f = −ϕ. This operator is linear and its range Ran(Pλ ) is a linear manifold in Nλ . Similarly, if Pλ¯ f = ψ, then Ran(Pλ¯ ) is a linear manifold in Nλ¯ .

1.3. The second von Neumann formula

7

Now consider operator U : Ran(Pλ ) → Ran(Pλ¯ ) deﬁned by U ϕ = ψ, where the vectors ϕ and ψ are deﬁned in (1.14) (and thus ϕ + ψ ∈ Dom(A)). Then by (1.12) we have that U ϕ = ϕ, and U is an isometric operator from Nλ into Nλ¯ . We should also mention that both f and Af are given in the form (1.13). Now let U be an isometric operator acting from Nλ into Nλ¯ and A be deﬁned by (1.13). It is clear that A˙ ⊂ A and all we have to check is that A is symmetric. Repeating the argument we used in the proof of (1.12) we get that, for any f1 and f2 in Dom(A), ¯ − λ) [(U ϕ1 , U ϕ2 ) − (ϕ1 , ϕ2 )] = 0, (Af1 , f2 ) − (f1 , Af2 ) = (λ and hence the operator A is symmetric.

Remark 1.3.2. If in the formulas (1.13) the isometric operator U is replaced by U = −U then the von Neumann formulas take the form f = g + ϕ + Uϕ, ¯ ˙ + λϕ + λUϕ, Af = Ag

(1.15)

that is often seen in some literature on the subject. Corollary 1.3.3. If one of the deﬁciency indices of a closed symmetric operator A˙ is zero, then A˙ does not have symmetric extensions. Proof. Indeed, if one of the deﬁciency numbers is zero we can not possibly construct an isometric operator U with non-zero domain acting from one deﬁciency subspace to the other. A symmetric operator A˙ is called maximal if it does not have a non-trivial ˙ Using the above argument we symmetric extension acting in the same space as A. conclude that a closed symmetric operator is maximal if and only if at least one of its deﬁciency indices is zero. Theorem 1.3.4. A symmetric operator A˙ in a Hilbert space H has self-adjoint extensions in the same space if and only if its deﬁciency indices are equal. Proof. Let A be a non-trivial symmetric extension of the operator A˙ deﬁned by (1.13). In order to ﬁnd the deﬁciency subspaces of A˙ we use (1.13) to see that ¯ ¯ ¯ ∈ M ¯ ⊕ Dom(U ), (A − λI)f = (A˙ − λI)g + (λ − λ)ψ λ where Dom(U ) ⊂ Nλ . On the other hand, if h ∈ Mλ¯ ⊕Dom(U ), then h = (A−λI)f , where f is a vector from Dom(A). Thus, (A − λI)Dom(A) = Mλ¯ ⊕ Dom(U ),

8

Chapter 1. Extensions of Symmetric Operators

and deﬁciency subspace Nλ of the operator A is deﬁned as ¯ Nλ = H (A − λI)Dom(A) = Nλ Dom(U ). Similarly we ﬁnd that Nλ¯ = Nλ¯ Ran(U ), where Ran(U ) is the range of the operator U . Consequently, if Dom(U ) = Nλ (or Ran(U ) = Nλ¯ ), then Nλ = {0} (or Nλ¯ = {0}), and one of the deﬁciency indices of A is zero. This means that under the above conditions our operator A is maximal. Moreover, if Dom(U ) = Nλ ,

Ran(U ) = Nλ¯ ,

(1.16)

then both deﬁciency indices of the operator A are zeros implying that A is a self-adjoint operator. It also means that in this case operator U is unitary. Therefore, an extension A of a symmetric operator A˙ deﬁned by (1.13) is self-adjoint if and only if operator U in (1.13) is a unitary operator mapping the deﬁciency subspace Nλ onto the deﬁciency subspace Nλ¯ . Hence the dimensions of Nλ and Nλ¯ are the same and the deﬁciency indices are equal. Remark 1.3.5. Combining Theorems 1.3.4 and 1.3.1 one can see that formulas (1.13) set up a one-to-one correspondence between all the unitary operators U satisfying (1.16) and the set of all self-adjoint extensions A of the symmetric ˙ operator A.

1.4 The Cayley transform ˙ Let A˙ be a symmetric operator in H and λ be such that Im λ = 0 and h ∈ Dom(A). We set ¯ (A˙ − λI)h = f, (A˙ − λI)h = g,

(1.17) (1.18)

where f ∈ Mλ¯ and g ∈ Mλ , respectively. It is easy to see that (1.17) sets a ˙ onto Mλ¯ , while (1.18) deﬁnes a one-to-one one-to-one mapping from Dom(A) ˙ onto Mλ . Thus for every f ∈ Mλ¯ there is a unique element mapping from Dom(A) ˙ satisfying (1.17). Once we have deﬁned this element h, we ﬁnd g ∈ Mλ h ∈ Dom(A) ˙ with the domain using (1.18). Therefore we can deﬁne a linear operator Uλ (A) ˙ = M ¯ and the range Ran(Uλ (A)) ˙ = Mλ such that Dom(Uλ (A)) λ ˙ g = Uλ (A)f, where

¯ −1 . ˙ = (A˙ − λI)(A˙ − λI) Uλ (A) ˙ is an isometric operator. It is linear and It is easy to see that Uλ (A) f 2 = (A˙ − (Reλ)I)h2 + (Imλ)2 h2 = g2.

(1.19)

1.4. The Cayley transform

9

˙ is called a Cayley transform of a symmetric operator A. ˙ This operator Uλ (A) ˙ Solving (1.19) for A we get ¯ λ (A) ˙ − λI Uλ (A) ˙ − I −1 . A˙ = λU (1.20) We note that the Cayley transform of a self-adjoint operator is a unitary operator. It is very important that the deﬁciency indices (n+ , n− ) of the operator A˙ ˙ Indeed are the same as the deﬁciency indices of the operator Uλ (A). n+ = dim(H Mλ¯ ), Im λ > 0, ˙ = M ¯ , so n+ = def (Dom(Uλ (A))). ˙ but Dom(Uλ (A)) Similarly, if Im λ < 0, then λ ˙ n− = def (Ran(Uλ (A))). Here by “def” we denote the deﬁciency of a set, that is the dimension of its orthogonal complement. Theorem 1.4.1. If U is an isometric operator such that Ran(U − I) is dense in H, then the operator A˙ which is deﬁned by the formula ¯ − λI U − I −1 A˙ = λU is symmetric, densely-deﬁned operator, and the operator U is its Cayley transform, ˙ i.e., U = Uλ (A). Proof. Since Ran(U − I) is dense in H, the inverse operator (U − I)−1 exists, and therefore, the operator ¯ − λI U − I −1 A˙ = λU exists and its domain is dense in H. We show that this operator is symmetric. Let ˙ = Ran(U − I) so that f and g be arbitrary elements of Dom(A) f = U ϕ − ϕ, Then

g = U ψ − ψ,

ϕ, ψ ∈ Dom(U ).

¯ − λI)ϕ = λU ¯ ϕ − λϕ, ˙ = (λU Af ¯ − λI)ψ = λU ¯ ψ − λψ. ˙ = (λU Ag

Therefore, ¯ ϕ − λϕ, U ψ − ψ) = (λ ¯ + λ)(ϕ, ψ) − λ(U ¯ ϕ, ψ) − λ(ϕ, U ψ), ˙ g) = (λU (Af, and ¯ ψ − λψ) = (λ + λ)(ϕ, ¯ ¯ ϕ, ψ) − λ(ϕ, U ψ), ˙ = (U ϕ − ϕ, λU (f, Ag) ψ) − λ(U ˙ g) = (f, Ag). ˙ so that (Af, The proof of the relation ¯ −1 U = (A˙ − λI)(A˙ − λI) ˙ is not diﬃcult. Thus, the operator U is the Cayley transform of the operator A, ˙ i.e., U = Uλ (A).

10

Chapter 1. Extensions of Symmetric Operators The next theorem immediately follows from the proposition proved above.

Theorem 1.4.2. Let A1 and A2 be symmetric operators and let Uλ (A1 ) and Uλ (A2 ) be their Cayley transforms. Then A2 is an extension of A1 if and only if Uλ (A2 ) is an extension of Uλ (A1 ). The theorem above allows us reduce the problem of ﬁnding extensions of a symmetric operator to the problem of ﬁnding isometric extensions of its Cayley transform. Since the closed linear manifolds F and G can be the domain and range of an isometric operator if and only if their dimensions coincide, we can construct ˙ as follows. In the deﬁciency spaces isometric extensions of Cayley transform Uλ (A) ˙ ˙ HDom(Uλ (A)) and HRan(Uλ (A)) we choose two subspaces of equal dimensions F and G and construct an arbitrary isometric operator W deﬁned on F with G as ˙ ⊕F its range. Further, we deﬁne a linear operator U on Dom(U ) = Dom(Uλ (A)) ˙ and Ran(U ) = Ran(Uλ (A)) ⊕ G by the formula ˙ ˙ Uλ (A)f, for f ∈ Dom(Uλ (A)); Uf = W f, for f ∈ F . ˙ and also changing F , G, and W Obviously, U is an isometric extension of Uλ (A) ˙ In order to ﬁnd a symmetric we will obtain all isometric extensions U of Uλ (A). ˙ ˙ extend as prescribed extension A of A we need to take the Cayley transform of A, above, and ﬁnally invert the resulting extensions U to get A.

1.5 Non-densely deﬁned symmetric operators and semi-deﬁciency subspaces In the present section we will describe the deﬁciency structure of a symmetric operator A˙ with non-dense domain. According to [3] the aperture of two linear manifolds M1 and M2 in H is denoted by Θ(M1 , M2 ) and deﬁned as Θ(M1 , M2 ) := P1 − P2 , where P1 , P2 are the orthogonal projection operators on the subspaces M1 and M2 , respectively. It follows from the deﬁnition of aperture that Θ(M1 , M2 ) = Θ(M1 , M2 ) = Θ(H M1 , H M2 ). Applying the relation P2 − P1 = P2 (I − P1 ) − (I − P2 )P1 , to an element h ∈ H yields (P2 − P1 )h = P2 (I − P1 )h − (I − P2 )P1 h.

1.5. Non-densely deﬁned operators and semi-deﬁciency subspaces

11

Then the orthogonality of P2 (I − P1 )h and (I − P2 )P1 h implies (P2 − P1 )h2 = P2 (I − P1 )h2 + (I − P2 )P1 h2 ≤ (I − P1 )h2 + P1 h2 = h2 . The last inequality shows that the aperture of two linear manifolds does not exceed 1, i.e., Θ(M1 , M2 ) ≤ 1. Moreover, one can see that the aperture always equals 1 if one of the manifolds contains a non-zero vector orthogonal to the other manifold. It is known that the relation Θ(M1 , M2 ) = max

sup f ∈M1 ,||f ||=1

||(I − P1 )f ||,

sup f ∈M2 ,||f ||=1

||(I − P2 )f ||

(1.21)

holds. In what follows we denote by [H1 , H2 ] the class of all bounded linear operators from H1 into H2 . Lemma 1.5.1. Let M1 and M2 be subspaces of a Hilbert space H. Then the following conditions are equivalent (i) P1 M2 = M1 and M2 ∩ M1⊥ = {0}, (ii) P2 M1 = M2 and M1 ∩ M2⊥ = {0}, (iii) Θ(M1 , M2 ) < 1, where Mk⊥ := H Mk , k = 1, 2. Proof. Suppose P1 M2 = M1 and M2 ∩ M1⊥ = {0}. Then ker(P1 M2 ) = {0} and by Banach’s inverse mapping theorem there exists a number c > 0 such that ||P1 h|| ≥ c||h||

for all

h ∈ M2 .

Let the operator S ∈ [M2 , M2 ] be deﬁned as S := P2 P1 M2 . Then

(Sh, h) = (P2 P1 h, h) = ||P1 h||2 ≥ c2 ||h||2 , h ∈ M2 .

It follows that ||Sh|| ≥ c2 ||h|| for all h ∈ M2 . Hence ||P2 P1 h|| ≥ c||h|| ≥ c2 ||P1 h||, h ∈ M2 ⇒ ||P2 g|| ≥ c2 ||g||, g ∈ M1 . It follows that M1 ∩ M2⊥ = {0}. Since M1 ∩ M2⊥ = {0} we get P2 M1 = M2 . In addition, the right-hand side of (1.21) is less then 1. So, (i)⇒(ii) and (i)⇒(iii). Similarly, (ii)⇒(i) and (ii)⇒(iii).

12

Chapter 1. Extensions of Symmetric Operators Now suppose Θ(M1 , M2 ) = γ < 1. Then from (1.21) we get that ||P1 h||2 ≥ (1 − γ 2 )||h||2 for all h ∈ M2 , ||P2 g||2 ≥ (1 − γ 2 )||g||2 for all g ∈ M1 .

Hence M2 ∩ M1⊥ = {0}, M1 ∩ M2⊥ = {0}, and P1 M2 = M1 , P2 M1 = M2 .

˙ and A˙ ∗ be the adjoint to the operator A˙ (we consider A˙ Let H0 = Dom(A), ˙ as acting from H0 into H). It is easy to see that for the symmetric operator A, ∗ ˙ ˙ Dom(A) ⊂ Dom(A ), and ˙ A˙ ∗ g = P Ag,

˙ g ∈ Dom(A),

where P is an orthogonal projection of H onto H0 . Following our notation for the dense case we set L := H H0 ,

˙ Mλ := (A˙ − λI)Dom(A),

Nλ := (Mλ¯ )⊥ .

Lemma 1.5.2. If Im λ = 0, then L ∩ Mλ = {0}. ¯ ∈ L ∩ Mλ , g ∈ Dom(A). ¯ and, since ˙ − λg ˙ Then Ag ˙ = h + λg Proof. Let h = Ag (h, g) = 0, ˙ = Imλ · (g, g). Im(g, Ag) But for symmetric operators ˙ g) = 0, Im(Ag,

and therefore g = 0 and h = 0. Let PNλ be the orthogonal projection operator onto Nλ . We set Bλ = PNλ L,

(1.22)

Nλ = Nλ Bλ .

(1.23)

˙ The subspace Nλ is called the semi-deﬁciency subspace of the operator A. Recall that P is the orthogonal projection in H onto H0 . We deﬁne the operator A˙ 0 by the formula ˙ A˙ 0 = P A,

˙ Dom(A˙ 0 ) = Dom(A).

(1.24)

Then A˙ 0 is a densely deﬁned symmetric operator in the Hilbert space H0 . Indeed, ˙ we have for f, g ∈ Dom(A) ˙ g) = (Af, ˙ g) = (f, Af ˙ ) = (f, P Ag) ˙ = (f, A˙ 0 g). (A˙ 0 f, g) = (P Af,

1.5. Non-densely deﬁned operators and semi-deﬁciency subspaces

13

Theorem 1.5.3. The semi-deﬁciency subspace Nλ of a symmetric operator A˙ is the defect subspace Nλ (A˙ 0 ) of the operator A˙ 0 of the form (1.24). Therefore the dimensions of semi-deﬁciency subspaces of a symmetric operator A˙ are the same for all values of λ from the open upper (resp. lower) half-plane. ˙ Thus Proof. Let ϕ ∈ Nλ . Then ϕ is orthogonal to L and therefore ϕ ∈ Dom(A). P ϕ = ϕ and ¯ = (ϕ, Ag ¯ = 0, ˙ − λg) ˙ − λg) (ϕ, P Ag and ϕ ∈ Nλ (A˙ 0 ). Thus,

˙ g ∈ Dom(A),

Nλ ⊂ Nλ (A˙ 0 ).

Conversely, if ψ ∈ Nλ (A˙ 0 ), then ψ = P ψ and hence ¯ = (ψ, P Ag ¯ = 0, ˙ − λg) ˙ − λg) (ψ, Ag

˙ g ∈ Dom(A).

Since ψ ∈ Nλ and ψ ∈ H0 , we get ψ ⊥ Bλ and therefore Nλ (A˙ 0 ) ⊂ Nλ .

The numbers dimNλ and dimNλ¯ (Im λ = 0) are called the semi-defect num˙ bers or the semi-deﬁciency indices of the operator A. Theorem 1.5.4. The following statements are equivalent: (i) Bλ is a subspace at least for one λ, Im λ = 0, (ii) Θ(L, Bλ ) < 1 at least for one λ, Im λ = 0, (iii) the operator A˙ 0 is closed. Proof. (i)⇒(ii)⇒(iii). Suppose Bλ is a subspace for some λ, Im λ = 0. From Lemma 1.5.1 and Lemma 1.5.2 we get that Θ(L, Bλ ) < 1. Hence Θ(H0 , H Bλ ) < 1. Again by Lemma 1.5.1 we get P (H Bλ ) = H0 . We have the equality ¯ H Bλ = (A˙ − λI)Dom( A˙ 0 ) ⊕ Nλ . Since Nλ ⊂ H0 , and ¯ ¯ P (A˙ − λI)Dom( A˙ 0 ) = (A˙ 0 − λI)Dom( A˙ 0 ),

(1.25)

14

Chapter 1. Extensions of Symmetric Operators

¯ we get that the linear manifold (A˙ 0 − λI)Dom( A˙ 0 ) is a subspace (in H0 ). Hence ˙ the operator A0 is closed. (iii)⇒(ii). Let A˙ 0 be a closed operator. Then the linear manifold (A˙ 0 − ¯ λI)Dom(A˙ 0 ) is a subspace for all non-real λ. Since ¯ ¯ P (A˙ − λI)Dom( A˙ 0 ) = (A˙ 0 − λI)Dom( A˙ 0 ), ¯ and (A˙ 0 − λI)Dom( A˙ 0 ) ⊕ Nλ = H0 , we get from (1.25) that P (H Bλ ) = H0 , and (H Bλ ) ∩ L = {0}. Now Lemma 1.5.1 yields Θ(H0 , H Bλ ) < 1. Hence Θ(L, Bλ ) < 1. (ii)⇒(i). Let Θ(L, Bλ ) < 1. Then Θ(L, Bλ ) < 1. Let PBλ be the orthogonal projection in H onto Bλ . From Lemma 1.5.1 we get PBλ L = Bλ . On the other hand, from (1.22) and the inclusion Bλ ⊆ Nλ we have PBλ L = Bλ . Therefore, Bλ is a subspace.

Corollary 1.5.5. The linear sets Bλ for any λ (Im λ = 0) are either subspaces or non-closed linear sets in the Hilbert space H.

1.6 Symmetric extensions of a non-densely deﬁned symmetric operator The Cayley transform of a symmetric non-densely deﬁned operator A˙ is deﬁned via (1.19), that is ¯ −1 , ˙ = (A˙ − λI)(A˙ − λI) Uλ (A) for (Imλ = 0). Equivalently, ¯ ˙ − λg, f = Ag ˙ = Ag ˙ − λg, Uλ (A)f

˙ g ∈ Dom(A) ˙ . f ∈ Dom(Uλ (A)),

¯ (Im λ = 0) are the points of a regular type for A, ˙ then Since both λ and λ, ¯ = M ¯ and the range Ran(Uλ (A)) ˙ = Ran(A˙ − λI) ˙ = the domain Dom(Uλ (A)) λ Ran(A˙ − λI) = Mλ are the subspaces. Similarly to the dense case in the Section ˙ is an isometry. 1.4 one can show that the operator Uλ (A) Lemma 1.6.1. For any element h ∈ L we have ˙ M ¯ h = PM h, Uλ (A)P λ λ

(Im λ = 0).

1.6. Symmetric extensions of a non-densely deﬁned symmetric operator

15

˙ Proof. Since Uλ (A)M ¯ = Mλ , we only need to show that the element h − λ ˙ Uλ (A)PMλ¯ h is orthogonal to the subspace Mλ . Indeed, if ψ ∈ Mλ , then ψ = ˙ Uλ (A)ϕ, where ϕ ∈ Mλ¯ . Thus, ˙ M ¯ h, ψ = h, Uλ (A)ϕ ˙ −ϕ , h − Uλ (A)P λ but by (1.20) ˙ − ϕ = g ∈ Dom(A), ˙ Uλ (A)ϕ ˙ M ¯ h, ψ = (h, g) = 0. h − Uλ (A)P λ

and hence

We introduce a new linear operator Vλ deﬁned by the formula Vλ PNλ h = PNλ¯ h,

h ∈ L, Im λ = 0.

(1.26)

By Lemma 1.5.2 this operator Vλ is well deﬁned. It is called the exclusion operator and plays an important role in the extension theory of operators with non-dense domain. The domain and the range of the operator Vλ are the sets Bλ and Bλ¯ respectively. Lemma 1.6.1 implies that Vλ PNλ¯ h2 = PNλ¯ h2 = h2 − PMλ¯ h2 = h2 − PMλ h2 = PNλ h2 , and operator Vλ is an isometry. This means that the dimensions of the subspaces Bλ and Bλ¯ are the same. Now we can conclude that the equality of the semideﬁciency numbers of a symmetric operator A˙ implies the equality of its deﬁciency numbers. The inverse statement is not true when dim L = ∞, i.e., one can easily construct a non-densely deﬁned symmetric operator with equal deﬁciency indices but with nonequal semi-deﬁciency indices. We should note several basic properties of the operator Vλ . According to Theorem 1.5.4, the operator Vλ is closed if and only if Θ(L, Bλ ) < 1. The above inequality also implies that the operator Vλ is closed if the symmetric operator A˙ is bounded. ˙ Let F be the orthogonal complement of the linear span of the sets Dom(A) ˙ in H. One may notice that F is the set of elements ϕ ∈ Dom(Vλ ) for and Ran(A) ¯ ˙ − λϕ) which Vλ ϕ = ϕ. Indeed, if ϕ ∈ F, then (ϕ, Aϕ = 0, and hence ϕ ∈ Nλ , ϕ ∈ Nλ¯ , and Vλ ϕ = ϕ. Conversely, if Vλ ϕ = ϕ then ϕ ∈ Nλ , ϕ ∈ Nλ¯ , and ¯ = 0, ˙ − λg) (ϕ, Ag

˙ − λg) = 0, (ϕ, Ag

˙ g ∈ Dom(A).

The latter implies that ϕ ∈ F. λ of the Cayley transform Uλ (A) ˙ of the symmetric An isometric extension U operator A˙ is called an admissible extension if λ f = f, U implies that f = 0.

λ ), f ∈ Dom(U

16

Chapter 1. Extensions of Symmetric Operators

λ is an admissible extension of Uλ (A), ˙ then the operator Lemma 1.6.2. If U ¯ U λ − λI)( λ − I)−1 , A = (λU

(1.27)

˙ is a symmetric extension of the symmetric operator A. ˙ All we have to show is that A is Proof. It is obvious that A is an extension of A. symmetric. Note that operator A is deﬁned on the elements g such that λ f − f, g=U

λ ), f ∈ Dom(U

(1.28)

on which it takes the values ¯ λ f − λf. Ag = λU

(1.29)

Consequently, ¯ U λ f − λf, λ f − f ) Im(Ag, g) = Im(λU ¯ ¯ U λ f, f ) − λ(f, λ f ) = 0, = Im (λ + λ)(f, f ) − λ(U

which proves the lemma.

It is easy to see that the inverse statement takes place as well: for every symmetric extension A of operator A˙ there exists an admissible isometric extenλ = Uλ (A) (the Cayley transform of A) of the isometric operator Uλ (A). ˙ sion U Therefore, there is a one-to-one correspondence between symmetric extensions of ˙ In paroperator A˙ and admissible isometric extensions of the operator Uλ (A). ˙ ticular, in order to obtain the self-adjoint extensions of A we need to construct ˙ In other words we should have admissible unitary extensions of Uλ (A). ¯ = H, Ran(Uλ (A)) = Ran(A − λI) = H. Dom(Uλ (A)) = Ran(A − λI)

(1.30)

λ of the We can summarize this as follows: An isometric (unitary) extension U ˙ ˙ Cayley transform Uλ (A) of symmetric operator A corresponds to the symmetric (self-adjoint) extension A of the symmetric operator A˙ if and only if the operator λ is an admissible isometric (unitary) extension of the operator Uλ (A). ˙ U ˙ Now we focus on the properties of non-admissible extensions of Uλ (A). λ of the operator Lemma 1.6.3. If for an isometric non-admissible extension U ˙ we have that, for some element h ∈ Dom(U λ ), Uλ (A) λ h = h, U then h ∈ L.

1.6. Symmetric extensions of a non-densely deﬁned symmetric operator

17

λ h = h, then for all f ∈ Dom(U λ ), Proof. If U λ h − h, Uλ (A)f ˙ U = 0. λ f = Uλ (A)f ˙ the last equality implies Since U ˙ h, f − Uλ (A)f = 0. ˙ By (1.28) we have (h, g) = 0 for g ∈ Dom(A).

Corollary 1.6.4. If operator A˙ is densely deﬁned, then all isometric extensions of ˙ are admissible. the operator Uλ (A) λ of the operator Theorem 1.6.5. If for some isometric non-admissible extension U ˙ Uλ (A) and for some element h ∈ Dom(Uλ ) we have that λ h = h, U then

λ PN h = Vλ PN h = PN ¯ h. U λ λ λ

λ is an Proof. It follows from Lemma 1.6.3 that h ∈ L. Since the operator U ˙ we have extension of Uλ (A) λ PM ¯ h = Uλ (A)P ˙ M ¯ h, U λ λ and thus by Lemma 1.6.1 Then

λ PM ¯ h = PM h. U λ λ

λ PN h = U λ (h − PM ¯ h) = h − PM h = PN ¯ h. U λ λ λ λ

The last theorem allows us to describe the subclass of all admissible exten˙ Evidently, the sions in the class of all isometric extensions of the operator A. admissible extensions are the ones that do not take the same value as operator Vλ λ of the Cayley transform on any element of Bλ . Thus an isometric extension U ˙ Uλ (A) coincides with the Cayley transform Uλ (A) of a symmetric extension A of λ ϕ = Vλ ϕ, where Vλ is the exclusion the operator A˙ if and only if the equality U operator (1.26), implies ϕ = 0. It follows from the deﬁnition of the operator Vλ and Lemma 1.6.1 that oper˙ Vλ is isometric 1 and leaves every element of L invariant, ator Uλ (A) ˙ Vλ h = h, Uλ (A) h ∈ L. (1.31) ˙ ˙ The operator Uλ (A)V λ is the minimal isometric extension of Uλ (A) leaving every element of L invariant. 1 Here

we denote by a direct sum of two operators [169]. Let S and T be linear operators with Dom(S) ∩ Dom(T ) = {0}. Then (D T )(ϕ + ψ) = Sϕ + T ψ, ϕ ∈ Dom(S), ψ ∈ Dom(T ).

18

Chapter 1. Extensions of Symmetric Operators

1.7 Indirect decomposition and the Krasnoselski˘i formulas λ of the operator Uλ (A) ˙ is A process of construction of an isometric extension U equivalent to the construction of an isometric operator U with domain Dom(U ) ⊂ Nλ and range Ran(U ) ∈ Nλ¯ so that λ = Uλ (A) ˙ U. U We are particularly interested in a possibility of constructing admissible unitary λ of the operator Uλ (A). ˙ Obviously, these extensions exist only if our extensions U ˙ operator A has equal deﬁciency indices. Moreover, the equality of deﬁciency indices is also a suﬃcient condition of the existence of unitary admissible extensions of ˙ the Cayley transform of the operator A. Let us assume ﬁrst that semi-deﬁciency indices of A˙ are zeros. This means Bλ = Nλ ,

Bλ¯ = Nλ¯ .

λ of Uλ (A) ˙ is determined by the formula In this case every admissible extension U λ = Uλ (A) ˙ U, U where U is an isometric operator deﬁned on Bλ with the range in Nλ¯ that does not take the same value as Vλ on any element of Bλ and U is the closure of operator U in H. For example, one can use U = −Vλ . Now let us assume that our operator A˙ has equal deﬁciency indices. Let also ˙ We U be some, generally speaking non-admissible, unitary extension of Uλ (A). will show that this extension can be corrected, i.e., one can construct an admissible λ using U . Let L be a subspace invariant under U . By Lemma 1.6.3, extension U L ⊂ L. Here, by Bλ we denote the projection of the subspace L onto the deﬁciency subspace Nλ . Consider the part U 0 of the operator U deﬁned on H Bλ . The ˙ By Lemma 1.6.2 and (1.27) operator U 0 is a closed admissible extension of Uλ (A). ˙ Also we should note that there exists a closed symmetric extension A0 of A. Dom(A0 ) ⊕ L = H.

(1.32)

Indeed, let Vλ be a part of the operator Vλ deﬁned on Bλ . The operator U 0 has ˜ 0 leaving every element of H Dom(A0 ) invariant. To an isometric extension U construct such an extension one needs to construct an operator Vλ ﬁrst and then ˜ 0 will be an isometric extension of the operator U 0 V . use (1.31). The operator U λ Since U is the unitary closure of the operator U 0 Vλ , then U is an extension of ˜ 0 . But the subspace L is only invariant with respect to U which implies (1.32). U

1.7. Indirect decomposition and the Krasnoselski˘i formulas

19

According to (1.32) the operator U 0 is a Cayley transform of the operator A whose semi-deﬁciency indices are zeros. In this case, as it was shown above, ˜ . Evidently these extensions the operator U 0 has admissible unitary extensions U ˙ are admissible unitary extensions of Uλ (A). If we summarize the above reasoning and the formulas (1.28) and (1.29), we get the Krasnoselski˘i formulas that generalize the von Neumann Theorem 1.3.1. However, unlike the von Neumann formulas (1.13) for the dense case, in the representation (1.33) below one must choose an admissible (not arbitrary) isometric operator U . 0

Theorem 1.7.1. A closed symmetric operator A˙ acting in the Hilbert space H has self-adjoint extensions if and only if its deﬁciency indices are equal. In this case a self-adjoint extension A of the operator A˙ is deﬁned by the formula f = g + ϕ − U ϕ, ¯ ϕ, Af = Ag + λϕ − λU

(1.33)

˙ ϕ ∈ Nλ , and U is an admissible isometric operator with where g ∈ Dom(A), Dom(U ) = Nλ ,

Ran(U ) = Nλ¯ .

Similar argument brings us to the generalization of the second von Neumann Theorem 1.3.1. Theorem 1.7.2. Any closed symmetric operator A˙ in the Hilbert space H has maximal symmetric extensions in H. ˙ This theorem can be proved even faster if we ﬁrst extend the operator Uλ (A) to U in a way that the symmetric operator

¯ A = (λU − λI)(U − I)−1

has a dense in H domain. To do this all we need is ˙ (−Vλ ). U = Uλ (A) Then all isometric extensions of U are going to be admissible. Now we consider symmetric extensions A of a symmetric operator A˙ with non-equal deﬁciency indices that exit to the space H ⊕ H . Choosing the space H of suﬃciently high dimension we can achieve the equality of the deﬁciency indices of A˙ by considering it as acting in a wider space H ⊕ H . This follows from the fact that H is included in both deﬁciency subspaces of A˙ (acting in H ⊕ H ). Thus we can now construct self-adjoint extensions of A˙ in H ⊕ H . Theorem 1.7.3. Every closed symmetric operator has self-adjoint extensions (possibly with exit to the wider space).

20

Chapter 1. Extensions of Symmetric Operators

Now we raise the question of the linear dependence of the sets Nλ , Nλ¯ , and ˙ as deﬁned in Section 1.2. The following theorems by M. Naimark partially Dom(A) answer it. ˙ are linearly dependent if Theorem 1.7.4. The linear sets Nλ , Nλ¯ , and Dom(A) ˙ is not dense in H. In this case and only if Dom(A) ϕ + ψ + g = 0,

˙ (ϕ ∈ Nλ , ψ ∈ Nλ¯ , g ∈ Dom(A)),

holds if and only if there exists an element h ∈ L such that ¯ ˙ − λg Ag , ¯ λ−λ (2) ϕ = PNλ h, (1) PMλ¯ h =

(3) ψ = −PNλ¯ h = −Vλ ϕ. ˙ = H. Then according to Lemma 1.6.1 and the deﬁnition Proof. Suppose Dom(A) ˙ for every element h ∈ L there are elements ϕ ∈ Nλ , ψ ∈ Nλ¯ , and of Uλ (A) ˙ such that (1), (2), and (3) hold and g ∈ Dom(A) h = h =

¯ ˙ − λg Ag ¯ , λ−λ ˙ − λg Ag −ψ + ¯ . λ−λ ϕ+

This implies that ϕ + ψ + g = 0, ˙ are linearly dependent. i.e., the linear sets Nλ , Nλ¯ , and Dom(A) To show the necessity of (1), (2), and (3) we note that ϕ + ψ + g = 0,

˙ (ϕ ∈ Nλ , ψ ∈ Nλ¯ , g ∈ Dom(A)),

is equivalent to ¯ ˙ − λg ˙ − λg Ag Ag + ϕ = (1.34) ¯ ¯ − ψ. λ−λ λ−λ ˙ − λg, ¯ ϕ) = 0, (Ag ˙ − λg, ψ) = 0, and Ag ˙ − λg ¯ = Ag ˙ − λg, we have Since (Ag ϕ = ψ.

(1.35)

Let W be a linear operator deﬁned on elements {kϕ, k ∈ C} by the formula W (kϕ) = −kψ. ˙ W is an isometric extension of Uλ (A). ˙ According to (1.35) the operator Uλ (A) ˙ ˙ Also Uλ (A) W is not an admissible extension of Uλ (A) because ¯ ˙ − λg ˙ − λg Ag Ag ˙ W) (Uλ (A) + ϕ = ¯ ¯ − ψ, λ−λ λ−λ

1.7. Indirect decomposition and the Krasnoselski˘i formulas

21

and, as it follows from (1.34), the element h=

¯ ˙ − λg Ag +ϕ ¯ λ−λ

˙ W . From Lemma 1.6.3 it follows then that is invariant for the operator Uλ (A) ˙ Dom(A) = H. We should also note that the element h ∈ L in the conditions of Theorem 1.7.4 can be deﬁned uniquely. Theorem 1.7.5. Let f be an element in the domain Dom(A) of a symmetric ex˙ Then according to Theorem 1.7.1 it has a tension A of the symmetric operator A. representation f = g + ϕ − U ϕ,

˙ ϕ ∈ Dom(U ) ⊂ Nλ , U ϕ ∈ N ¯ , g ∈ Dom(A), λ

˙ and ϕ ∈ Nλ . and uniquely deﬁnes the elements g ∈ Dom(A) Proof. Suppose f has two representations f

= g1 + ϕ1 − U ϕ1 ,

f

= g1 + ϕ2 − U ϕ2 ,

˙ and ϕ1 , ϕ2 ∈ Nλ . Then where g1 , g2 ∈ Dom(A) (g1 − g2 ) + (ϕ1 − ϕ2 ) − U (ϕ1 − ϕ2 ) = 0, and the previous theorem implies that U (ϕ1 − ϕ2 ) = V (ϕ1 − ϕ2 ). By Theorem 1.7.1 we have ϕ1 = ϕ2 and thus g1 = g2 .

Chapter 2

Geometry of Rigged Hilbert Spaces In this chapter we study extensions of symmetric non-densely deﬁned operators in the triplets H+ ⊂ H ⊂ H− of rigged Hilbert spaces. The Krasnoselski˘i formulas discussed in Section 1.7 are based upon the indirect decomposition (1.33), where deﬁciency subspaces and the domain of symmetric operator may be linearly dependent. Introduction of the rigged Hilbert spaces allows us to obtain the direct decomposition and parameterization for the domain of the adjoint operator. This direct decomposition is written in terms of the semi-deﬁciency subspaces and is an analogue of the von Neumann formulas (1.7) and (1.13) for the case of the symmetric operator A˙ whose domain is not dense in H.

2.1 The Riesz-Berezansky operator In this section we are going to equip our Hilbert space H with spaces H+ and H− , called spaces with positive and negative norms, respectively. We start with a Hilbert space H with inner product (x, y) and norm · . Let H+ be a dense in H linear set that is a Hilbert space itself with respect to another inner product (x, y)+ generating the norm · + . We assume that x ≤ x+ , (x ∈ H+ ), i.e., the norm · + generates a stronger than · topology in H+ . The space H+ is called the space with positive norm. Now let H− be a space dual to H+ . It means that H− is a space of linear functionals deﬁned on H+ and continuous with respect to · + . By the · − we denote the norm in H− that has a form h− = sup

u∈H+

|(h, u)| , h ∈ H. u+

Y. Arlinskii et al., Conservative Realizations of Herglotz–Nevanlinna Functions, Operator Theory: Advances and Applications, DOI 10.1007/978-3-7643-9996-2_2, © Springer Basel AG 2011

23

24

Chapter 2. Geometry of Rigged Hilbert Spaces

The value of a functional f ∈ H− on a vector u ∈ H+ is denoted by (u, f ). The space H− is called a space with negative norm. Further on in this chapter we will need to consider an embedding operator σ : H+ → H that embeds H+ into H. Since σf ≤ f + for all f ∈ H+ , then σ ∈ [H+ , H]. The adjoint operator σ ∗ maps H into H− and satisﬁes the condition σ ∗ f − ≤ f for all f ∈ H. Since σ is a monomorphism with a (·)-dense range, then σ ∗ is a monomorphism with (−)-dense range. By identifying σ ∗ f with f (f ∈ H) we can consider H embedded in H− as a (−)-dense set and f − ≤ f . Also, the relation (σf, h) = (f, σ ∗ h),

f ∈ H+ , h ∈ H,

implies that the value of the functional σ ∗ h ∈ H calculated at a vector f ∈ H+ as (f, σ∗ h) corresponds to the value (f, h) in the space H. It follows from the Riesz representation theorem that there exists an isometric operator R which maps H− onto H+ such that (f, g) = (f, Rg)+ (∀f ∈ H+ , g ∈ H− ) and Rg+ = g− . Now we can turn H− into a Hilbert space by introducing (f, g)− = (Rf, Rg)+ . Thus, (f, g)− = (f, Rg) = (Rf, g) = (Rf, Rg)+ , −1

(u, v)+ = (u, R

−1

v) = (R

−1

u, v) = (R

−1

u, R

v)− ,

(f, g ∈ H− ), (u, v ∈ H+ ).

(2.1)

The operator R (or R−1 ) will be called the Riesz-Berezansky operator. Applying the above reasoning, we deﬁne a triplet H+ ⊂ H ⊂ H− to be called the rigged Hilbert space. In what follows we use symbols (+), (·), and (−) to indicate the norms · + , ·, and ·− by which geometrical and topological concepts are deﬁned in H+ , H, and H− . When considering continuity or closeness of an operator, we ﬁrst indicate the topology of its domain and then the topology of the range. For instance, an operator B is called (−, ·)-continuous, if Dom(B) ⊂ H− ,

Ran(B) ⊂ H,

Bh < ∞. h∈Dom(B) h− sup

Similarly, the closure of a set L using the norms · + , · , and · − is denoted (+)

(−)

by L , L, and L , respectively. If L is a subset from H+ , then its orthogonal complement L⊥ will be a set of those functionals from H− that annihilate L. Thus, L⊥ ⊂ H− . Likewise, if L ⊂ H− then its orthogonal complement L⊥ ⊂ H+ is a set of those elements x ∈ H+ that annihilate all the functionals from L. If L ⊂ H, then L⊥ ⊂ H and is deﬁned as a set of elements from H that are (·)-orthogonal to L. The following theorem establishes the relationship between the orthogonal complements of the set L.

2.1. The Riesz-Berezansky operator

25

1. If L is a subspace in H, then

Theorem 2.1.1.

(−) ⊥ L = L⊥ ∩ H+ ,

⊥

(L ∩ H+ ) = L⊥

(−)

.

(2.2)

2. If L is a subspace in H+ , then ⊥ L = L⊥ ∩ H.

(2.3)

3. If L is a subspace in H− , then ⊥

(L ∩ H) = L⊥ . (−)

Proof. Let f ∈ L (−)

(2.4)

. Then there is a sequence of elements {fn } ⊂ L such that

fn −→ f . Therefore (h, fn ) → (h, f ) for any h ∈ H+ . In particular, if h ∈ L⊥ ∩H+ , then (h, fn ) = 0, and consequently, (h, f ) = 0. Thus, L⊥ ∩ H+ ⊂ (L (−) ⊥

(−) ⊥

) . (−)

Conversely, let h ∈ (L ) . Then h ∈ H+ and (h, f ) = 0 for any f ∈ L , and, in particular, for any f ∈ L. Hence h ∈ L⊥ and h ∈ L⊥ ∩ H+ . This proves the ﬁrst part of (2.2). The second part is being proved similarly by substituting L⊥ for L. Statements (2.3) and (2.4) can be proved in a similar way. We just note that one can obtain (2.4) from (2.3) by the respective substitution of L(⊂ H+ ) for L⊥ (⊂ H− ). Theorem 2.1.1 can be interpreted as asserting the commutativity of the following diagram: ←− R −−−−→ R ⏐ ⏐ ⏐↑ ↓⏐ −→

R+ ←−−−− R− Here R+ , R, and R− are the classes of all (+)-closed, (·)-closed, and (−)-closed linear manifolds in H+ , H, and H− , respectively. The horizontal arrows denote the passage to the orthogonal complement. The short down arrow ↓ denotes intersection with H+ , and the long down arrow stands for the (−)-closure. The long up arrow on the left represents (·)-closure whenever the short up arrow ↑ on the right denotes intersection with H. If B is an operator in the class [H+ , H− ], then its adjoint operator B ∗ is deﬁned by the formula (Bf, g) = (f, B ∗ g) (∀f, g ∈ H+ ). This operator B ∗ acts from H+ into H− , is bounded, and therefore B ∗ ∈ [H+ , H− ] as well. Thus, the class [H+ , H− ] is invariant under taking adjoint. The class [H− , H+ ] has a similar property. The concept of a bounded self-adjoint operator is, therefore, well deﬁned in both of these classes. For instance, for the class [H+ , H− ] such an operator is characterized by the quadratic functional (Bf, f ) (f ∈ H+ ) taking real values

26

Chapter 2. Geometry of Rigged Hilbert Spaces

only. If (Bf, f ) ≥ 0 for all f ∈ H+ , then B is called non-negative. For an operator B ∈ [H+ , H− ] we introduce a new operator ˆ = B Dom(B), ˆ B

ˆ = {f ∈ H+ : Bf ∈ H}. Dom(B)

(2.5)

ˆ is called a quasi-kernel of the operator B. This operator B For the remainder of this text we will need the following theorem. Theorem 2.1.2. Let H1 , H2 , and H be Hilbert spaces and let B and C be operators in [H1 , H] and [H2 , H], respectively. The following conditions are equivalent: (i) Ran(B) ⊂ Ran(C); (ii) ker(C ∗ ) ⊂ ker(B ∗ ) and B ∗ h < ∞; ∗ ∗ ) C h h∈H,h∈ker(C / sup

(iii) there exists an operator W ∈ [H1 , H2 ] such that B = CW . Proof. First we show that (i)⇒(ii). The ﬁrst part of condition (ii) can be derived from (i) by passing to the orthogonal complements. Now let us assume that the second part of the condition (ii) is not true. Then there exists a sequence of fn ∈ H such that B ∗ fn → ∞ and C ∗ fn → 0. By condition (i) for any h1 ∈ H1 there is an element h2 ∈ H2 such that Bh1 = Ch2 . We have (B ∗ fn , h1 ) = (fn , Bh1 ) = (fn , Ch2 ) = (C ∗ fn , h2 ) → 0. Therefore, B ∗ fn converges weakly to zero which contradicts B ∗ fn → ∞. (ii)⇒(iii). To every vector ϕ = C ∗ f ∈ Ran(C ∗ ) ⊂ H2 we assign a vector ψ ∈ B ∗ f ∈ Ran(B ∗ ) ⊂ H1 . According to condition (ii), the operator ψ = U ϕ is well deﬁned and bounded. We extend U onto H2 to an operator U ∈ [H2 , H1 ] for which B ∗ = U C ∗ . Then, B = CU ∗ and we can deﬁned W = U ∗ . It is very easy to see that (iii)⇒(i). Remark 2.1.3. Theorem 2.1.2 can be stated equivalently in the form: For every A, B ∈ [H, H] the following statements are equivalent: (i) Ran(A) ⊂ Ran(B); (ii) A = BC for some C ∈ [H, H]; (iii) AA∗ ≤ λBB ∗ for some λ ≥ 0. In this case there is a unique C satisfying C2 = inf{ λ : AA∗ ≤ λBB ∗ } and Ran(C) ⊂ Ran(B ∗ ), in which case ker C = ker A. The following three results are based upon Theorem 2.1.2.

2.2. Construction of the operator generated rigging

27

Theorem 2.1.4. Let H+ ⊂ H ⊂ H− be a rigged Hilbert space and G be a Hilbert space. If B ∈ [G, H− ] and Ran(B) ⊂ H, then B ∈ [G, H]. Moreover, if B ∈ [G, H− ] or B ∈ [G, H] and Ran(B) ⊂ H+ , then B ∈ [G, H+ ]. Proof. For the embedding operator σ deﬁned in the beginning of this section we have that σ ∗ ∈ [H, H− ] and Ran(B) ⊂ Ran(σ). According to Theorem 2.1.2 there exists an operator W ∈ [G, H] such that B = σ ∗ W . Since σ ∗ is an embedding of H into H− , then B ∈ [G, H]. The proof of the second statement is similar. Theorem 2.1.5. Let C ∈ [H+ , H− ]. Then C is a monomorphism and C −1 is (−, ·)continuous if and only if Ran(C ∗ ) ⊃ H. Proof. The existence and (−, ·)-continuity of operator C −1 are equivalent to inf

h∈H+

Ch− > 0, h

σh i.e., ker(C) = {0} and sup Ch < ∞. Applying Theorem 2.1.2 we see that the − latter is equivalent to Ran(σ ∗ ) ⊂ Ran(C ∗ ) that means H ⊂ Ran(C ∗ ).

The following theorem can be proven similarly. Theorem 2.1.6. Let C ∈ [H+ , H− ]. Then C is a monomorphism and C −1 is (−, −)continuous if and only if Ran(C ∗ ) ⊃ H+ .

2.2 Construction of the operator generated rigging ˙ is not assumed Let now A˙ be a closed symmetric operator whose domain Dom(A) ˙ = H0 , we can consider A˙ as a densely deﬁned to be dense in H. Setting Dom(A) operator from H0 into H. Clearly, Dom(A˙ ∗ ) is dense in H and Ran(A˙ ∗ ) ⊂ H0 . We introduce a new Hilbert space H+ = Dom(A˙ ∗ ) with inner product (f, g)+ = (f, g) + (A˙ ∗ f, A˙ ∗ g),

(f, g ∈ H+ ),

(2.6)

and then construct the rigged Hilbert space H+ ⊂ H ⊂ H− . Theorem 2.2.1. Let A˙ be a closed symmetric operator in H. Then 1. The operator A˙ is (·, −)-continuous. 2. If A˙ is an extension of A˙ by (·, −)-continuity to H0 , then the Riesz-Berezansky operator is given by the formula R−1 = I + A˙ A˙ ∗ . 3. RH = Dom(A˙ A˙ ∗ ).

(2.7)

28

Chapter 2. Geometry of Rigged Hilbert Spaces

Proof. (1) Since A˙ ∗ h ≤ h+, (∀h ∈ H+ ), then ˙ ˙∗ ˙∗ ˙ − = sup |(Ag, h)| = sup |(g, A h)| ≤ sup g · A h ≤ g, Ag h+ h+ h∈H+ h+ h∈H+ h∈H+ ˙ This yields the (·, −)-continuity of A. ˙ Now let A˙ be an extenfor all g ∈ Dom(A). ˙ ˙ sion of A onto Dom(A) = H0 using (·, −)-continuity. We will show that ˙ f ) = (g, A˙ ∗ f ), (Ag,

(g ∈ H0 , f ∈ H+ ).

(2.8)

˙ such that gn → g in Let g ∈ H0 . Then there is a sequence {gn } ⊂ Dom(A) ˙ ˙ (·)-metric. Hence Agn → Ag in (−)-metric. Letting n → ∞ in ˙ n , f ) = (gn , A˙ ∗ f ), (Ag we get (2.8). We should note that (2.8) indicates that A˙ ∈ [H0 , H− ] is the adjoint to A˙ ∗ ∈ [H+ , H0 ] operator. ˙ ∈ H for some g ∈ H0 implies g ∈ Dom(A). ˙ Indeed, The condition that Ag it follows from (2.8) that for an arbitrary f ∈ H+ we have g ∈ Dom(A˙ ∗∗ ) and ˙ = A˙ ∗∗ g. Since A˙ = A˙ ∗∗ we have that g ∈ Dom(A). ˙ Ag (2) For any g, f ∈ H+ , (R−1 g, f ) = (g, f )+ = (g, f ) + (A˙ ∗ g, A˙ ∗ f ) = (g, f ) + (A˙ A˙ ∗ g, f ) = ((I + A˙ A˙ ∗ )g, f ), which implies (2.7). (3) Obviously, for g ∈ Dom(A˙ A˙ ∗ ) R−1 g = (I + A˙ A˙ ∗ )g ∈ H. Conversely, if g ∈ H+ , R−1 g ∈ H. Then A˙ A˙ ∗ g = R−1 g − g ∈ H. As we have shown above ˙ and thus g ∈ Dom(A˙ A˙ ∗ ). So, the conditions g ∈ Dom(A˙ A˙ ∗ ) and A˙ ∗ g ∈ Dom(A) −1 R g ∈ H are equivalent. Theorem 2.2.2. Let f ∈ H0 and A˙ be an extension of A˙ by (·, −)-continuity to H0 . ˙ belongs to H if and only if f ∈ Dom(A). ˙ Then Af ˙ = Af ˙ ∈ H for f ∈ Dom(A). ˙ Proof. The suﬃciency part is obvious because Af ∗ ∗ ˙ ). Since A˙ is closed and Assume that for a g ∈ H+ = Dom(A˙ ), (A˙ g, f ) = (g, Af ∗∗ ˙ ˙ ˙ ˙ ˙ A = A, we have that f ∈ Dom(A) and Af = Af .

2.3 Direct decomposition and an analogue of the ﬁrst von Neumann’s formula We call an operator A˙ regular, if P A˙ is a closed operator in H0 . Here P is an ˙ Obviously, any densely deﬁned orthogonal projection in H onto H0 = Dom(A).

2.3. Direct decomposition and analogue of the ﬁrst von Neumann’s formula

29

closed symmetric operator is regular. For a regular operator A˙ we construct a rigged Hilbert space H+ ⊂ H ⊂ H− using the technique from the previous section. If A˙ is densely deﬁned, then by the ﬁrst von Neumann formula (1.7) we have ˙ Nλ Nλ¯ . H+ = Dom(A)

(2.9)

This decomposition is (+)-orthogonal for λ = ±i. When the domain of A˙ is not dense in H, Theorem 1.7.1 implies an indirect decomposition ˙ + Nλ + Nλ¯ , H+ = Dom(A)

(2.10)

˙ Nλ , and Nλ¯ may be linearly dependent (see Theorem 1.7.4). Now where Dom(A), we are going to derive an analogue of the ﬁrst von Neumann’s formula that has a direct decomposition of the involved linear manifolds. Deﬁne two subspaces of H+ : (+) ˙ D∗ := H+ ∩ H0 , D := Dom(A) . (2.11) Clearly, D is the domain of the closure of a densely deﬁned in H0 symmetric ˙ ∗ ). Hence operator P A˙ and D∗ = Dom((P A) ˙ λ +N ˙ λ¯ , D∗ = D+N

Im λ = 0,

˙ where Nλ and Nλ¯ are deﬁned by (1.23). If A˙ is regular, then D = Dom(A). ∗ According to Theorem 2.1.1 the orthogonal complement of the subspace D is (−) L where L = H H0 . (2.12) This makes N = RL

(−)

(2.13)

∗

a (+)-orthogonal complement of D . Thus we have H+ = D Nλ Nλ¯ N,

(Imλ = 0).

(2.14)

This is a generalization of the ﬁrst von Neumann’s formula. For λ = ±i we obtain the (+)-orthogonal decomposition

Let

H+ = D ⊕ Ni ⊕ N−i ⊕ N.

(2.15)

M = Ni ⊕ N−i ⊕ N,

(2.16)

˙ (⊂ H+ ), i.e., and let F (⊂ H− ) be the (·)-orthogonal complement of Dom(A) ˙ . F = ϕ ∈ H− : (ϕ, f ) = 0 for all f ∈ Dom(A) (2.17) It is clear that

(−)

F = R−1 M = R−1 Ni ⊕ R−1 N−i ⊕ L

,

(2.18)

30

Chapter 2. Geometry of Rigged Hilbert Spaces

and the last decomposition is (−)-orthogonal. Here and below by PG+ we denote the orthogonal projection in H+ onto a subspace G of H+ . Respectively, PG would represent the orthogonal projection in H onto a subspace G of H. Theorem 2.3.1.

1. The operator A˙ ∗ N is symmetric in H and A˙ ∗ N ⊂ D.

2. The operator A˙ ∗ ±iI maps N (+, ·)-isometrically on B±i , where Bλ is deﬁned in (1.22). 3. The operator B˙ given by the relations ˙ = Dom(A) ˙ ⊕ N, Dom(B)

˙ + ˙∗ + B˙ = AP ˙ + A PN , Dom(A)

(2.19)

is closed, densely deﬁned and symmetric in H, and ˙ ⊕ N ⊕ N , B˙ ∗ = AP + ˙∗ + Dom(B˙ ∗ ) = Dom(B) i −i ˙ + A PM . Dom(A)

(2.20)

˙ and f ∈ M. Then Proof. (1) Let g ∈ Dom(A) ˙ A˙ ∗ f ) = (g, f ) + (Ag, ˙ A˙ ∗ f ). 0 = (g, f )+ = (g, f ) + (A˙ ∗ g, A˙ ∗ f ) = (g, f ) + (P Ag, Hence, ˙ A˙ ∗ f ) = (g, −P f ), (Ag,

˙ ∀g ∈ Dom(A).

Consequently, A˙ ∗ f ∈ Dom(A˙ ∗ ) = H+ and (A˙ ∗ )2 f = −P f . On the other hand, since Ran(A˙ ∗ ) ⊂ H0 , we get A˙ ∗ f ∈ D∗ , i.e., A˙ ∗ M ⊂ D∗ . Let ψ, f ∈ M. Then (ψ, A˙ ∗ f )+ = (ψ, A˙ ∗ f ) + (A˙ ∗ ψ, (A˙ ∗ )2 f ) = (ψ, A˙ ∗ f ) + (A˙ ∗ ψ, −P f ) = (ψ, A˙ ∗ f ) − (A˙ ∗ ψ, f ). In particular, if ψ ∈ N, then ψ is (+)-orthogonal to A˙ ∗ f ∈ D∗ . This implies that (ψ, A˙ ∗ f ) = (A˙ ∗ ψ, f ),

∀ψ ∈ N, ∀f ∈ M.

Therefore, the operator A˙ ∗ N is symmetric. Now let ψ ∈ N, f ∈ N±i . Then (A˙ ∗ ψ, f )+ = (A˙ ∗ ψ, f ) + (−P ψ, A˙ ∗ f ) = (A˙ ∗ ψ, f ) − (ψ, A˙ ∗ f ) = 0. Thus, A˙ ∗ N is (+)-orthogonal to Ni and N−i . It follows from (2.15) that A˙ ∗ N ⊂ D. (2) Using the symmetric property of A˙ ∗ N for ψ ∈ N we get (A˙ ∗ ± iI)ψ2 = A˙ ∗ ψ2 + ψ2 = ψ2+ .

2.3. Direct decomposition and analogue of the ﬁrst von Neumann’s formula

31

This implies that (A˙ ∗ ± iI) N is an (+, ·)-isometry. Letting f ∈ H+ and g ∈ ˙ we have Dom(A) ˙ + i(f, Ag) ˙ − i(A˙ ∗ f, g) + (f, g) (A˙ ∗ + iI)f, (A˙ + iI)g = (A˙ ∗ f, Ag) ˙ + (f, g) = (f, g)+ . = (A˙ ∗ f, P Ag) In particular, if f ∈ M, then (A˙ ∗ + iI)f, (A˙ + iI)g = 0 and hence (A˙ ∗ + iI)M ⊂ Ni . Now we will show that for any φ ∈ L, Pi φ = −i(A˙ ∗ + iI)Rφ,

(2.21)

where R is the Riesz-Berezansky operator and Pi φ = φ− .

(2.22)

Here and below Pλ = PNλ . Since Rφ ∈ RL ⊂ M, (A˙ ∗ + iI)Rφ ∈ Ni . On the other hand, by Theorem 2.2.1, we have Rφ ∈ Dom(A˙ A˙ ∗ ) and φ = (I + A˙ A˙ ∗ )Rφ = (A˙ + iI)A˙ ∗ − i(A˙ ∗ + iI) Rφ = (A˙ + iI)A˙ ∗ Rφ − i(A˙ ∗ + iI)Rφ, so φ + i(A˙ ∗ + iI)Rφ ∈ M−i . This proves (2.21). The latter also implies via (2.1) that Pi φ = (A˙ + iI)Rφ = Rφ+ = φ− , (φ ∈ L). Thus, the operator (A˙ ∗ + iI) maps a linear (+)-dense in N set RL (+, ·)-isometrically onto Pi L = Bi . That is why (A˙ ∗ + iI)N = Bi . Similarly we can show the same for (A˙ ∗ − iI). Also P−i φ = i(A˙ ∗ − iI)Rφ,

P−i φ = φ− ,

(φ ∈ L),

(2.23)

so that Pi φ = P−i φ. (3) Let the operator B˙ be given by (2.19). If a vector h ∈ H is orthogonal to ˙ then h ∈ L and (h, f ) = 0 for all f ∈ N. From deﬁnition (2.13) of N we Dom(B), get 0 = (h, Rh) = ||h||2− .

32

Chapter 2. Geometry of Rigged Hilbert Spaces

˙ is dense in H. Therefore, h = 0 and Dom(B) ∗ ˙ Since A N is symmetric in H, the operator B˙ is also symmetric in H. The relations (A˙ ∗ ± iI)N = B±i yield ˙ = (A˙ ± iI)Dom(A) ˙ ⊕ B±i . (B˙ ± iI)Dom(B) ˙ are closed in H. It follows that the Hence, the linear manifolds (B˙ ± iI)Dom(B) ˙ operator B is closed, and ˙ = N±i . H (B˙ ± iI)Dom(B) Thus, semi-deﬁciency subspace N±i of the operator A˙ coincides with the deﬁciency subspace of B˙ corresponding to the number ±i. In accordance with the ﬁrst von Neumann formula for B˙ ∗ we get (2.20). Notice that the operator B˙ is a symmetric extension of the operator A˙ and ˙ B is self-adjoint if and only if the semi-deﬁciency indices of A˙ are zero. Corollary 2.3.2. The operator A˙ ∗ ± iI maps N±i ⊕ N with (+)-metric homeomorphically onto the subspace N±i with either (·)- or (+)-metric. Moreover, (A˙ ∗ ± iI)M = N±i . Proof. Indeed, using (1.23) we get ˙ A˙ ∗ ± iI)N = N±i +B ˙ ±i = N±i . (A˙ ∗ ± iI)(N±i ⊕ N) = N±i +( (+, ·)-continuity of (A˙ ∗ ± iI) follows from A˙ ∗ h ± ih ≤ A˙ ∗ h + h ≤ 2h+,

h ∈ H+ .

Let φ ∈ N±i ⊕ N, i.e., φ = ϕ + ψ, where ϕ ∈ N±i and ψ ∈ N. Then φ2+ = ϕ2+ + ψ2+ . Further, (A˙ ∗ ± iI)φ = 2iϕ + (A˙ ∗ ± iI)ψ, and the terms ±2iϕ ∈ N±i and (A˙ ∗ ±iI)ψ ∈ B±i are (·)-orthogonal. Consequently, (A˙ ∗ ± iI)φ2 = 4ϕ2 + (A˙ ∗ ± iI)ψ2 = 2ϕ2+ + ψ2+ ≥ φ2+ , which implies the continuity of the inverse mapping.

We should note that the operator A˙ ∗ ± iI maps N∓i to zero and acts like ±2iI on N±i . Therefore, it maps M onto N±i , the mapping is one-to-one, and mutually (+, +)-continuous on N±i ⊕ N. Following Section 1.6 we use (1.26) to introduce an isometric exclusion operator V = Vi : Bi → B−i deﬁned by the formula V Pi f = P−i f,

f ∈ L, P±i = PN±i .

(2.24)

2.3. Direct decomposition and analogue of the ﬁrst von Neumann’s formula

33

Its closure V maps Bi isometrically onto B−i . It follows from (2.21) and (2.23) that V (A˙ ∗ + iI)Rf = −(A˙ ∗ − iI)Rf, (f ∈ L), and hence

V (A˙ ∗ + iI)Rf = −(A˙ ∗ − iI)Rf, (−)

for all f ∈ L

. Thus V (A˙ ∗ + iI)ψ = −(A˙ ∗ − iI)ψ,

ψ ∈ N.

+ is a bijection and a homeomorphism of N±i with Theorem 2.3.3. The operator PM the (·)-metric onto (N±i ⊕ N) with the (+)-metric.

Proof. Let φ ∈ Ni . Then φ = ϕ + ψ, where ϕ ∈ Ni and ψ ∈ Bi . According to Theorem 2.3.1, there exists such an element h ∈ N that A˙ ∗ h + ih = ψ and ˙ then P + ψ = ih, and hence, P + φ = ϕ + ih. ψ = h+ . Since A˙ ∗ h ∈ Dom(A), M M + Thus PM (Ni ) = N±i ⊕ N. Furthermore, + φ2+ = ϕ2+ + h2+ = 2ϕ2 + ψ2 ≥ ϕ2 + ψ2 = φ2 , φ2+ ≥ PM

implies the conclusion of the theorem for Ni . The proof of the theorem for N−i is similar. + Let us now denote by PN , the orthogonal projection operator from H+ onto N. We introduce a new inner product (·, ·)1 deﬁned by + + f, PN g)+ (f, g)1 = (f, g)+ + (PN

(2.25)

for all f, g ∈ H+ . The obvious inequality f 2+ ≤ f 21 ≤ 2f 2+ shows that the norms · + and · 1 are topologically equivalent. It is easy to ˙ N , N , N are (1)-orthogonal. We write M1 for the see that the spaces Dom(A), i −i Hilbert space M = Ni ⊕ N−i ⊕ N with inner product (f, g)1 . We denote by H+1 the space H+ with norm · 1 , and by R the corresponding Riesz-Berezansky operator related to the triplet H+1 ⊂ H ⊂ H−1 . Both operators R and R act according to Fig. 2.1 below. One can also see that 1 + + R−1 = R−1 (I + PN ), R = (I − PN )R. (2.26) 2 1

1

+ + Note also that the operators 2− 2 PM Ni and 2− 2 PM N−i mentioned in Theorem 2.3.3 are (·, 1)-isometries from Ni and N−i onto Ni ⊕ N and N−i ⊕ N, respectively. + N±i It follows from the proof of Theorem 2.3.1 that the explicit expression for PM is of the form + + + PM φ = ±i(2PN + PN )(A˙ ∗ ± iI)−1 φ, ±i

φ ∈ N±i .

(2.27)

34

Chapter 2. Geometry of Rigged Hilbert Spaces

H+ ⊂ H ⊂ H−

H+1 ⊂ H ⊂ H−1

R

R Figure 2.1: Operators R and R

2.4 Regular and singular symmetric operators At the beginning of Section 2.3 we introduced the deﬁnition of a regular operator ˙ In this section we will provide a criteria for an operator A˙ to be regular. It was A. shown in Corollary 1.5.5 that for all λ (Im λ = 0) the manifolds Bλ are either all closed or all non-closed. We will show that operator A˙ is regular in the ﬁrst case and is called singular in the second. Theorem 2.4.1. The following statements are equivalent for a closed symmetric ˙ operator A: (1) The manifolds Bλ are (·)-closed for all λ (Imλ = 0). ˙ is (+)-closed. (2) Dom(A) (3) A˙ is regular (P A˙ is closed). (4) A˙ is (+, ·)-bounded. (5) L is (−)-closed. (6) N is (·)-closed. (7) Dom(B˙ ∗ ) = Dom(A˙ ∗ ), where B˙ is deﬁned by (2.19). Proof. The equivalence (1) ⇐⇒ (3) is already proved (see Theorem 1.5.4 and Corollary 1.5.5). The equivalence (2) ⇐⇒ (3) follows from the deﬁnition of (+)norm. (2) ⇒ (4) follows from the Closed Graph Theorem and the inequality ||f || ≤ ||f ||+ , f ∈ H+ . Since A˙ is closed we get (4)⇒ (2). (1) ⇐⇒ (5). Because of (2.22) the operator Pi = PNi maps the set L isometrically onto the set Bi . Thus (−)-closure of L is equivalent to the (·)-closure of Bi . (5) ⇐⇒ (6). Since f 2− = (Rf, f ) = Rf 2+ for f ∈ H− , and hence f 2− ≤ Rf f for f ∈ H, we get γf ≤ f − ≤ f for all ⇐⇒ Rf ≥ γRf + .

f ∈ L and for some γ ∈ (0, 1)

Comparing (2.15) with (2.19) and (2.20) we get that the equivalence (2) and (7) holds true.

2.5. Closed symmetric extensions

35

This theorem immediately implies the following independently suﬃcient conditions for a closed symmetric operator A˙ to be regular: ˙ has a ﬁnite codimension (dim L < ∞); • Dom(A) • L ⊂ H+ , where L is deﬁned in (2.12). Proposition 2.4.2. If A˙ is a regular symmetric operator, then the direct decomposition ˙ H = H0 +N holds. Proof. Since A˙ is regular, by Theorem 2.4.1 and Theorem 1.5.4 the linear manifold Bi is (·)-closed and Θ(L, Bi ) < 1. Now from Lemma 1.5.1 we get the equality PL Bi = L. On the other hand, from Theorem 2.3.1 we have that Bi = (A˙ ∗ + iI)N. Hence, ˙ ⊂ H0 . Therefore, PL (A˙ ∗ + iI)N = L. But A˙ ∗ N ⊂ Dom(A) PL N = L. Taking into account that N ∩ H0 = {0} and H = H0 ⊕ L, we get the equality ˙ H = H0 +N. A closed symmetric operator A˙ is said to be an O-operator if both its semideﬁciency indices equal zero. For such an operator M = N. Theorem 2.4.3. A closed symmetric operator A˙ is a regular O-operator if and only if F ⊂ H, where F is of the form (2.17). The proof easily follows from the deﬁnition of a regular O-operator.

2.5 Closed symmetric extensions ˙ Then Let A be a closed symmetric extension of a symmetric operator A. (Af, g) = (f, Ag),

(∀f, g ∈ Dom(A)),

and, in particular ˙ g) = (f, Ag) = (f, P Ag), (Af,

˙ g ∈ Dom(A)). (∀f ∈ Dom(A),

It follows then that g ∈ Dom(A˙ ∗ ) and P Ag = A˙ ∗ g, and thus Dom(A) ⊂ H+ and P Af = A˙ ∗ f, f ∈ Dom(A),

(2.28)

˙ The next where P is an orthogonal projection operator in H onto H0 = Dom(A). theorem is an immediate consequence of (2.28) and the Closed Graph Theorem.

36

Chapter 2. Geometry of Rigged Hilbert Spaces

Theorem 2.5.1. For a closed symmetric extension A of an operator A˙ the following conditions are equivalent: (1) The set Dom(A) is (+)-closed. (2) The operator A is (+, ·)-bounded. (3) The operator A˙ ∗ Dom(A) is (·, ·)-closed. Moreover, under the conditions (1)–(3) the operator A˙ is regular. A closed symmetric extension A of a closed symmetric operator A˙ satisfying the conditions (1)–(3) of Theorem 2.5.1 is called a regular symmetric extension. Let us recall some aspects of the theory of extensions of closed symmetric operators A˙ with non-dense domain developed in Chapter 1. Once again we denote the exclusion operator (2.24) by V . Following Section 1.6 we call an isometric operator U (Dom(U ) = Dom(U ) ⊂ Ni , Ran(U ) ⊂ N−i ) an admissible operator if U x = V x only for x = 0. The general form of a closed symmetric extension A of an operator A˙ follows from (1.33) and is given by formulas ˙ Ran(I − U ), Dom(A) = Dom(A) ˙ + i(ϕ + U ϕ), A(g + ϕ − U ϕ) = Ag ˙ ϕ ∈ Dom(U ), g ∈ Dom(A),

(2.29)

where U is an admissible operator. The operator A is self-adjoint if and only if Dom(U ) = Ni and Ran(U ) = ˙ Conversely, if ϕ − ψ ∈ Dom(A), ˙ N−i . Also, if ϕ ∈ Bi then ϕ − V ϕ ∈ Dom(A). where ϕ ∈ Ni , ψ ∈ N−i , then ϕ ∈ Bi , and ψ = V ϕ ∈ B−i . Hence, in particular, for admissible U the equality x = U x holds only for x = 0. Thus, the operator I − U is injective and (·, +)-continuous. Now we are going to prove an auxiliary result of a general geometrical nature. First let us recall the notion of the minimal angle α(L, M) between subspaces L and M of a Hilbert space H: cos α(L, M) :=

sup x∈L,x2 ∈M,

|(x, y)|

(2.30)

x=y=1

Deﬁnition (2.30) yields the equalities cos α(L, M) = ||PL M|| = ||PM L||,

(2.31)

where PL and PM are orthogonal projections onto L and M, respectively. Lemma 2.5.2. Let L1 be a subspace of a Hilbert space H and L2 be a linear manifold in H that is a range of a bounded operator T mapping a Banach space into H. Let also PL1 be an orthogonal projection from H onto L1 and PL⊥ = I − PL1 . Then 1 the following statements are equivalent: (1) L1 ∩ L2 = {0} and the linear set L1 L2 is closed.

2.5. Closed symmetric extensions

37

(2) L1 ∩ L2 = {0} and the linear set PL⊥ L2 is closed. 1 (3) L2 is closed and the minimal angle between spaces L1 and L2 is positive. (4) L2 is closed and PL⊥ L2 is a homeomorphism. 1 Proof. (1) ⇔ (2). It is clear that L1 L2 = L1 ⊕ PL⊥ L2 , 1 and hence, the linear manifolds L1 L2 and PL⊥ L2 are closed simultaneously. 1 (1) ⇔ (3). Assume that L2 = T L0 , where L0 is a Banach space, T : L0 → H is bounded and injective. Consider a Banach space M = L1 × L0 with the norm || f, g ||M := ||f ||L1 + ||g||L0 , f ∈ L1 , g ∈ L0 . ˙ 2 given by Then the mapping Z : M → L1 +L Z f, g = f + T g is continuous and bijective. Applying the Banach inverse mapping theorem, we get that L2 = Z 0 × L0 is closed in H. Let us show that the minimal angle between L1 and L2 is positive, i.e., cos α(L1 , L2 ) < 1. Since L2 is closed, L1 ∩ L2 = {0}, and PL⊥ L2 is closed, from Banach’s inverse mapping theorem we get that there 1 exists a constant γ ∈ (0, 1) such that ||PL⊥ h|| ≥ γ||h|| 1

for all

h ∈ L2 .

(2.32)

It follows that ||PL1 L2 || < 1. This means α(L1 , L2 ) > 0. Thus (1)⇒(3). Now suppose that (3) holds. Due to (2.31) we have cos α(L1 , L2 ) < 1 ⇐⇒ (2.32). Therefore, the operator PL⊥ L2 is a homeomorphism. Taking into account the 1 relation ||f + h||2 = ||f + PL1 h||2 + ||PL⊥ h||2 , f ∈ L1 , h ∈ L2 , 1 ˙ 2 is closed. So, (3) ⇐⇒ (4) and (3)⇒(1). we get that L1 +L

Now let A˙ be a closed symmetric operator and A be its closed symmetric extension with the corresponding admissible isometric operator U . Applying Lemma ˙ and L2 = Dom(U ) and taking into account that 2.5.2 to H = H+ , L1 = Dom(A), Dom(U ) is the range of a (·, +)-continuous operator I − U , we obtain the following theorem: Theorem 2.5.3. The following statements are equivalent: (1) A is a regular symmetric extension of A˙ (that is, Dom(A) is a (+)-closed set).

38

Chapter 2. Geometry of Rigged Hilbert Spaces

+ (2) PM (Ran(I − U )) is a (+)-closed set.

˙ and (3) Ran(I − U ) is (+)-closed, and the minimal angle between Dom(A) Ran(I − U ) is positive (with respect to the (+)-metric). + (4) Ran(I − U ) is (+)-closed, and PM Ran(I − U ) is a homeomorphism.

We note that under the conditions of Theorem 2.5.3 there is a constant c > 0 such that g + ϕ − U ϕ+ ≥ cϕ,

˙ ∀ϕ ∈ Dom(U )). (∀g ∈ Dom(A),

(2.33)

˙ and Dom(U ) implies Indeed, the positivity of the minimal angle between Dom(A) ˙ is (+)-continuous, i.e., that orthoprojection on Dom(U ) parallel to Dom(A) c1 ϕ − U ϕ+ ≤ g + ϕ − U ϕ+ , for some c1 > 0. On the other hand, it follows from the Closed Graph Theorem that ϕ − U ϕ+ ≥ c2 ϕ, (ϕ ∈ Dom(U )), for some c2 > 0. This proves (2.33). Theorem 2.5.4. If A is a regular symmetric extension of a regular closed symmetric ˙ then there is a constant c > 0 such that operator A, V ϕ − U ϕ+ ≥ cϕ,

∀ϕ ∈ Bi ∩ Dom(U ),

(2.34)

where V is of the form (2.24). The converse is valid if Dom(U ) ⊃ Bi , in particular, for self-adjoint A. Proof. For ϕ ∈ Bi ∩ Dom(U ) we have V ϕ − U ϕ = −(ϕ − V ϕ) + (ϕ − U ϕ). ˙ then Theorem 2.5.3 yields (2.34). Since −(ϕ − V ϕ) ∈ Dom(A), Conversely, let Dom(U ) ⊃ Bi and for all ϕ ∈ Bi V ϕ − U ϕ+ ≥ cϕ, + for some c > 0. We will show that PM Dom(U ) is (+)-closed. Let (+)

+ PM (gn − U gn ) −→ 0,

for gn = gn + gn , (gn ∈ Ni , gn ∈ Bi ). Then (+)

+ gn + PM (gn − U gn ) −→ 0.

It was shown in Theorem 2.3.3 that + PM Bi = N,

+ PM N−i = N ⊕ Ni .

2.5. Closed symmetric extensions

39

+ + Therefore, PM (gn − U gn ) ⊂ N ⊕ Ni , i.e., the elements gn and PM (gn − U gn) are (+)

(+)

+ (+)-orthogonal. Hence, gn −→ 0, and thus PM (gn − U gn ) −→ 0. Furthermore,

gn − U gn = gn − V gn + V gn − U gn . ˙ is (+)-orthogonal to M, then Since the vector gn − V gn ∈ Dom(A) (+)

+ PM (V gn − U gn ) −→ 0. (+)

According to Theorem 2.3.3 we have V gn − U gn −→ 0. It follows from (2.34) (+)

(+)

+ that gn −→ 0. Therefore, gn −→ 0 which implies that PM (Dom(U )) is (+)-closed. Then, by Theorem 2.5.3, A is a regular closed symmetric extension of the operator ˙ A.

In the case of a regular closed symmetric operator A˙ we can describe its symmetric (in particular, self-adjoint) extensions in terms other than those in Chapter 1. The following theorem gives a characterization of the regular extensions ˙ for a regular closed symmetric operator A. Theorem 2.5.5. I. For each closed symmetric extension A of a regular operator A˙ there exists a (1)-isometric (see formula (2.25) for the deﬁnition of (1)-metric) operator U = U(A) on M1 with the properties: (a) Dom(U) is (+)-closed and belongs to N ⊕ Ni , Ran(U) ⊂ N ⊕ N−i ; (b) Uψ = ψ only for ψ = 0, and ˙ ⊕ (I − U )Dom(U), Dom(A) = Dom(A) ˙ + A˙ ∗ (I − U )ψ + i(A˙ A˙ ∗ + I)P + (I + U)ψ, A(g + (I − U )ψ)) = Ag N (2.35) ˙ ψ ∈ Dom(U). where g ∈ Dom(A), Conversely, for each (1)-isometric operator U with the properties (a) and (b) ˙ the operator A deﬁned by (2.35) is a closed symmetric extension of A. II. The extension A is regular if and only if the manifold Ran(I −U ) is (1)-closed. III. The operator A is self-adjoint if and only if Dom(U) = N ⊕ Ni , Ran(U) = N ⊕ N−i . Proof. Let A˙ be a regular closed symmetric operator and A be its closed symmetric extension whose domain is deﬁned by (2.29) via the corresponding operator U . Besides, Dom(A) admits (+)-orthogonal decomposition ˙ ⊕ P + (Ran(I − U )). Dom(A) = Dom(A) M

40

Chapter 2. Geometry of Rigged Hilbert Spaces

We introduce a new operator U = U(A) by the formula + + UPM ϕ = PM U ϕ,

Therefore,

ϕ ∈ Dom(U ).

(2.36)

+ Dom(U) = PM Dom(U ) ⊂ N ⊕ Ni ,

+ Ran(U) = PM (Ran(U )) ⊂ N ⊕ N−i .

It follows from Theorem 2.3.3 that Dom(U) is (+)-closed and thus the deﬁnition 1 + (2.36) makes sense. Since the operator 2− 2 PM N±i is a (·, 1)-isometry and U is a (·, ·)-isometry, the operator U is isometric with respect to · 1 metric, i.e., (1,1)+ + isometry. Let us assume that Uψ = ψ for some ψ ∈ Dom(U). Then PM ϕ−PM Uϕ = ˙ 0 for some ϕ ∈ Dom(U ). This implies that ϕ−U ϕ ∈ Dom(A). Consequently, ϕ = 0 and ψ = 0. From (2.27) we get the parametric expression for U ⎧ + + −1 ˙∗ ⎨ ψ = i(2PN ϕ + PN )(A + iI) i , ϕ ∈ Dom(U ). (2.37) + + ∗ ˙ ⎩ Uψ = −i(2PN + PN )(A − iI)−1 U g −i

It follows that ⎧ + + ⎨ ϕ = −i(A˙ ∗ + iI)( 1 PN + PN )ψ 2 i

+ + ⎩ U ϕ = i(A˙ ∗ − iI)( 12 PN + PN )Uψ

, ψ ∈ Dom(U).

(2.38)

−i

Hence,

+ ϕ − U ϕ = ψ − Uψ − iA˙ ∗ PN (I + U)ψ, + ∗ i(ϕ + U ϕ) = i(I + U)ψ + A˙ P (I − U)ψ. N

˙ ϕ ∈ Dom(U ). Then Let g ∈ Dom(A), ˙ + i(I + U )ϕ. A(g + (I − U )ϕ)) = Ag Therefore + A(g + (I − U )ψ)) = A(g + iA˙ ∗ PN (I + U)ψ + ϕ − U g) + ∗ ˙ + iA˙ A˙ P (I + U)ψ + i(I + U )ϕ = Ag N

˙ + iA˙ A˙ ∗ P + (I + U)ψ + i(I + U)ψ + A˙ ∗ P + (I − U)ψ = Ag N N + ∗ ∗ ˙ ˙ ˙ ˙ = Ag + A (I − U )ψ + i(AA + I)P (I + U)ψ. N

Conversely, let U be a (1,1)-isometry in the subspace M with (1)-closed domain Dom(U) ⊂ N ⊕ Ni , and Ran(U) ⊂ N ⊕ N−i such that Uψ = ψ only for ψ = 0. Let the operator A be deﬁned by (2.35). Then A is an extension of A˙ and by direct calculations one can check that A is symmetric. Relations (2.38) deﬁne

2.5. Closed symmetric extensions

41

(·, ·)-isometry U with Dom(U ) = Dom(U ) ⊂ Ni and Ran(U ) ⊂ N−i such that (2.36) and consequently (2.29) hold. It remains to show that U is an admissible operator. Suppose U ϕ = V ϕ for ˙ is (+)-orthogonal to some ϕ ∈ Bi ∩ Dom(U ). Since the element ϕ − V ϕ ∈ Dom(A) + + + + + + M, then PM ϕ = PM V ϕ and thus PM U ϕ = PM ϕ, and U(PM ϕ) = PM ϕ. Therefore, + PM ϕ = 0 and ϕ = 0. Theorem 2.5.6. A regular closed symmetric operator A˙ has a regular self-adjoint extension if and only if its semi-deﬁciency indices are equal. ˙ Then by Theorem 2.5.5 the Proof. Let A be a regular self-adjoint extension of A. corresponding operator U = U(A) maps N ⊕ Ni (1)-isometrically onto N ⊕ N−i . Hence it is clear that the dimensions dim N−i and dim Ni are the same and the semi-deﬁciency indices of A˙ are equal. Conversely, suppose dim N−i =dim Ni . It is not hard to construct a (1)isometry U in M1 that maps N ⊕ Ni onto N ⊕ N−i with number 1 as a regular point. For example, we can take U Ni to be arbitrary (1)-isometric operator with the range N−i and set Uh = εh, h ∈ N, where |ε| = 1, ε = 1. Then the corresponding operator A will become a regular self-adjoint extension of the operator ˙ A. Theorem 2.5.7. Let A be a regular self-adjoint extension of a regular symmetric ˙ The following statements are valid: operator A. 1. the operator P A (Dom(A) ∩ H0 ) is self-adjoint in H0 ; 2. if U ∈ [Ni , N−i ] is an admissible operator that determines A by formulas (2.29) and if ˜ i := {ϕ ∈ Ni , (U − I)ϕ ∈ H0 }, N (2.39) then ˜ i. H+ = Dom(A) (U + I)N Proof. Let us set

(2.40)

N0i := (A − iI)−1 L.

˙ then Then N0i ⊂ Ni . Indeed, if f ∈ L and g ∈ M−i = (A˙ + iI)Dom(A), (g, (A − iI)−1 f ) = ((A + iI)−1 g, f ) = ((A˙ + iI)−1 g, f ) = 0, ˙ then the operator A is (+, ·)Since A is a regular self-adjoint extension of A, −1 bounded and therefore the resolvent (A − iI) has the estimate c1 ||f || ≤ ||(A − iI)−1 f ||+ ≤ c2 ||f ||

for all f ∈ H,

with c > 0. Therefore N0i is a subspace in H+ (and simultaneously in H). ˜ i be deﬁned by (2.39). We are going to prove that Let N ˜ i = Ni , N0i ⊕ N

(2.41)

42

Chapter 2. Geometry of Rigged Hilbert Spaces

where the sum is (·)-orthogonal. By (2.29) we have (A + iI)(U − I)ϕ = −iϕ − iU ϕ + iU ϕ − iϕ = −2iϕ, and hence (A + iI)−1 ϕ = −

ϕ ∈ Ni ,

1 (U − I)ϕ. 2i

This yields the equivalence relations (U − I)ϕ ∈ H0 ⇐⇒ (A + iI)−1 ϕ ⊥ L ⇐⇒ ϕ ⊥ N0i , and thus the decomposition (2.41) holds true. Thus, it is proved that ˜ i. ˙ +(U ˙ Dom(A) ∩ H0 = Dom(A) − I)N Suppose the vector ψ ∈ H0 is (·)-orthogonal to (P A + iI)(Dom(A) ∩ H0 ). Then ψ ∈ Ni ∩ N0i . Since N0i ⊂ Dom(A), we get that ψ ∈ Dom(A) and (see (2.28)) P Aψ = A˙ ∗ ψ = iψ. Because P A (Dom(A) ∩ H0 is symmetric, we get that ψ = 0. This means (P A+iI)(Dom(A)∩H0 ) is dense in H0 . Similarly it can be proved that (P A−iI)(Dom(A)∩H0 ) is dense in H0 . Thus the operator (P A+iI)(Dom(A)∩H0 ) is essentially self-adjoint in H0 . On the other hand, since A is a regular extension, Theorem 2.5.1 yields that the set Dom(A) ∩ H0 is (+)-closed. Consequently, the operator (P A + iI)(Dom(A) ∩ H0 ) is (·)-closed and hence is self-adjoint in H0 . The ﬁrst statement is proved. ˜ i , (U + I)φ = (U − I)φ + 2φ, then (U + I)φ ∈ Dom(A) Since for all φ ∈ N implies φ ∈ Dom(A). But then P Aφ = A˙ ∗ φ = iP φ and thus P (A − iI)φ = 0 or ˜ i = 0. equivalently (A − iI)φ ∈ L. Therefore, φ = 0 and hence Dom(A) ∩ (U + I)N 0 It is easy to see that (U + I)Ni ⊂ Dom(A). According to (2.10) Dom(A˙ ∗ ) = Dom(A) + Ni + N−i . Let f = g + ϕ + ψ, ˙ ϕ ∈ Ni , and ψ ∈ N−i . Then where g ∈ Dom(A), 1 1 f = g + (U − I)(U −1 ψ − ϕ) + (U + I)(U −1 ϕ + ϕ) = x + (U + I)y, 2 2 where x = g + 12 (U − I)(U −1 ψ − ϕ) and y = 12 (U + I)(U −1 ϕ + ϕ). This implies that Dom(A˙ ∗ ) ⊆ Dom(A) + (U + I)Ni . Since the inverse inclusion is obvious, we conclude that Dom(A˙ ∗ ) = Dom(A) + (U + I)Ni . (2.42) ˜ i = 0, and (U + I)N0 ⊂ Dom(A) we obtain Combining (2.42), Dom(A) ∩ (U + I)N i ˜ i. Dom(A˙ ∗ ) = Dom(A) (U + I)N This completes the proof of statement 2.

2.5. Closed symmetric extensions

43

Theorem 2.5.8. Let A˙ be a regular O-operator. Then all its regular self-adjoint extensions are of the form Dom(A) = H+ ,

+ −1 ˙ + ˙∗ ˙ ˙∗ A = AP S)PN , ˙ + (A + (AA + I) Dom(A)

where S is an arbitrary (+)-self-adjoint and (+)-bounded operator in N. Proof. We apply Theorem 2.5.5 for the case Ni = N−i = {0}. Then there is a one-to-one correspondence between all (1)-unitary operators U in N having the ˙ which number 1 as a regular point and all regular self-adjoint extensions A of A, take the form (2.35). Let S = i(I + U)(I − U )−1 . Then S ∈ [N, N], S = S ∗ , and (2.35) can be re-written as + −1 ˙ ⊕ N = H+ , A = AP ˙ + ˙∗ ˙ ˙∗ Dom(A) = Dom(A) S)PN . ˙ + (A + (AA + I) Dom(A)

Chapter 3

Bi-extensions of Closed Symmetric Operators In this chapter we introduce and consider a new type of extensions of a given symmetric operator with its exit into the triplets of Hilbert spaces. These new extensions are called bi-extensions. We present a complete description and parameterizations of these bi-extensions. Special attention is paid to so-called twice self-adjoint (t-self-adjoint ) bi-extensions that will play an important role in the remaining part of this text.

3.1 Bi-extensions Deﬁnition 3.1.1. An operator A ∈ [H+ , H− ] is called a bi-extension of a symmetric ˙ If, in addition, A = A∗ , then the bi-extension A operator A˙ if A ⊃ A˙ and A∗ ⊃ A. is called self-adjoint. ˙ By deﬁnition if A ∈ The class of all bi-extensions of A˙ is denoted by E(A). ∗ ˙ ˙ ˙ E(A), then A ∈ E(A). It is easy to see that E(A) is a convex set1 in [H+ , H− ]. In ˙ , then its real part Re A = (1/2)(A + A∗ ) also belongs to particular, if A ∈ E(A) ˙ E(A). ˙ and A ∈ [H+ , H− ]. Then A belongs to E(A) ˙ if and only if Let A ∈ E(A) ˙ ⊂ ker(A − A) Dom(A)

˙ ⊂ ker(A∗ − A∗ ). and Dom(A)

The last inclusion is equivalent to Ran(A − A) ⊆ F, 1 That means for any A , A ∈ E(A) ˙ all the operators of the form λA1 + (1 − λ)A2 belong to 1 2 ˙ E(A) for any λ ∈ C.

Y. Arlinskii et al., Conservative Realizations of Herglotz–Nevanlinna Functions, Operator Theory: Advances and Applications, DOI 10.1007/978-3-7643-9996-2_3, © Springer Basel AG 2011

45

46

Chapter 3. Bi-extensions of Closed Symmetric Operators

˙ given by (2.17) and where F (⊂ H− ) is the (·)-orthogonal complement of Dom(A) (2.18). Hence, in particular, the imaginary part Im A = (1/2i)(A − A∗ ) of an ˙ satisﬁes the condition operator A ∈ E(A) ˙ ⊂ ker(Im A) ⇐⇒ Ran(Im A) ⊂ F . Dom(A)

(3.1)

˙ The operator (A − A) M is (+, −)-continuous and its Let A, A ∈ E(A). range belongs to F . Hence the operator S = R(A − A) M,

(3.2)

is (+, +)-continuous as acting from M into M, i.e., S ∈ [M, M]. Here R is the + Riesz-Berezansky operator of the triplet H+ ⊂ H ⊂ H− . Let PM be the (+)˙ orthogonal projection in H+ onto M described by (2.16). We know that Dom(A) is (+)-orthogonal to M, and thus + A − A = R−1 SPM .

It is easy to check that the inverse statement is also true. That is, if S in (3.2) + ˙ Consequently, in order to describe belongs to [M, M], then A + R−1 SPM ∈ E(A). ˙ we need only one representative operator of this class. Below the entire class E(A) ˙ we construct two “particular” operators of the class E(A). Let us recall the spaces D and D∗ introduced in (2.11) (+)

˙ D = Dom(A)

,

D∗ = H+ ∩ H0 ,

˙ (H0 = Dom(A)),

and A˙ be the (·, −)-continuity extension of the operator A˙ onto H0 described in Theorem 2.2.1. Let us show that ˙ g) = (f, Ag), ˙ (Af,

(f ∈ D, g ∈ D∗ ).

(3.3)

(+) (·) ˙ such that fn − There exists a sequence {fn } ⊂ Dom(A) −→ f , or fn −→ f and (·) ˙ n , g) = (fn , A˙ ∗ g) and since g ∈ H0 , A˙ ∗ fn −→ A˙ ∗ f . We have that (Af

˙ n , g) = (P Af ˙ n , g) = (A˙ ∗ fn , g). (fn , A˙ ∗ g) = (Af Letting n → ∞ we get (f, A˙ ∗ g) = (A˙ ∗ f, g). Applying (2.8) we obtain (3.3). Now we use formulas (2.9) and (2.10) with λ = i. Following our convention above we will denote by PG+ a (+)-orthoprojection operator in H+ onto a subspace G of H+ . We set ˙ + + A˙ ∗ P + , A0 = AP D M

˙ +∗ + A˙ ∗ P + . A1 = AP D N

(3.4)

Since the operator A˙ is (·, −)-continuous while A˙ ∗ is (+, ·)-continuous, then both ˙ Relying on A0 and A1 belong to [H+ , H− ]. It is clear that A0 ⊃ A˙ and A1 ⊃ A.

3.1. Bi-extensions

47

(3.4) and using (2.8) and (3.3) we have, for arbitrary f, g ∈ H+ , ˙ + f + A˙ ∗ P + f, P +∗ g + P + g) (A0 f , g) = (AP D D M N ˙ + f, P +∗ g) + (AP ˙ + f, P + g) + (A˙ ∗ P + f, P + g) + (A˙ ∗ P + f, P + g) = (AP D D D D∗ N M M N ˙ +∗ g) + (P + f, A˙ ∗ P + g) + (P + f, AP ˙ +∗ g) + (P + f, A˙ ∗ P + g) = (PD+ f, AP D D D N M M N + ˙ +∗ g + A˙ ∗ P + g) = (f, A1 g). = (PD+ f + PM f, AP D N

This implies the equality

A∗0 = A1 ,

(3.5)

˙ All the above reasoning can be summarized in the following hence, A0 , A1 ∈ E(A). theorem: Theorem 3.1.2. Let A˙ be a symmetric operator in H. The relations + A = A0 + R−1 SA PM ,

SA = R(A − A0 ),

(3.6)

˙ and [M, M]. The adjoint set a bijective mapping A ↔ SA between the classes E(A) operator A∗ is given by + A∗ = A1 + R−1 SA∗ PM . ˙ We recall from Chapter 2 that L = H H0 (see (2.12)), where H0 = Dom(A) ˙ and F (⊂ H− ) is the (·)-orthogonal complement of Dom(A) (⊂ H+ ) given by (2.18). It follows from Theorem 2.1.1 that F ∩ H = L, and hence, F ∩ H0 = {0}. The linear set B = H + F = H0 F, (3.7) is (−)-dense in H− . Let Π be a projection operator in B onto H0 parallel to F. It is easy to see that Π H coincides with P , the orthogonal projection operator from H onto H0 . One can also show that Π is a (−, ·)-closed operator. Lemma 3.1.3. 1. A vector f ∈ H− belongs to the set B of the form (3.7) if and ˙ only if the functional Ff (ϕ) = (ϕ, f ) is (·)-continuous on Dom(A). ˙ belongs to the set B if and only if f ∈ D∗ = 2. Let f ∈ H0 . Then an element Af H+ ∩ H0 . Moreover, ˙ = A˙ ∗ f. ΠAf Proof. 1. If f ∈ B, f = f1 + f2 , where f1 = Πf ∈ H0 and f2 ∈ F , then (ϕ, f ) = ˙ (ϕ, f1 ) is a (·)-continuous functional with respect to ϕ ∈ Dom(A). ˙ functional. Conversely, let Ff (ϕ) = (ϕ, f ) be a (·)-continuous on Dom(A) ˙ = H0 such that Ff (ϕ) = (ϕ, f1 ). Then there exists an element f1 ∈ Dom(A) Thus, f − f1 ∈ F and f = f1 + f2 ∈ B. ˙ ∈ B is equivalent to (Af, ˙ ϕ) 2. According to statement 1. the condition Af ˙ being a (·)-continuous with respect to ϕ ∈ Dom(A) functional. Since ˙ ϕ) = (f, A˙ ∗ ϕ) = (f, A∗ ϕ) = (f, ΠAϕ) ˙ = (f, Aϕ), ˙ (Af,

48

Chapter 3. Bi-extensions of Closed Symmetric Operators

then the (·)-continuity of this functional is equivalent to f ∈ H+ . Consequently, ˙ = (A˙ ∗ f, ϕ). Thus, Af ˙ ∈ B if and only if f ∈ H+ ∩H0 and besides (Af, ˙ ϕ) = (f, Aϕ) ∗ ∗ ∗ ˙ That is why Af ˙ − A˙ f ∈ F and ΠAf ˙ = A˙ f . (A˙ f, ϕ) for all ϕ ∈ Dom(A). ˙ then Ran(A) ⊂ B and ΠA = A˙ ∗ . Conversely, Theorem 3.1.4. If A belongs to E(A), if A ∈ [H+ , H− ], Ran(A) ⊂ B, Ran(A∗ ) ⊂ B, and ΠA = ΠA∗ = A˙ ∗ , then A ∈ ˙ E(A). ˙ = A˙ ∗ f for all f ∈ D∗ . Taking Proof. By Lemma 3.1.3 we have AD∗ ⊂ B and ΠAf ∗ ˙ into account that Ran(A ) ⊂ H0 and relying on (3.4) we get ˙ +∗ f + A˙ ∗ P + f ∈ B, A1 f = AP D N and

˙ +∗ f + A˙ ∗ P + f = A˙ ∗ (P +∗ f + P + f ) = A˙ ∗ f. ΠA1 f = ΠAP D N D N ˙ Now let A be an arbitrary operator in E(A). The inclusion Ran(A − A1 ) ⊂ F (see (3.1)) and the equality ΠF = 0 imply that Ran(A) ⊂ B + F = B, and

ΠAf = ΠA1 f = A˙ ∗ f, f ∈ H+ .

Conversely, let A ∈ [H+ , H− ], Ran(A) ⊂ B, and ΠA = A˙ ∗ . Then Af − A˙ ∗ f ∈ ˙ then F for all f ∈ H+ . If g ∈ Dom(A), ˙ (Af, g) = (A˙ ∗ f, g) = (f, Ag). ˙ i.e., A∗ ⊃ A. ˙ Similarly conditions Ran(A) ⊂ B and This implies that A∗ g = Ag, ∗ ˙ ˙ ΠA = A imply A ⊃ A. That is why, if Ran(A), Ran(A∗ ) ⊂ B and ΠA = ΠA∗ = ˙ A ⊃ A, ˙ and thus A ∈ E(A). ˙ A˙ ∗ then A∗ ⊃ A, Let A ∈ [H+ , H− ]. We recall (see (2.5)) that the operator ˆ = {f ∈ H+ : Af ∈ H} , Aˆ = A Dom(A) ˆ Dom(A) is called the quasi-kernel of A. ˙ then the quasi-kernel Aˆ of the Theorem 3.1.5. If A belongs to the class E(A), operator A is (·, ·)-closed. (·) ˆ and fn − Proof. Let {fn } ⊂ Dom(A) → f , while (·) ˆ n− Afn = Af → g.

We recall that Π H = P . According to Theorem 3.1.4 A˙ ∗ fn → P g. Since A˙ ∗ is (+) (−) (·, ·)-closed, f ∈ H+ and P y = A˙ ∗ f . Therefore, fn −−→ f and hence Afn −−→ Af . (·) ˆ n− ˆ and On the other hand Afn = Af → g ∈ H and so Af = g ∈ H, f ∈ Dom(A), ˆ ˆ g = Af . This proves that the operator A is closed.

3.1. Bi-extensions

49

˙ in (+, ·)- and Now we will study continuity of operators from the class E(A) (·, −)-topologies. Theorem 3.1.6. The following statements about a closed symmetric operator A˙ and its bi-extensions A0 and A1 are equivalent: (i) A˙ is a regular operator; (ii) A0 is (+, ·)-continuous; (iii) A1 is (·, −)-continuous. Proof. First we will show that (i) implies (ii). For a regular operator A˙ we have ˙ and thus in the formula (3.4) for A0 one can replace A˙ by A. ˙ Then D = Dom(A) Ran(A0 ) ⊂ H and hence A0 ∈ [H+ , H]. Taking into account that A0 ⊃ A˙ we conclude that (ii) implies (i). The equivalence of (ii) and (iii) follows from the fact that A0 = A∗1 . We note that if A˙ is regular, then A0 = B˙ ∗ (see (2.20)), where B˙ is a closed symmetric operator with dense domain in H deﬁned by (2.19). The following theorem reﬁnes the result of Theorem 3.1.6. ˙ = H, then E(A) ˙ conTheorem 3.1.7. If A˙ is densely deﬁned in H, i.e., Dom(A) ∗ ˙ tains a unique (+, ·)-continuous operator A0 = A and a unique (·, −)-continuous operator ˙ D ∗ = A ˙ H+ . A1 = A Proof. According to Theorem 2.4.1 a closed densely deﬁned symmetric operator ˙ N = 0, D∗ = H+ , and from (3.5) A˙ is automatically regular, D = Dom(A), ˙ + + A˙ ∗ P + = A˙ ∗ , A0 = AP D M

˙ D∗ = A ˙ H+ . A1 = A∗0 = A

It follows that A0 is (+, ·)-continuous while A1 is (·, −)-continuous. ˙ is (·, −)-continuous, then, being an If an operator A from the class E(A) ˙ it coincides with A˙ on H+ , i.e., A = A1 . Consequently, A1 is the extension of A, ˙ If an operator A ∈ E(A) ˙ is unique (·, −)-continuous operator in the class E(A). ˙ is (·, −)-continuous. Then A∗ = A1 and A = A0 , (+, ·)-continuous, then A∗ ∈ E(A) ˙ i.e., A0 is unique (+, ·)-continuous of the class E(A). We should also note that if a closed symmetric operator A˙ is not self-adjoint, then A0 and A1 are not self-adjoint in [H+ , H− ]. Thus, in the class of all biextensions of the operator A˙ that are self-adjoint operators in [H+ , H− ] there are no (+, ·)- and (·, −)-continuous operators. ˙ and Aˆ Theorem 3.1.8. Let A˙ be a regular closed symmetric operator, A ∈ E(A), ˆ ˆ is is a quasi-kernel of A. Then A is (+, ·)-continuous and its domain Dom(A) (+)-closed.

50

Chapter 3. Bi-extensions of Closed Symmetric Operators

Proof. According to Theorem 3.1.2 the operator A admits the representation (3.6) for some SA ∈ [M, M]. If A˙ is a regular operator then, according to Theorem 3.1.6, + Ran(A0 ) ⊂ H. That is why the expression Af ∈ H is equivalent to R−1 SA PM f∈ −1 H. Since R M ∩ H = F ∩ H we have that + R−1 SA PM f ∈ L,

+ SA PM f ∈ RL, f ∈ H+ .

In the case of a regular operator A˙ we have RL = N. Therefore, Af ∈ H if and + only if PM f ∈ SA−1 (N) = G, where G = {f ∈ M | SA f ∈ N}. The latter linear manifold is (+)-closed as a complete pre-image of the (+)-closed linear manifold N. Thus the set of such f ∈ H+ that Af ∈ H coincides with the (+)-closed linear ˙ ⊕ G. Using Theorem 3.1.5 we get that the operator Aˆ is (·, ·)manifold Dom(A) closed and hence (+, ·)-closed. But it is deﬁned on a (+)-closed linear manifold and hence (+, ·)-continuous. Theorem 3.1.9. The formula ˙ + + (A˙ ∗ + R−1 S)P + − i R−1 P + + i R−1 P + , A = AP D M Ni N−i 2 2

(3.8)

˙ and all opestablishes a bijective correspondence between all operators A ∈ E(A) ∗ erators S ∈ [M, M]. The adjoint operator A is given by ˙ + + (A˙ ∗ + R−1 S ∗ )P + − i R−1 P + + i R−1 P + , A∗ = AP D M Ni N−i 2 2 ˙ is where S ∗ ∈ [M, M] is (+)-adjoint to S. In particular, the operator A ∈ E(A) ∗ self-adjoint if and only if S = S . Proof. From (3.4) and (3.5) we have ˙ + − P + ) − A˙ ∗ (P + − P + ) = (A˙ − A˙ ∗ )(P + + P + ). A1 − A0 = A(P D∗ D M N N N i

−i

By virtue of Theorem 2.2.1, if ϕ ∈ N±i , then ˙ = ±i(Aϕ ˙ ∓ iϕ) = ±i(Aϕ ˙ − A˙ ∗ ϕ). R−1 ϕ = ϕ + A˙ A˙ ∗ ϕ = ϕ ± iAϕ Therefore + + A1 − A0 = −iR−1 (PN − PN ). i

−i

(3.9)

Now we will establish a connection between SA and SA∗ given by (3.6). Let S ∗ be the operator adjoint to S ∈ [M, M] with respect to the (+)-metric. Let J be the operator in M deﬁned by the formula + + J = (PN − PN ) M, i

−i

(3.10)

3.1. Bi-extensions

51

+ + Notice that J is (+)-self-adjoint and J2 = PN + PN . By Theorem 3.1.2 each i −i ˙ takes the form operator A ∈ E(A) + , A = A0 + R−1 SA PM + where S ∈ [M, M] and A∗ = A1 + R−1 SA∗ PM . Deﬁne the operator S ∈ [M, M]:

i S = SA + J, 2 + . Using (3.9) one has where J is deﬁned by (3.10). Then A = A0 + R−1(S − 2i J)PM

i + A∗ = A0 + R−1 (S ∗ − J)PM . 2 Clearly, SA = S − 2i J, SA∗ = S ∗ − 2i J, and A∗ = A if and only if S = S ∗ .

Observe that relation (3.9) for the case of a densely deﬁned operator A˙ can be rewritten as + (A˙ − A˙ ∗ )f = −R−1 A˙ ∗ PM f, f ∈ H+ . Due to Theorem 2.4.1 in the case of a regular operator A˙ we can replace A˙ by A˙ in the formula (3.8) and obtain i i −1 + + + −1 ˙ + ˙∗ R PN . A = AP S)PM − R−1 PN + ˙ + (A + R Dom(A) i −i 2 2

(3.11)

In addition, using the operator B˙ and its adjoint B˙ ∗ given by (2.19) and (2.20) we can rewrite the right-hand side of (3.11) as i + i + + ∗ −1 ˙ S − PN + PN PM A=B +R . (3.12) 2 i 2 −i Remark 3.1.10. If the (+)-inner product in H+ is replaced by the inner product + (f, g)1 = ((I + PN )f, g)+ , then the operator R must be replaced by the operator 1 + R = (I − 2 PN )R (see ﬁgure 2.1, (2.25) and (2.26)). It follows that if A is given by (3.8), then one has to replace R−1 by R−1 and the operator S in M by the + operator S1 = (I − PN /2)S. Therefore A and A∗ are of the form ˙ + + (A˙ ∗ + R−1 S1 )P + − i R−1 P + + i R−1 P + , A = AP D M N N 2 2 i

−i

˙ + + (A˙ ∗ + R−1 S ∗ )P + − i R−1 P + + i R−1 P + , A∗ = AP 1 D M N N 2 2 i

S1∗

−i

∗

where ∈ [M, M] is (1)-adjoint to S1 . Moreover, A = A ⇐⇒ S1 = S1∗ . In sequel we will use the parametrization of all bi-extensions of a regular symmetric operator A˙ with non-zero semi-deﬁciency numbers in the form ˙ + ˙ ∗ + R−1 (S − i J) P + , A = AP + A (3.13) ˙ M Dom(A) 2

52

Chapter 3. Bi-extensions of Closed Symmetric Operators

where J is given by (3.10). Clearly, if A˙ is densely deﬁned, then N = {0} and R = R. Proposition 3.1.11. Let A˙ be a regular closed symmetric operator and let A be a bi-extension of A˙ given by (3.13). Deﬁne a subspace GS in M as i + + GS = ker (PN + P )S − J . (3.14) N−i i 2 Then the domain of the quasi-kernel Aˆ of A can be described as ˆ = Dom(A) ˙ ⊕ GS , Dom(A) + ˆ ˙ A(g + f ) = Ag + A˙ ∗ f + R−1 PN Sf, ˙ f ∈ GS . g ∈ Dom(A),

(3.15)

Proof. From (3.13) it follows that if f ∈ M, then i Af ∈ H ⇐⇒ R−1 S − J f ∈ H. 2 ˙ On the other hand, every vector from F = R−1 (M) is (·)-orthogonal to Dom(A). (−)

Taking into account (2.18) and the equality L = L (see Theorem 2.4.1) we have F ∩ H = L and therefore i i −1 R S − J f ∈ H ⇐⇒ (S − J)f ∈ N ⇐⇒ f ∈ GS . 2 2

3.2 Bi-extensions of O-operators Now we will consider the bi-extensions of O-operators. If A˙ is a regular O-operator, then all its bi-extensions take the form + −1 ˙ + ˙∗ ˙ ˙∗ A = AP S)PN , ˙ + (A + (AA + I) Dom(A)

where S ∈ [N, N] and A = A∗ iﬀ S = S ∗ . Then it follows that Ran(A) ⊂ H. Theorem 3.2.1. If A˙ is a regular O-operator, then all its bi-extensions are (+, ·)continuous and (·, −)-continuous. Moreover, if a closed symmetric operator A˙ has a (+, ·)-continuous or (·, −)-continuous self-adjoint bi-extension A, then A˙ is a regular O-operator. Proof. As mentioned above, an arbitrary bi-extension A of a regular O-operator A˙ satisﬁes the condition Ran(A) ⊂ H. This implies (+, ·)-continuity of operators A and A∗ . It follows that A and A∗ are (+, ·)-continuous. In order to prove the second statement of the theorem we note that an operator B ∈ [H+ , H− ] is (+, ·)-continuous if and only if B ∗ is (·, −)-continuous.

3.2. Bi-extensions of O-operators

53

Thus we can consider a bi-extension of A˙ that is a self-adjoint operator in [H+ , H− ] and both (+, ·)- and (·, −)-continuous. Since A˙ is a restriction of A, then the operator A˙ is (+, ·)-continuous, i.e., it is regular. We prove that A˙ is also an Ooperator. Using the fact that A ⊃ A˙ and (·, −)-continuity of both A and A˙ we apply Theorem 2.2.1 and obtain ˙ Af = Af,

˙ f ) = (f, A˙ ∗ f ), (Af, f ) = (Af,

∀f ∈ H+ ∩ H0 ,

where A˙ is the (·, −)-continuity extension of the operator A˙ onto H0 . It is proved that f ∈ N±i . Then A˙ ∗ f = ±if,

(Af, f ) = ∓i(f, f ).

˙ f ) is real we get f = 0. Therefore, N = N = 0 Taking into account that (Af, i −i and A˙ is an O-operator. ˙ One can consider operator Now let A˙ be a regular O-operator and A ∈ E(A). A as an operator B : H → H with the domain Dom(B) = H+ that is dense on H. Similarly, by C we denote the operator A∗ as acting from H into H and such that ˙ C ⊃ A˙ and hence Dom(B ∗ ) ⊂ Dom(A˙ ∗ ) = H+ , Dom(C) = H+ . Obviously, B ⊃ A, ∗ ∗ and Dom(C ) ⊂ Dom(A˙ ) = H+ . We will show that B = C ∗ and the operators B and C are (·)-closed. Indeed, for f, g ∈ H+ we have (Bf, g) = (Af, g) = (f, A∗ g) = (f, Cg), which implies that B ⊂ C ∗ . Since Dom(B) ⊂ Dom(C ∗ ) ⊂ H+ , then Dom(B) = ˙ then Dom(C ∗ ) and B = C ∗ . In particular, if A (A = A∗ ) is a bi-extension of A, ˙ considering A as an operator in H we obtain a usual self-adjoint extension of A. Theorem 3.2.2. If A˙ is a regular O-operator, then any of its bi-extensions A (A = A∗ ) admits a (−, −)-closure. Moreover, if some bi-extension A (A = A∗ ) of a closed symmetric operator A˙ admits a (−, −)-closure, then A˙ is an O-operator. Proof. As we mentioned above, A can be considered as a usual self-adjoint extension of A˙ and in this case A is (+, ·)-continuous. It is known that (A + iI) maps Dom(A) = H+ onto H bijectively. According to the Banach inverse mapping theorem the operator (A + iI)−1 maps H onto H+ bijectively and (·, −)-continuously. Consequently, this operator maps the (·)-dense in H linear manifold H+ onto some (+)-dense in H+ linear manifold L0 . Consider an operator B0 = A L0 acting in H+ and (+)-densely deﬁned in H+ . Its adjoint B0∗ operates in H− and is (−, −)-closed. Since A ⊃ B0 , then A ⊂ B0∗ , i.e., B0∗ is a (−, −)-closed extension of A. The second part of the theorem is proved by repeating the argument of Theorem 3.2.1.

54

Chapter 3. Bi-extensions of Closed Symmetric Operators

3.3 Self-adjoint and t-self-adjoint bi-extensions In this sections we will study in detail self-adjoint bi-extensions of a regular closed ˙ First, let us note that if GS is deﬁned by (3.14), then symmetric operator A. i i + + GS = ker PN (S − I) ∩ ker (P (S + I) . (3.16) N−i i 2 2 Let S be a (1)-self-adjoint operator in M and let −1 i i V= S− I S+ I 2 2

(3.17)

be the Cayley transform of S. Then V is (1)-unitary in M. In what follows the operator V will play an important role. Deﬁne LV = {h ∈ Ni ⊕ N : Vh ∈ N−i ⊕ N}.

(3.18)

It follows from (3.16) that

f ∈ GS ⇐⇒

h = (S + 2i I)f ∈ LV . Vh = (S − 2i I)f

Hence GS = (I − V)LV .

(3.19)

The next statement is well known. Proposition 3.3.1. Let F be a bounded linear operator in a Hilbert space K deﬁned on a subspace K1 of K. If F is symmetric, then there exists a bounded self-adjoint extension of F in K. Proof. Let K2 = K K1 and let F11 = PK1 F , F21 = PK2 F . According to the decomposition K = K1 ⊕ K2 , the operator F˜ ∈ [K, K] given by the block-operator matrix ∗ F11 F21 ˜ F = F21 F22 ∗ is a bounded extension of F . Then F˜ is self-adjoint iﬀ F22 = F22 .

Theorem 3.3.2. If the semi-deﬁciency indices of a regular closed symmetric operator A˙ with non-dense domain are inﬁnite, then A˙ admits a self-adjoint bi-extension ˙ A with the quasi-kernel Aˆ = A. Proof. Let dim Ni = dim N−i = ∞ and U be a (+)-isometric operator from Ni onto N−i . Then (U + I)Ni is a (1)-subspace in M. Since H is separable then H+ is separable as well (see [81]). Consequently, dim(Ni ⊕ N) = dim(U + I)Ni .

3.3. Self-adjoint and t-self-adjoint bi-extensions

55

Let W be a (1)-isometric operator from Ni ⊕N onto (U +I)Ni . Then for ϕ ∈ Ni ⊕N there exists a unique ψ ∈ Ni such that W ϕ = U ψ + ψ. Furthermore, + + 2 W ϕ − ϕ21 = U ψ + ψ − ϕ21 = U ψ21 + PN ϕ21 + ψ − PN ϕ1 i

+ + 2 = ψ21 + PN ϕ21 + ψ − PN ϕ1 i 1 + 2 + + 2 = PN ϕ21 + PN ϕ1 + 2ψ − PN ϕ1 i i 2 1 1 + 2 + 2 ≥ PN ϕ1 + 2ψ − PN ϕ1 = ϕ21 . i i 2 2

The last estimate implies that (I − W )(Ni ⊕ N) is a subspace in M. We deﬁne a linear operator S in M as follows: Dom(S ) = Ran(I − W ),

S (I − W )ϕ =

i (W + I)ϕ. 2

Then Dom(S ) is a subspace in M and therefore S is a bounded operator. In addition S is (1)-symmetric. Then by Proposition 3.3.1 S admits a bounded (1)self-adjoint extension S in M. Let V be the Cayley transform (3.17) of S. Then V is a (1)-unitary extension of W . Let A be a self adjoint bi-extension of A determined by S, i.e., ˙ + ˙ ∗ + R−1 (S − i J) P + . A = AP + A ˙ M Dom(A) 2 ˙ Indeed, if there is a vector Let us show that its quasi-kernel Aˆ coincides with A. h ∈ M such that i + + (PN + P )S − J h = 0, N−i i 2 then,

Because

i i S − I h ∈ N−i ⊕ N, S + I h ∈ Ni ⊕ N. 2 2

i i S − I h = V S + I h, 2 2

and ϕ = S + 2i I h ∈ Ni ⊕ N, we get Vϕ = W ϕ ∈ N−i ⊕ N. By construction Ran(W ) ∩ (N−i ⊕ N) = {0}. It follows that ϕ = 0 and h = 0. Thus i + + ker (PN + PN )S − J = {0}. i −i 2 ˙ Then Proposition 3.1.11 yields Aˆ = A.

Theorem 3.3.3. If one of the semi-deﬁciency indices of a regular closed symmetric non-densely deﬁned operator A˙ is ﬁnite, then A˙ does not admit self-adjoint bi˙ extensions A with Aˆ = A.

56

Chapter 3. Bi-extensions of Closed Symmetric Operators

Proof. Let p = dim Ni < ∞. Assume that A˙ admits a bi-extension A deﬁned by ˆ the formula (3.13) that is a self-adjoint operator in [H+ , H− ] and with A˙ = A. According to Proposition 3.1.11we have GS = {0}. Hence for every φ =

0, φ ∈ M + + at least one of the vectors PN S − 2i I φ and PN S + 2i I φ is diﬀerent from i −i zero. Put i + M0 := ker PN (S + I) . −i 2 Then

i S + I φ ∈ Ni ⊕ N, 2

(∀φ ∈ M0 ).

+ Besides, for every φ = 0, φ ∈ M0 we have PN S − 2i I φ = 0. Hence, i

i S− I φ∈ / N−i ⊕ N, 2

(φ = 0, φ ∈ M0 ).

Let V be the Cayley transform (3.17) of S. It follows from the above reasoning that (N−i ⊕ N) ∩ V(Ni ⊕ N) = {0}. + + Let ψ ∈ V(Ni ⊕ N) and ψ = 0. Then PN ψ = 0 or otherwise the equality PN ψ = + 0 would imply ψ = PN

−i

i

i

⊕N ψ which means that ψ ∈ N−i ⊕ N. The latter is

+ impossible and we get a contradiction. Therefore, the operator PN maps the i subspace V(Ni ⊕ N) bijectively onto some subspace of Ni . Because dim Ni = p, we get dim V(Ni ⊕ N) ≤ p. On the other hand,

dim V(Ni ⊕ N) = dim(Ni ⊕ N) > p. The contradiction has arrived. This proves the theorem.

Proposition 3.3.4. Let S be a (1)-self-adjoint and bounded operator in M. Deﬁne the subspace ˜ S := ker S(P + + P + ) + i J . G (3.20) Ni N−i 2 Then the operator ˙ + ˙ ∗ P + (Dom(A) ˙ ⊕G ˜S ) A˜ =: AP + A ˙ M Dom(A) is a closed symmetric extension of the operator B˙ deﬁned by (2.19). Proof. Recall that ˙ = Dom(A) ˙ ⊕ N, Dom(B)

˙ + ˙∗ + B˙ = AP ˙ + A PN , Dom(A)

(3.21)

3.3. Self-adjoint and t-self-adjoint bi-extensions

57

˙ and the subspaces B˙ is closed, densely deﬁned and symmetric in H, B˙ ⊃ A, ˙ Ni , N−i are deﬁciency subspaces of B, corresponding to the numbers i and −i, ˜ S ⊇ N. Let f ∈ G ˜ S N. Then the relation respectively. Clearly, G i i + + (S + I)PN I)PN f + (S − f = 0 i −i 2 2 + + + + ˜ ˜ is equivalent to PN f = −VPN f. Consequently, VPN GS = PN GS and i

−i

−i

i

˜ S N = (I − V)(P + G ˜ S ). G N −i

(3.22)

Let the operator U be given by + ˜ Dom(U ) = PN (GS N) ⊆ Ni , i

U = V −1 Dom(U ).

˜ S = N ⊕ (I − U )Dom(U ). Because G ˜ S is a Then U is (1)- and (·)-isometric and G subspace in M, the linear manifold Dom(U ) is closed in Ni . Therefore, the operator A˜ given by (3.21) coincides with B˙ or is its closed symmetric extension. Deﬁnition 3.3.5. A self-adjoint bi-extension A of a regular symmetric operator A˙ is called twice-self-adjoint (t-self-adjoint) if its quasi-kernel Aˆ is a self-adjoint operator in H. Theorem 3.3.6. Let ˙ + ˙ ∗ + R−1 (S − i J) P + A = AP + A ˙ M Dom(A) 2 be a self-adjoint bi-extension of a regular symmetric operator A˙ with equal semideﬁciency numbers. Then the following statements are equivalent: 1) the operator A is t-self-adjoint; ˙ 2) the operator A˜ given by (3.21) is a self-adjoint extension of B; ˜ S N) = M; ˙ G 3) GS +( 4) the Cayley transform V = (S − 2i I)(S + 2i I)−1 possesses the property V(N−i ) = Ni . Proof. 1)⇒ 2), 1)⇒ 3), and 1)⇒ 4). Suppose the quasi-kernel Aˆ of A is a selfadjoint operator. By Theorem 2.5.5 and Proposition 3.1.11 we have LV = Ni ⊕ N = (S + 2i I)GS , GS = (I − V)LV ,

VLV = N−i ⊕ N = (S − 2i I)GS ,

58

Chapter 3. Bi-extensions of Closed Symmetric Operators

ˆ = Dom(A) ˙ ⊕ (I − V)LV , where V and LV are deﬁned by (3.17) and and Dom(A) (3.18). Since V is (1)-unitary in M and maps Ni ⊕ N onto N−i ⊕ N, the inverse operator V −1 maps Ni onto N−i . Therefore V(N−i ) = Ni . Let U = V −1 Ni . Since i S(I − V −1 )f = − (I + V −1 )f, 2

f ∈ M,

˜ S , deﬁned we have S + 2i J (I −U )ϕ = 0, for all ϕ ∈ Ni . Hence, for the subspace G by (3.20), we get ˜ S = (I − U )Ni ⊕ N G ˙ Since (I − U )N = and, as a result, the operator A˜ is a self-adjoint extension of B. i ˜ S N) = M. ˙ G (I − V)N−i and Ran(I − V) = M, we get GS +( ˙ Then there exists a (1)2)⇒1). Suppose A˜ is a self-adjoint extension of B. isometric operator U such that Dom(U ) = Ni , It follows that

Ran(U ) = N−i ,

˜ S = N ⊕ (I − U )Ni . G

i + + Ran (PN + P )S − J = (I + U )Ni , N−i i 2

where the bar stands for (+)-closure. In particular, i i + + ||PN I)f ||1 = ||PN I)f ||1 (S + (S − −i i 2 2

for all f ∈ M.

(3.23)

ˆ = Dom(A) ˙ ⊕ Let GS be deﬁned by (3.14). By Proposition 3.1.11 we have Dom(A) GS and i i (S + I)GS ⊆ Ni ⊕ N, (S − I)GS ⊆ N−i ⊕ N. 2 2 Assume (S + 2i I)GS = Ni ⊕ N. Then there exists a non-zero vector h ∈ / GS such that i (S + I)h ∈ Ni ⊕ N. 2 Then i + PN I)h = 0. (S + −i 2 + i It follows from (3.23) that PN (S − 2 I)h = 0. Therefore h ∈ GS and we have a i contradiction. Hence, i (S + I)GS = Ni ⊕ N. 2

Similarly (S − 2i I)GS = N−i ⊕N. From Theorem 2.5.5 and relations (3.17), (3.18), ˙ i.e., A is a t-self-adjoint and (3.19) we get that Aˆ is a self-adjoint extension of A, ˙ bi-extension of A.

3.3. Self-adjoint and t-self-adjoint bi-extensions

59

˜ S N) = M. Then from (3.22) ˙ G 3)⇒1). Suppose GS +( + ˜ ˙ − V)(PN (I − V)LV +(I GS ) = M. −i

+ ˜ Since the number 1 is a regular point for V, we get LV ⊕ (PN GS ) = M. This −i yields the equalities

LV = Ni ⊕ N,

+ ˜ PN GS = N−i . −i

Because V is (1)-unitary, we have + ˜ ˙ − V −1 )(VPN (I − V −1 )(VLV )+(I GS ) = M. −i

+ ˜ Similarly, we have that the equalities VLV = N−i ⊕ N and VPN GS = Ni hold. −i ˙ Therefore, the operator Aˆ is a self-adjoint extension of A. 4)⇒1). Let V(N−i ) = Ni . Since V is (1)-unitary, the operator V maps Ni ⊕ N onto N−i ⊕ N. Deﬁne W = V (Ni ⊕ N). Then

S(I − W )f =

i (I + W )f, 2

f ∈ Ni ⊕ N.

Therefore GS = (I − W )(Ni ⊕ N) and the quasi-kernel Aˆ of A is a self-adjoint ˙ i.e., A is a t-self-adjoint extension of A. ˙ extension of A, The next statement easily follows from Proposition 3.3.4 and Theorem 3.3.6. Corollary 3.3.7. If i −1 + + ∗ ˙ ˙ A = APDom(A) (S − J) PM , ˙ + A +R 2 is a t-self-adjoint bi-extension of a regular symmetric operator A˙ with equal nonzero semi-deﬁciency numbers and if Aˆ is the quasi-kernel of A, then ˆ + (Dom(A) ˜ ∩ H0 ) = H+ , Dom(A) where A˜ is given by (3.21). ˜ (Dom(A)∩H ˜ Note that by Theorem 2.5.7 the operator P A 0 ) is a self-adjoint ˙ extension of the operator P A in the Hilbert space H0 . Now we give a parametrization of all t-self-adjoint bi-extensions whose quasikernel is a ﬁxed regular self-adjoint extension Aˆ of a regular symmetric operator with equal semi-deﬁciency numbers. Theorem 3.3.8. Let A˙ be a regular symmetric operator in H with non-zero equal ˙ Then semi-deﬁciency numbers and let Aˆ be a regular self-adjoint extension of A.

60

Chapter 3. Bi-extensions of Closed Symmetric Operators

ˆ Moreover, there exists a t-self adjoint bi-extension A of A˙ whose quasi-kernel is A. ˆ if Dom(A) is given by ˆ = Dom(A) ˙ ⊕ (I − U)(N ⊕ N), Dom(A) i where U ∈ [Ni ⊕N, N−i ⊕N] is (1)-isometric operator, ker(I −U) = {0}, Ran(U) = N−i ⊕ N, and Ran(I − U) is a subspace in M (see Theorem 2.5.5), then there is a bijective correspondence between all t-self-adjoint bi-extensions A with quasi-kernel Aˆ and all (1)-isometric operators U mapping Ni onto N−i , satisfying the condition ˙ − U )Ni = M. (I − U )(Ni ⊕ N)+(I

(3.24)

This correspondence is given by the formula ˙ + ˙ ∗ + R−1 (S − i J) P + , A = AP + A ˙ M Dom(A) 2 where S is (1)-self-adjoint operator in M of the form S ((I − U)f + (I − U )ϕ) i i = (I + U)f − (I + U )ϕ, f ∈ Ni ⊕ N, ϕ ∈ Ni . 2 2 Proof. Deﬁne an operator S by the formula ⎧ ⎨ f = (I − U )ϕ, Dom(S ) = (I − U)(Ni ⊕ N), i ⎩ S f = (I + U)ϕ, 2

(3.25)

ϕ ∈ Ni ⊕ N.

Then S is (1)-symmetric, (1)-bounded, closed, and non-densely deﬁned in M. Moreover, i + + (PN J f = 0, f ∈ Dom(S ). + PN )S − i −i 2 Let S ∈ [M, M] be a bounded (1)-self-adjoint extension of S and let i −1 + + ∗ ˙ ˙ A = APDom(A) (S − J) PM ˙ + A +R 2 ˙ Applying Proposition 3.1.11 we get that Aˆ is be a self-adjoint bi-extension of A. the quasi-kernel of A. On the other hand, the Cayley transform i i V = (S − I)(S + I)−1 , 2 2 is a (1)-unitary extension of U on M and Ran(I − V) = M. It follows that V −1 (1)-unitarily maps N−i onto Ni . Deﬁne U := V −1 Ni . Since (I − U )Ni = (I − V −1 )Ni = (I − V)N−i ,

3.4. The case of a densely deﬁned symmetric operator

61

relation (3.24) holds. Because V is the Cayley transform of S, we have (3.25). If U ∈ [Ni , N−i ] is (1)-isometric, Ran(U ) = N−i , and (3.24) holds, then one can verify that S deﬁned by (3.25) is (1)-self-adjoint in M, GS = (I − U)(Ni ⊕ N), ˜ S = (I − U )N ⊕ N. Hence, Aˆ is a quasi-kernel of A. and G i Notice that equality (3.24) is equivalent to ˆ +(I ˆ +(Dom( ˜ Dom(B)) ˙ = H+ , ˙ − U )Ni = Dom(A) ˙ Dom(A) A) where A˜ is deﬁned by (3.21).

3.4 The case of a densely deﬁned symmetric operator If A˙ is a densely deﬁned closed symmetric operator, then N = {0}, the semideﬁciency subspaces Nλ become the deﬁciency subspaces Nλ , M = Ni ⊕ N−i , the operator J deﬁned by (3.10) becomes + + J = PN − PN , i −i

and Theorem 3.1.9 takes the following form: Theorem 3.4.1. Let A˙ be a densely deﬁned closed symmetric operator in the Hilbert space H. Then the formula i + A = A˙ ∗ + R−1 (S − J)PM , 2

(3.26)

establishes a bijective correspondence between all bi-extensions of A˙ and all operators S ∈ [M, M]. The adjoint operator A∗ is of the form i + A∗ = A˙ ∗ + R−1 (S ∗ − J)PM , 2 and A is self-adjoint (A = A∗ ) if and only if S = S ∗ . Notice that the subspace GS deﬁned by (3.14) for a densely deﬁned A˙ takes the form i GS = ker S − J , 2 ˜ S (3.20) becomes G−S . It is easy to see that A˙ admits self-adjoint biand G ˙ Actually, if S = 0, then the operator extensions with quasi-kernel A. i + A = A˙ ∗ − R−1 JPM , 2 ˙ is self-adjoint and G0 = ker(− 2i J) = {0}. Hence, Aˆ = A.

62

Chapter 3. Bi-extensions of Closed Symmetric Operators

Theorem 3.4.2. Let Aˆ be an arbitrary closed densely deﬁned symmetric operator ˙ Then there is a self-adjoint bi-extension A of A˙ whose quasi-kernel extending A. ˆ is A. Proof. According to the von Neumann Theorem 1.3.1 we have ˆ = Dom(A) ˙ ⊕ (I − U )Dom(U ), Dom(A)

(3.27)

where U

Ni ⊇ Dom(U ) → Ran(U ) ⊆ N−i ,

(3.28)

is a (·) (and (+))-isometric operator, Dom(U ) is (·) (and (+))-closed. Deﬁne G := (I − U )Dom(U ),

G⊥ := M G.

Since U is (+)-isometric, the inclusion (I + U )Dom(U ) ⊆ G⊥ . Let S be deﬁned as i Dom(S ) = G, S (I − U )f = (I + U )f, f ∈ Dom(U ). 2 Then S is (+)-symmetric, and non-densely deﬁned in M. In addi (+)-bounded, tion, Ran(S) ⊂ G⊥ and S − 2i J f = 0 for f ∈ Dom(S ) = G. It follows from Proposition 3.3.1 that each (+)-bounded and (+)-self-adjoint extension S of S on M with respect to the decomposition M = G ⊕ G⊥ takes the block-operator matrix form ⎛ ⎞ i + ⊥ − PG J G ⎟ ⎜ 0 2 S=⎝ (3.29) ⎠, i J G L 2 + + where J = PN − PN , PG+ is (+)-orthogonal projection onto G in M, and L is i −i an arbitrary (+)-bounded and (+)-self-adjoint operator in G⊥ . Clearly i i GS = G ⇐⇒ L − PG+ J − J f = 0 for all f ∈ G⊥ \ {0}. 2 2

Let L be a zero operator in G⊥ . If i + i − PG J − J f = 0, 2 2 for some f ∈ G⊥ , then Jf = 0. Hence, f = 0. Thus, if ⎛ ⎜ S0 = ⎝

0 i J G 2

⎞ i − PG+ J G⊥ ⎟ 2 ⎠, 0

3.4. The case of a densely deﬁned symmetric operator

63

then GS0 = G. Thus the quasi-kernel of the self-adjoint bi-extension i + A0 = A˙ ∗ + R−1 (S0 − J)PM , 2

ˆ coincides with A.

Below we establish in terms of aperture a criterion for a self-adjoint biextension A of a closed densely deﬁned symmetric operator to be t-self-adjoint. + Theorem 3.4.3. Let A = A˙ ∗ + R−1 (S − 2i J)PM be a self-adjoint bi-extension of a ˙ closed densely deﬁned symmetric operator A with equal deﬁciency numbers. Then A is t-self-adjoint if and only if the apertures below satisfy Θ{GS , Ni } < 1

and

Θ{GS , N−i } < 1.

(3.30)

Proof. Let Aˆ be the quasi-kernel of A. Then by Proposition 3.1.11, ˆ = Dom(A) ˙ ⊕ GS , Dom(A) (3.27), and (3.28) hold. Moreover, Aˆ is a self-adjoint extension of A˙ if and only if Dom(U ) = Ni and Ran(U ) = N−i . By Lemma 1.5.1 conditions (3.30) are equivalent to + + PN GS = Ni and PN GS = N−i . i −i The latter are equivalent to Dom(U ) = Ni and Ran(U ) = N−i , i.e., to the operator ˙ Aˆ being a self-adjoint extension of A. Notice that when Aˆ is self-adjoint, we have GS = (I − U )Ni and hence, + + PN f 2+ = PN f 22 = i −i

1 f 2+, f ∈ GS . 2

Since M GS = (I + U )Ni , ϕ= and

1 1 (I − U )ϕ + (I + U )ϕ, ϕ ∈ Ni , 2 2

1 1 ψ = − (I − U )U −1 ψ + (I + U )U −1 ψ, ψ ∈ N−i , 2 2

we get 1 (I − PG+S )ϕ2+ = ϕ2+ , ϕ ∈ Ni , 2 1 + 2 (I − PGS )ψ+ = ψ2+ , ψ ∈ N−i . 2 It follows from (1.21) that 1 Θ{GS , Ni } = Θ{GS , N−i } = √ . 2 Therefore Theorem 3.4.3 yields

64

Chapter 3. Bi-extensions of Closed Symmetric Operators

Corollary 3.4.4. Under the conditions of Theorem 3.4.3 formula (3.30) implies 1 Θ{GS , Ni } = Θ{GS , N−i } = √ . 2 The next theorem is an analogue of Theorem 3.3.6. Theorem 3.4.5. Let A˙ be a densely deﬁned closed symmetric operator with equal deﬁciency numbers. Then the following statements are equivalent: 1) the operator i + A = A˙ ∗ + R−1 (S − J)PM 2 ˙ is a t-self-adjoint bi-extension of A; 2) the operator ˜ = A˙ ∗ + R−1 (−S − i J)P + A M 2 ˙ is a t-self-adjoint bi-extension of A; ˙ −S = M; 3) GS +G 4) the Cayley transform V = (S − 2i I)(S + 2i I)−1 possesses the property V(N−i ) = Ni . ˜ in Theorem 3.4.5 are It should be mentioned that the operators A and A connected by the relation −1 ˙ ∗ + ˜ + A = 2A˙ ∗ + R−1 −iP + + iP + ˙∗ A A PM . Ni N−i = 2A − R The latter is equivalent to the equality ˜ + A)f, g) = (A˙ ∗ f, g) + (f, A˙ ∗ g), f, g ∈ H+ . ((A

(3.31)

Suppose that a (+)-self-adjoint operator S ∈ [M, M] satisﬁes condition 3) of Theorem 3.4.5 and let V be the Cayley transform (3.17) of S. Then the quasi-kernels ˜ are self-adjoint extensions of A˙ and Aˆ and A˜ of self-adjoint-bi-extensions A and A posses the properties (see proof of Theorem 3.3.6 and Corollary 3.3.7): ˆ = Dom(A) ˙ ⊕ (I − V)Ni , Dom(A) ˜ ˙ ⊕ (I − V)N−i = Dom(A) ˙ ⊕ (I − V −1 )Ni , Dom(A) = Dom(A) ∗ ˆ ˜ ˙ Dom(A) + Dom(A) = Dom(A ) = H+ . Now we reformulate Theorem 3.3.8 for the case of a densely deﬁned symmetric operator.

3.4. The case of a densely deﬁned symmetric operator

65

Theorem 3.4.6. Let A˙ be a densely deﬁned closed symmetric operator in H with equal deﬁciency numbers and let Aˆ be a self-adjoint extension of A˙ given by the von Neumann formula ˆ = Dom(A) ˙ ⊕ (I − U )Ni . Dom(A) Then there is a bijective correspondence between all (·)-isometric operators W mapping Ni onto N−i , satisfying the condition Ran(I − W −1 U ) = Ni ,

(3.32)

ˆ This correspondence is and all t-self-adjoint bi-extensions A with quasi-kernel A. given by the formula i + A = A˙ ∗ + R−1 (S − J)PM , 2 where S is (+)-self-adjoint operator in M that is the inverse Cayley transform of the form S=

−1 i + + −1 + −1 + IM + U PN + W P I − U P − W P . M N−i Ni N−i i 2

Notice that condition (3.32) is equivalent to ˙ − W )Ni = M, (I − U )Ni +(I and i i S ((I − U )f + (I − W )ϕ) = (I + U )f − (I + W )ϕ, f ∈ Ni , ϕ ∈ Ni . 2 2 Hence

Ran(A − A˙ ∗ ) = R−1 (I + W )N−i .

(3.33)

(3.34)

Now we give a parametrization in a slightly diﬀerent form of all t-self-adjoint ˆ We need a bi-extension whose quasi-kernel is a ﬁxed self-adjoint extension A. notion of mutually transversal self-adjoint extensions of a given densely deﬁned symmetric operator. Deﬁnition 3.4.7. Let A˙ be a closed densely deﬁned symmetric operator with equal defect numbers. Two self-adjoint extensions A1 and A2 of A˙ are called relatively ˙ and transversal if prime (disjoint) if Dom(A1 ) ∩ Dom(A2 ) = Dom(A) Dom(A1 ) + Dom(A2 ) = Dom(A˙ ∗ ). Proposition 3.4.8. Let A1 and A2 be self-adjoint extensions of a closed densely ˙ Then the following statements are equivalent: deﬁned symmetric operator A. 1) A1 and A2 are transversal;

66

Chapter 3. Bi-extensions of Closed Symmetric Operators

2) Ran (A1 − λI)−1 − (A2 − λI)−1 = Nλ for some (and then for all) non-real λ. Proof. Clearly, for all λ, Im λ = 0, Ran (A1 − λI)−1 − (A2 − λI)−1 ⊆ Nλ . If Ran (A1 − λI)−1 − (A2 − λI)−1 = Nλ , then Nλ ⊂ Dom(A1 )+Dom(A2 ). This yields Dom(A1 ) + Dom(A2 ) = Dom(A˙ ∗ ). Conversely, if A1 and A2 are transversal, then every vector f ∈ Nλ can be represented as f = f1 + f2 , f1 ∈ Dom(A1 ), f2 ∈ Dom(A2 ). Then it follows that (A1 − λI)−1 − (A2 − λI)−1 (A1 − λI)f1 = f. Therefore, Ran (A1 − λI)−1 − (A2 − λI)−1 = Nλ .

Let A1 and A2 be self-adjoint extensions of A˙ and let ˙ ⊕ (I − Uk )Ni , Dom(Ak ) = Dom(A)

k = 1, 2.

(3.35)

Since Uk = (Ak − iI)(Ak + iI)−1 Ni and U1 − U2 = 2i A2 + iI)−1 − (A1 + iI)−1 Ni , it follows from Proposition 3.4.8 that A1 and A2 are mutually transversal if and only if Ran(U1 − U2 ) = N−i ⇐⇒ Ran(I − U1−1 U2 ) = Ni ˙ − U2 )Ni = Ni ⊕ N−i = M. ⇐⇒ (I − U1 )Ni +(I Besides, two transversal self-adjoint extensions are relatively prime. The transversality of A1 and A2 yields the direct decompositions of H+ , ˙ − U2 )Ni = Dom(A2 )+(I ˙ − U1 )Ni . H+ = Dom(A1 )+(I Set ˙ MAk := (I − Uk )Ni = Dom(Ak ) Dom(A),

k = 1, 2,

Then the following conditions are equivalent: ˙ 1. A1 and A2 are transversal self-adjoint extensions of A; ˙ A2 ; 2. H+ = Dom(A1 )+M ˙ A1 . 3. H+ = Dom(A2 )+M

(3.36)

3.4. The case of a densely deﬁned symmetric operator

67

Theorem 3.4.9. Let A˙ be a densely deﬁned closed symmetric operator in H with ˙ Then there is a equal deﬁciency numbers and let Aˆ be a self-adjoint extension of A. ˆ bijective correspondence between all transversal to A self-adjoint extensions A˜ and ˆ This correspondence all t-self-adjoint bi-extensions of A˙ whose quasi-kernel is A. is given by the formula A = A˙ ∗ − R−1 A˙ ∗ (I − PAˆA˜ ), ˆ corresponding to the decomposition where PAˆA˜ is a projector in H+ onto Dom(A) ˆ +M ˙ A˜ . H+ = Dom(A) Proof. Let ˆ = Dom(A) ˙ ⊕ (I − W )Ni = Dom(A) ˙ ⊕ M ˆ. Dom(A) A By Theorem 3.4.6 there is a bijective correspondence between all t-self-adjoint biextensions with quasi-kernel Aˆ and all (·)-isometric operators U , Dom(U ) = Ni , Ran(U ) = N−i such that ˙ − U )Ni = M. MAˆ +(I Deﬁne A˜ by ˜ = Dom(A) ˙ ⊕ (I − U )Ni , Dom(A)

˜ A˜ = A˙ ∗ Dom(A).

Then A˜ is a self-adjoint extension of A˙ transversal to Aˆ and MA˜ = (I − U )Ni . ˆ +M ˙ A˜ . Hence H+ = Dom(A) Deﬁne a (+)-self-adjoint operator S in M by (3.33). Then i (S − J)(I − U )ϕ = −i(I + U )ϕ = −A˙ ∗ (I − U )ϕ, 2 for all ϕ ∈ Ni and i (S − J)(I − W )f = 0, f ∈ Ni . 2 + Let A = A˙ ∗ + R−1 (S − 2i J)PM . Then A = A˙ ∗ − R−1 A˙ ∗ (I − PAˆA˜ ).

Let N1 = Ni , N2 = N−i , and let Pk (k = 1, 2) be a (+)-orthogonal projection operator from M onto Nk . If A is a self-adjoint bi-extension of a closed symmetric operator A˙ then it takes a form of (3.26), where Q is of the form i Q = S − J. 2 It is easy to see that Q can be written in the form Q11 Q12 Q= , Q21 Q22 where Qjk operates from Pk M into Pj M.

68

Chapter 3. Bi-extensions of Closed Symmetric Operators

Theorem 3.4.10. Let A be a self-adjoint extension of a closed symmetric operator A˙ with Dom(A) deﬁned by von Neumann’s formulas (1.15) using isometric operator U. Then A is a quasi-kernel of a bi-extension A (A = A∗ ) if and only if Q11 + Q12 U = 0, Q21 + Q22 U = 0.

(3.37)

The proof of this theorem immediately follows from Theorem 3.4.1. Let now A˙ be a closed densely deﬁned symmetric operator with deﬁciency indices (n, n). It is not hard to see that according to Theorem 3.4.1 any self-adjoint bi-extension A of A˙ takes a form ⎡ ⎤ n n n # # # ⎣ A = A˙ ∗ + akj (·, αj ) + bkj (·, βj )⎦ αk k=1

+

n # k=1

j=1

j=1

j=1

j=1

⎡ ⎤ n n # # ⎣ ckj (·, αj ) + dkj (·, βj )⎦ βk ,

where the coeﬃcient block-matrix

A B C D

(3.38)

satisﬁes the relations

akj = ajk , dkj = djk , (k = j), C = B ∗ , 1 1 Im ajj = − , Im djj = , (k, j = 1, . . . , n), 2 2 and {αj }, {βj }, (j = 1, . . . , n) are (−)-orthonormal bases in subspaces R−1 (Ni ) and R−1 (N−i ), respectively. Theorem 3.4.11. Let A be a self-adjoint bi-extension of a closed symmetric densely deﬁned operator with deﬁciency indices (n, n) given by (3.38). Then A is a t-selfadjoint bi-extension of A˙ if and only if A B rank = n. (3.39) C D Theorem 3.4.11 implies that if (3.39) holds, then any self-adjoint extension A of A˙ is described by the formula Ah = A˙ ∗ h, where h ∈ H+ satisﬁes the conditions n # j=1 n # j=1

akj (h, αj ) + ckj (h, αj ) +

n # j=1 n #

bkj (h, βj ) = 0, (3.40) dkj (h, βj ) = 0.

j=1

Formula (3.40) above provides the abstract set of boundary conditions for a selfadjoint extension A of A˙ under assumption (3.39).

Chapter 4

Quasi-self-adjoint Extensions In this chapter we consider quasi-self-adjoint extensions of, generally speaking, non-densely deﬁned symmetric operators and establish analogues of von Neumann’s and Krasnoselki˘i’s formulas in cases of direct and indirect decompositions of their domains. The quasi-self-adjoint bi-extensions and the so-called (∗)extensions (with exit into triplets of rigged Hilbert spaces) of symmetric operators will be introduced. We also present an analysis of these extensions together with their description and parametrization.

4.1 Quasi-self-adjoint extensions of symmetric operators Once again, let A˙ be a closed symmetric operator in a Hilbert space H and let H+ ⊂ H ⊂ H− be a rigged Hilbert space generated by A˙ with H+ = Dom(A˙ ∗ ) (see Section 2.2). A closed densely deﬁned linear operator T acting on a Hilbert space H is called a quasi-self-adjoint extension of a closed symmetric operator A˙ if ˙ T ⊃ A˙ and T ∗ ⊃ A. For such operators A˙ ∗ ⊃ P T,

A˙ ∗ ⊃ P T ∗ ,

(4.1)

˙ Therefore, where P is the (·)-orthogonal projection of H onto H0 = Dom(A). ∗ Dom(T ) ⊂ H+ , Dom(T ) ⊂ H+ . A quasi-self-adjoint extension T of a closed symmetric operator A˙ is called regular if P T and P T ∗ are closed linear operators in H. Deﬁnition 4.1.1. A quasi-self-adjoint extension T of a closed symmetric operator ˙ if the resolvent set ρ(T ) is not empty. A˙ belongs to the class Ω(A) Y. Arlinskii et al., Conservative Realizations of Herglotz–Nevanlinna Functions, Operator Theory: Advances and Applications, DOI 10.1007/978-3-7643-9996-2_4, © Springer Basel AG 2011

69

70

Chapter 4. Quasi-self-adjoint Extensions

˙ Clearly, self-adjoint extensions of A˙ (if they exist) belong to the class Ω(A) ∗ ˙ ˙ and T ∈ Ω(A) if and only if T ∈ Ω(A). Theorem 4.1.2. The following statements about a quasi-self-adjoint extension T of a closed symmetric operator A˙ are equivalent: (i) The linear manifolds Dom(T ) and Dom(T ∗ ) are (+)-closed. (ii) The operators T and T ∗ are (+, ·)-bounded. ˙ (iii) T is a regular quasi-self-adjoint extension of A. Proof. First we prove that statement (i) is equivalent to the statement (ii). Both operators T and T ∗ are closed and thus (+, ·)-closed. By the Closed Graph Theorem the operator T (T ∗ , respectively) is (+, ·)-bounded if and only if Dom(T ) (Dom(T ∗ ), respectively) is (+)-closed. Now let us show that (ii) ⇔ (iii). It follows from (4.1) that densely deﬁned in H operators P T and P T ∗ admit closure in H, that we will denote by P T and P T ∗ , respectively. It is enough to show that Dom(P T ) (Dom(P T ∗ )) coincides with the (+)-closure of Dom(T ) (Dom(T ∗ )). An element f ∈ H belongs to Dom(P T ) if and (·)

only if there exists such a sequence {fn } ⊂ Dom(P T ) = Dom(T ) that fn −→ f , (·) A˙ ∗ fn = P T fn −→ P T = A˙ ∗ f, (+)

or fn −−→ f . The latter is equivalent to that f belongs to the (+)-closure of Dom(T ). Similar reasoning can be used to show that the same is true about Dom(T ∗ ). Thus, a necessary condition for the existence of regular quasi-self-adjoint extensions of an operator A˙ is that it is (+, ·)-bounded, that is, regular. It is ˙ = H every quasi-self-adjoint extension T of obvious that in the case Dom(A) ∗ ˙ ˙ A is regular, since A f = T f , T f = A˙ ∗ f ≤ f + for all f ∈ Dom(T ), and, similarly, T ∗ f ≤ f + (∀f ∈ Dom(T ∗ ). The following proposition is more general. ˙ < ∞, then every quasi-self-adjoint extension T Theorem 4.1.3. If codim Dom(A) ˙ of A is regular. Proof. It is suﬃcient to show that T is (+, ·)-bounded. Since T f 2 < f 2 + T f 2 = (I + T ∗ T )1/2 f , 1

(f = 0),

then the operator Γ = T (I + T ∗ T )− 2 satisﬁes the condition Γf < f ,

(f ∈ H,

f = 0).

4.1. Quasi-self-adjoint extensions of symmetric operators

71

˙ and Q = I −P . Let P be, as above, the orthoprojection operator in H onto Dom(A) The operator QΓ then is ﬁnite-dimensional and QΓf < f (f ∈ H) and hence QΓ = q < 1. Consequently, we have that QΓf ≤ qf ,

f ∈ H.

1

Letting g = (I + T ∗ T )− 2 f we have that Γf = T g, 1

f 2 = (I + T ∗ T ) 2 g2 = g2 + T g2, and hence

QT g2 ≤ q 2 (g2 + T g2).

Since A˙ ∗ g = P T g, (g ∈ Dom(T )), then g2+ = g2 + A˙ ∗ g2 = g2 + T g2 − QT g2 ≥ (1 − q 2 )(g2 + T g2) > (1 − q 2 )T g2.

Following Section 1.7 we call the operators U ∈ [Ni , N−i ] and W ∈ [N−i , Ni ] admissible if Uφ = V φ, φ ∈ Bi ,

and

Wφ = V −1 φ, φ ∈ B−i ,

respectively, imply that φ = 0, where V is the exclusion operator deﬁned in Section 1.7 by (2.24). The admissibility of U and W is equivalent to ˙ ∩ (U − I)Ni = {0} and Dom(A) ˙ ∩ (W − I)N−i = {0}, Dom(A) respectively (see Theorem 1.7.4). The lemma below is the analogue of the Krasnoselski˘i formulas in the case of indirect decomposition of the domains of quasiself-adjoint extensions. ˙ and −i ∈ ρ(T ), then Lemma 4.1.4. If T ∈ Ω(A) ˙ (I − U )Ni , Dom(T ) = Dom(A) ˙ (U ∗ − I)N−i , Dom(T ∗ ) = Dom(A)

(4.2)

where U and U ∗ are admissible operators in [Ni , N−i ] and [N−i , Ni ], respectively. Further, if f ∈ Dom(T ) or f ∈ Dom(T ∗ ), then ⎧ ⎧ f = g + ϕ − U ϕ, f = g + U ∗ ψ − ψ, ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ˙ + iϕ + iUϕ, ˙ + iU ∗ ψ + iψ, T f = Ag T ∗ f = Ag or (4.3) ˙ ϕ ∈ Ni , ˙ ψ ∈ N−i , ⎪ ⎪ g ∈ Dom(A), g ∈ Dom(A), ⎪ ⎪ ⎩ ⎩ f ∈ Dom(T ), f ∈ Dom(T ∗ ). Moreover, the formulas (4.2) and (4.3) establish a bijection between the set of all ˙ with −i ∈ ρ(T ) and the set of all admissible quasi-self-adjoint extensions T ∈ Ω(A) linear operators U ∈ [Ni , N−i ] and U ∗ ∈ [N−i , Ni ].

72

Chapter 4. Quasi-self-adjoint Extensions

Proof. Let us consider the Cayley transform (1.19) of the operator T U = (T − iI)(T + iI)−1 = I − 2i(T + iI)−1 . Then

U ∗ = (T ∗ + iI)(T ∗ − iI)−1 . ˙ (see Chapter 1). Clearly, Recall the notation Mλ = (A˙ − λI)Dom(A) U : M−i → Mi , U ∗ : Mi → M−i , ˙ (U − I)M−i = (U ∗ − I)Mi = Dom(A). It follows from the orthogonal decompositions H = M−i ⊕ Ni = Mi ⊕ N−i that U : Ni → N−i and U ∗ : N−i → Ni . Let U = U Ni . Then U ∈ [Ni , N−i ] and for U ∗ ∈ [N−i , Ni ] one has U ∗ = U ∗ N−i . Since

f = i(U − I)x h = i(I − U ∗ )x , , x ∈ H, T f = (U + I)x T ∗ h = (U ∗ + I)x and because Dom(T ) and Dom(T ∗ ) are dense in H, we get the equalities ker(U − I) = ker(U ∗ − I) = {0}, ˙ ∩ (U − I)Ni = {0}, Dom(A) ˙ ∩ (U ∗ − I)Ni = {0}, Dom(A) and relations (4.2), (4.3). Now suppose that U ∈ [Ni , N−i ] and U ∗ ∈ [N−i , Ni ] are both admissible operators. Using formulas (4.2) we deﬁne the operator T . Then T ⊃ A˙ and ker(T + iI) = {0},

Ran(T + iI) = H.

Moreover, formulas (4.3) yield that T is a closed operator in H. Hence −i is a regular point of T . Let us show that Dom(T ) is dense in H. If x ∈ H is orthogonal ˙ and to Dom(T ), then x ∈ L = H Dom(A) (x, ϕ − U ϕ) = 0, for all ϕ ∈ Ni . It follows that PNi x = U ∗ PN−i x. Since U ∗ is admissible, we get that PNi x = PN−i x = 0 ⇐⇒ x = 0. ˙ = (A˙ − iI)(A˙ + iI)−1 be the Cayley Let U = (T − iI)(T + iI)−1 and Ui (A) ˙ respectively. Then, clearly, transforms of T and A, ˙ M + UPN . U = Ui (A)P −i i

4.1. Quasi-self-adjoint extensions of symmetric operators Hence

73

˙ Mi + U ∗ PN−i , U ∗ = Ui−1 (A)P

where U ∗ ∈ [N−i , Ni ] is the adjoint operator to U ∈ [Ni , N−i ]. Let T ∗ be the adjoint to T . Because U ∗ = (T ∗ + iI)(T ∗ − iI)−1 , we get that ˙ +(U ˙ ∗ − I)N−i , Dom(T ∗ ) = Dom(A) and

˙ + iU ∗ ψ + iψ, T ∗ (g + U ∗ ψ − ψ) = Ag

˙ ψ ∈ N−i . Thus T ∗ ⊃ A. ˙ for all g ∈ Dom(A),

˙ −i ∈ ρ(T ): Theorem 4.1.5. The following statements are equivalent for T ∈ Ω(A), ˙ 1. T is a regular extension of A; 2. (I−U)Ni and (U ∗ −I)N−i are (+)-closed, and the minimal angles (in the (+)˙ and (I−U)Ni and between Dom(A) ˙ and (U ∗ −I)N−i metric) between Dom(A) are positive; + + 3. PM ((I − U)Ni ) and PM ((U ∗ − I)N−i ) are (+)-closed linear manifolds; + + 4. (I −U)Ni and (U ∗ −I)N−i are (+)-closed, and PM (I −U )Ni and PM (U ∗ − I)N−i are homeomorphisms.

Proof. The proof of the theorem immediately follows from Lemma 2.5.2 if one sets ˙ and L2 = (I − U )Ni and Theorem 4.1.2. H = H+ , L1 = Dom(A), We note that when the conditions (1)–(4) of the theorem are satisﬁed, there ˙ are constants d1 > 0 and d2 > 0 such that for all ϕ ∈ Ni , ψ ∈ N−i , g ∈ Dom(A) we have g + ϕ − U ϕ+ ≥ d1 ϕ, g + U ∗ ψ − ψ+ ≥ d2 ψ. ˙ and λ ∈ ρ(T ). Then using the Cayley transform Remark 4.1.6. Let T ∈ Ω(A) −1 ¯ (T − λI)(T − λI) and its restriction ¯ Mλ = (T − λI)(T − λI)−1 Nλ¯ : Nλ¯ → Nλ , one can prove that the relations ˙ +(I ˙ − Mλ )Nλ¯ , Dom(T ) = Dom(A) ˙ +(I ˙ − Mλ∗ )Nλ , Dom(T ∗ ) = Dom(A)

(4.4)

and ¯ − λMλ ϕ, g ∈ Dom(A), ˙ + λϕ ˙ ϕ ∈ Nλ¯ , T (g + (I − Mλ )ϕ) = Ag ∗ ∗ ∗ ¯ ˙ ˙ ψ ∈ Nλ T (g + (I − Mλ )ψ) = Ag + λψ − λMλ ψ, g ∈ Dom(A), hold true.

(4.5)

74

Chapter 4. Quasi-self-adjoint Extensions

Recall that the operator T in the Hilbert space H is called dissipative if Im (T f, f ) ≥ 0 for all f ∈ Dom(T ). A dissipative operator T is called maximal dissipative (m-dissipative) if one of the equivalent conditions is satisﬁed: • T has no dissipative extensions in H; • the resolvent set ρ(T ) contains a point from the open lower half-plane; • T is densely deﬁned, closed, and −T ∗ is a dissipative operator. A closed dissipative operator T possess the properties: 1. the ﬁeld of regularity of T contains the open lower half-plane; 2. the numerical range W (T ) = {(T f, f ), ||f || = 1} of T is contained in the closed upper half-plane. The resolvent set ρ(T ) of m-dissipative operator T contains the open lower halfplane and 1 ||(T − λI)−1 || ≤ , Im λ < 0. |Im λ| The Cayley transform U (T ) = (T − iI)(T + iI)−1 of m-dissipative operator T is a ˙ then U (T ) contraction and if T is an extension of a closed symmetric operator A, is a contractive extension of the Cayley transform of A, which is an isometry from M−i onto Mi . The next statement is well known [219]. Proposition 4.1.7. Let A˙ be a closed symmetric operator in H. If T is an m˙ then T ∗ is also an extension of A. ˙ Moreover, if A˙ is dissipative extension of A, ˙ densely deﬁned, then each m-dissipative extension of A is a restriction of A˙ ∗ . We complete Lemma 4.1.4 by the following assertion. Lemma 4.1.8. Each closed symmetric operator A˙ admits m-dissipative extensions. Moreover, there is a one-to-one correspondence between all admissible contractions ˙ U : Ni → N−i and all m-dissipative extensions of A. It should be noted that in view of Proposition 4.1.7, if U : Ni → N−i is an admissible contraction, then U ∗ : N−i → Ni is automatically admissible. We recall that in Section 2.3 we introduced an inner product (·, ·)1 on H+ deﬁned by (2.25). Then the (1)-orthogonal decomposition ˙ ⊕ N ⊕ N ⊕ N, H+ = Dom(A) i −i holds true. The following theorem is a generalization of the von Neumann formulas to the case of quasi-self-adjoint extensions and direct decompositions of their domains.

4.1. Quasi-self-adjoint extensions of symmetric operators

75

Theorem 4.1.9. Let A˙ be a regular closed symmetric operator. ˙ −i ∈ ρ(T ), there corresponds an operator M ∈ [N ⊕ I. To each T ∈ Ω(A), i N, N−i ⊕ N] such that ker(M + I) = {0},

ker(M ∗ + I) = {0},

(4.6)

and ˙ ⊕ (M + I)(N ⊕ N), Dom(T ) = Dom(A) i ˙ + A˙ ∗ (I + M )ϕ + i(A˙ A˙ ∗ + I)P + (I − M )ϕ, T (g + (M + I)ϕ) = Ag N ˙ ϕ ∈ N ⊕ N, g ∈ Dom(A), i

(4.7)

˙ ⊕ (M ∗ + I)(N ⊕ N), Dom(T ∗ ) = Dom(A) −i ˙ + A˙ ∗ (I + M ∗ )ψ + i(A˙ A˙ ∗ + I)P + (M ∗ − I)ψ, T ∗ (g + (M ∗ + I)ψ) = Ag N ˙ ψ ∈ N ⊕ N. g ∈ Dom(A), −i (4.8) II. Conversely, for each operator M ∈ [Ni ⊕ N, N−i ⊕ N] satisfying (4.6), for˙ −i ∈ ρ(T ), and its mulas (4.7) and (4.8) determine an operator T ∈ Ω(A), adjoint T ∗ . III. An operator T is a regular extension of A˙ if and only if the linear manifolds Ran(M + I) and Ran(M ∗ + I) are (+)-closed. IV. The operator T is an m-dissipative extension of A˙ if and only if the operator M is a (1)-contraction. Proof. Deﬁne the operator M ∈ [Ni ⊕ N, N−i ⊕ N] as

+ h = PM ϕ, ϕ ∈ Ni , + M h = −PM Uϕ,

(4.9)

where U is an admissible operator deﬁning T via formulas (4.2) and (4.3). By (2.27) we have + + −1 ˙∗ h = i(2PN ϕ, + PN )(A + iI) i ϕ ∈ Ni . + + ∗ M h = i(2PN + PN )(A˙ − iI)−1 Uϕ, −i

Because

⎧ + 1 + ⎨ ϕ = −i A˙ ∗ + iI P + P N h, 2 Ni + + 1 ∗ ⎩ Uϕ = −i A˙ − iI P + PN M h, 2 N

(4.10)

−i

we obtain

+ (I − U)ϕ = (I + M )h − iA˙ ∗ PN (I − M )h.

+ ˙ (see Theorem 2.3.1), it follows that ker(I + M ) = Since A˙ ∗ PN (h − M h) ∈ Dom(A) {0} and ˙ ⊕ (I + M )(N ⊕ N). Dom(T ) = Dom(A) i

76

Chapter 4. Quasi-self-adjoint Extensions

+ From (4.10) we also have i(I + U)ϕ = A˙ ∗ (I + M )h + iPN (I − M )h. ˙ h ∈ Now let f ∈ Dom(T ). Then f = g + (I + M )h, where g ∈ Dom(A), (Ni ⊕ N). From (4.3) we obtain + T f = T (g + (I + M )h) = T (g + iA˙ ∗ PN (I − M )h) + (I − U )ϕ + = A˙ g + iA˙ ∗ PN (I − M )h + i(I + U)ϕ

˙ + A˙ ∗ (I + M )h + i(A˙ A˙ ∗ + I)P + (I − M )h. = Ag N Thus, (4.7) is proved. 1 1 + + Recall that the operators 2− 2 PM Ni and 2− 2 PM N−i are (·, 1)-isometries from Ni and N−i onto Ni ⊕ N and N−i ⊕ N, respectively. It follows from (4.9) that the (1)-adjoint operator M ∗ ∈ [N−i ⊕ N, Ni ⊕ N] is given by

+ x = PM ψ , ψ ∈ N−i . + ∗ ∗ M x = −PM U ψ

Arguing as above we obtain the equality ker(M ∗ + I) = {0} and relations (4.7). Let now M ∈ [Ni ⊕ N, N−i ⊕ N] satisfy conditions (4.6). Deﬁne the operator T by (4.7). Using (4.10) one can check that T is of the form (4.3). Hence, T is a quasi-self-adjoint extension of A˙ and for its adjoint T ∗ relations (4.8) are valid. Part III of the theorem follows directly from Theorem 4.1.2 and formulas (4.7). Part IV is a consequence of Lemma 4.1.8. Let us ﬁnd the (1)-orthogonal complements ⊥

[Ran(M + I)] = M Ran(M + I), and

⊥

[Ran(M ∗ + I)] = M Ran(M ∗ + I), ⊥

where M ∈ [Ni ⊕ N, N−i ⊕ N]. Suppose ψ ∈ [Ran(M ∗ + I)] . Then ((M ∗ + I)ϕ, ψ)1 = 0, ϕ ∈ N−i ⊕ N. Furthermore, using the (1)-orthogonality relation one can show that + + + 0 = ((M ∗ + I)ϕ, ψ)1 = (M ∗ + I)ϕ, PN ψ + PN ψ + PN ψ i −i 1 + + + + = ϕ, M (PN + PN )ψ + ϕ, (PN + PN )ψ . 1

i

−i

1

Therefore, we have that + + + + M (PN + PN )ψ. + PN )ψ = −(PN i

−i

(4.11)

4.1. Quasi-self-adjoint extensions of symmetric operators

77

+ + + Let us set φ = (PN + PN )ψ. Then (4.11) implies PN (M + I)φ = 0. Hence, if i

⊥

ψ ∈ [Ran(M ∗ + I)] , then + + + φ = (PN + PN )ψ ∈ Ker PN (M + I)φ i

+ and ψ = φ − PN M φ. −i

+ + Let now φ ∈ Ker PN (M + I) . Then the vector ψ = φ − PN M φ belongs −i

⊥

to [Ran(M ∗ + I)] . Indeed,

+ + + + + + −(PN + PN )ψ = −PN φ + PN M φ = PN M φ + PN M φ −i

−i

−i

+ + = M φ = M (PN + PN )ψ. i

Hence,

+ + [Ran(M ∗ + I)]⊥ = (I − PN M ){Ker [PN (M + I)]}. i

It can be shown similarly that + + ∗ ∗ [Ran(M + I)]⊥ = (I − PN + I)]}. M ){Ker [PN (M i

(4.12)

The next statement is an application of Theorem 4.1.9 to the case of an mdissipative extension (cf. Theorem 2.5.7). ˙ be m-dissipative and regular. Then the operator Theorem 4.1.10. Let T ∈ Ω(A) P T (Dom(T ) ∩ H0 ) ˙ where P is the (·)is an m-dissipative extension in H0 of the operator P A, ˙ orthogonal projection of H onto H0 = Dom(A). Proof. Relations (4.7) and (4.8) are valid for T with a contraction M ∈ [Ni ⊕ N, N−i ⊕ N]. Let us show that Ran(M ∗ + I) ∩ Ni = {0}. If ϕ ∈ Ni and ϕ = (M ∗ + I)h, then + PN (M ∗ + I)h = 0, h ∈ N. It follows that PN M ∗ h = −h. Since M ∗ is a (1)-contraction, we get + ||M ∗ h||1 ≤ ||h||1 = ||PN M ∗ h||1 ≤ ||M ∗ h||1 . + Consequently, PN M ∗ h = 0. This yields the equality M ∗ h + h = 0 ⇐⇒ h = 0. i Hence, ϕ = 0. Now because Ran(M ∗ + I) is (1)-closed we get that Ran(M ∗ + I) = M. It + follows from (4.12) that N := ker(PN (I + M )) = {0}. Let h ∈ N . Then

1 + 1 + + 2 ||PN ||P M h||21 = (||M h||21 − ||PN M h||21 ) M h|| = −i 2 N−i 2 1 1 1 + 2 + + + 2 ≤ (||h||21 − ||PN h||21 ) = (||h||21 − ||PN h||21 ) = ||PN h||1 = ||PN h|| . i i 2 2 2

78

Chapter 4. Quasi-self-adjoint Extensions

This yields that the operator given by + y = PN h i , h∈N + W y = PN Mh −i

is well deﬁned and is a (·)-contraction with values in N−i . Let us show that + + Dom(W ) = PN N = Ni . If {PN hn }, where {hn } ⊂ N , is a (1)-Cauchy sei

i

+ quence, then, since W is a contraction, we get that {PN M hn } and hence, −i {(M + I)hn } are (1)-Cauchy sequences as well. It follows that {hn } (1)-converges. + + Let h = limn→∞ hn = h. Then h ∈ N and limn→∞ PN hn = PN h ∈ Dom(W ). It i i follows from the (1)-orthogonal decomposition

(M ∗ + I)N ⊕ N = Ni ⊕ N that if the vector x ∈ Ni is orthogonal to Dom(W ), then x = (M ∗ +I)f , where f ∈ + + ∗ N. Consequently, PN M f = x and PN M f = −f . The last equality implies (since −i

M ∗ is a (1)-contraction) that M ∗ f = −f , and, hence, x = 0. Thus, Dom(W ) = Ni . This yields the relation ˙ +(I ˙ + W )Ni . Dom(T ) ∩ H0 = Dom(A) Therefore, P T (Dom(T ) ∩ H0 ) is an m-dissipative extension in H of densely deﬁned in H0 symmetric operator P A. Let T be a closed densely deﬁned operator in a Hilbert space H. The operator A˙ deﬁned as ˙ = {f ∈ Dom(T ) ∩ Dom(T ∗ ) : T f = T ∗ f } , Af ˙ = T f, f ∈ Dom(A) ˙ Dom(A) is called the maximal common symmetric part of T and T ∗ . ˙ −i ∈ ρ(T ) and let M be the operator deﬁning T Theorem 4.1.11. Let T ∈ Ω(A), by (4.7). If A˙ is a maximal common symmetric part of T and T ∗ , then ker(M M ∗ − I) = ker(M ∗ M − I) = {0}, and the linear manifold Dom(T ) + Dom(T ∗ ) is dense in H+ . Proof. Let us assume that ker(M M ∗ −I) = {0}. Then there is ϕ = 0, ϕ ∈ N−i ⊕N such that M M ∗ ϕ = ϕ. Let ψ = M ∗ ϕ and observe that M ψ = ϕ then. It is easy to see that + + + + + + + + ∗ ∗ PN M ϕ = PN ψ, PN M ϕ = PN ψ, PN M ψ = PN ϕ, PN M ψ = PN ϕ, i

i

−i

−i

which implies (M + I)ψ = (M ∗ + I)ϕ,

+ + PN (M ∗ ϕ − ϕ) = PN (ψ − M ψ).

4.1. Quasi-self-adjoint extensions of symmetric operators

79

It follows from (4.7) and (4.8) that for an element f = (M ∗ + I)ϕ = (M + I)ψ, ˙ the equality T f = T ∗ f holds true. But according to Theorem 4.1.9, f ∈ / Dom(A) and hence f = 0 yielding ϕ = 0. Let h ∈ M be (1)-orthogonal to the linear manifold Ran(M + I). Then from (4.12) one has + + + + M ∗ (PN + PN )h = −(PN + PN )h. −i

i

Similarly any g ∈ M (1)-orthogonal to Ran(M ∗ + I) satisﬁes the condition + + + + M (PN + PN )g. + PN )g = −(PN i

−i

It follows that if ψ ∈ M is (1)-orthogonal to Ran(M + I) + Ran(M ∗ + I), then + + + + M ∗ (PN + PN )ψ = −(PN + PN )ψ −i

i

+ (PN −i

and

+ + + + M (PN + PN )ψ. + PN )ψ = −(PN i

−i

+ PN )ψ

Hence, for f = + we have M M ∗ f = f . Since ker(M M ∗ − I) = {0}, we get f = 0 and ψ = 0. This yields the density of Ran(M + I) + Ran(M ∗ + I) in M. Consequently, Dom(T ) + Dom(T ∗ ) is dense in H+ . Observe that the equality ker(M ∗ M − I) = {0} (respect., ker(M M ∗ − I) = {0}) implies ker(M M ∗ − I) = {0} (respect., ker(M ∗ M − I) = {0}). Indeed, if (M M ∗ − I)f = 0, then from the equality M ∗ (M M ∗ − I)f = (M ∗ M − I)M ∗ f follows that M ∗ f = 0. But ||M ∗ f ||21 = ||f ||21 . Hence, f = 0. ˙ −i ∈ ρ(T ). Theorem 4.1.12. Let A˙ be a regular O-operator and let T ∈ Ω(A), Then Dom(T ) and Dom(T ∗ ) are dense in H+ . Moreover, if T is regular, then: 1. The equalities Dom(T ) = Dom(T ∗ ) = H+ , + −1 ˙ + ˙∗ ˙ ˙∗ T = AP S)PN , ˙ + (A + (AA + I) Dom(A) + ∗ ∗ ∗ −1 ∗ ˙ ˙ ˙ ˙ T = AP + (A + (AA + I) S )P + , ˙ Dom(A)

(4.13)

N

are valid, where S ∈ [N, N] and (−i) is a regular point of S; ˙ 2. Re T = (T + T ∗ )/2 is a regular self-adjoint extension of A; 3. the operator Im T = (T − T ∗ )/2i is (·, ·) bounded and (·)-essentially selfadjoint in H. Moreover, formulas (4.13) establish a one-to-one correspondence between all regu˙ −i ∈ ρ(T ) and all operators S ∈ [N, N] lar quasi-self-adjoint extensions T ∈ Ω(A), such that −i ∈ ρ(S).

80

Chapter 4. Quasi-self-adjoint Extensions

Proof. We have Ni = N−i = {0}. Suppose (−i) ∈ ρ(T ). Then the operator M corresponding to T by Theorem 4.1.9 belongs to [N, N] and possesses properties (4.6). It follows that (+)

Ran(M + I)

= Ran(M ∗ + I)

(+)

(+)

= N.

(+)

From (4.7) and (4.8) we get Dom(T ) = Dom(T ∗ ) = H+ . Suppose that T is regular. Then the linear manifolds (M + I)N and (M ∗ + I)N are (1)-closed. Equalities (4.6) now imply that (M + I)N = (M ∗ + I)N = N. It follows that Dom(T ) = Dom(T ∗ ) = H+ . Deﬁne the operator S ∈ [N, N] via S = i(I − M )(I + M )−1 . Then (−i) ∈ ρ(S), S ∗ = −i(I − M ∗ )(I + M ∗ )−1 , and formulas (4.7) and (4.8) become (4.13). Hence, Dom(Re T ) = H+ ,

+ −1 ˙ + ˙∗ ˙ ˙∗ Re T = AP Re S)PN , ˙ + (A + (AA + I) Dom(A)

where Re S = (S + S ∗ )/2 ∈ [N, N]. By Theorem 2.5.8 the operator Re T is a ˙ For Im T we have regular self-adjoint extension of A. + Dom(Im T ) = H+ , Im T = ((A˙ A˙ ∗ + I)−1 Im S)PN ,

˙ Using where Im S = (S − S ∗ )/2i. Consequently, Im T g = 0 for all g ∈ Dom(A). ˙ ˙ decompositions H+ = Dom(A) ⊕ N, H = H0 +N (see Proposition 2.4.2), and symmetry of the operator Im T we obtain that Im T is essentially self-adjoint and (·, ·)-bounded. If S ∈ [N, N] and −i ∈ ρ(S), then M := (iI − S)(S + iI)−1 ∈ [N, N], Ran(I + M ) = N,

ker(I + M ) = {0},

S(I + M ) = i(I − M ).

From Theorem 4.1.9 we get that T given by (4.13) is a regular quasi-self-adjoint ˙ and −i ∈ ρ(T ). extension of A˙ of the class Ω(A) ˙ is an arbitrary It should be noted that if A˙ is an O-operator and T ∈ Ω(A) ∗ regular quasi-self-adjoint extension, then Dom(T ) = Dom(T ). Indeed, let λ = a + ib ∈ ρ(T ), where a, b ∈ R, and b = 0. Then the operator Dom(T˜) = Dom(T ),

T˜ := −b−1 (T − aI)

is a regular quasi-self-adjoint extension of the regular symmetric operator ˜˙ = Dom(A), ˙ Dom(A)

A˜˙ = −b−1 (A˙ − aI),

˙ the operator and −i ∈ T˜ . Clearly, Dom(T˜ ∗ ) = Dom(T ), T˜ ∗ = −b−1 (T ∗ − aI)A, ˜˙ ˜˙∗ A is an O-operator, and Dom(A ) = Dom(A˙ ∗ ) = H+ . It follows from Theorem 4.1.12 that Dom(T˜ ) = Dom(T˜ ∗ ) = H+ .

4.2. Quasi-self-adjoint bi-extension

81

Theorem 4.1.13. If a quasi-self-adjoint extension T of a regular O-operator A˙ is (+, −)-continuous, then T is (+, ·)-continuous (and, consequently, a regular ˙ extension of A). Proof. If T is (+, −)-continuous, then so is T ∗ . According to Theorem 4.1.12 both Dom(T ) and Dom(T ∗ ) are (+)-dense in H+ . Hence both T and T ∗ can be extended to H+ by (+, −)-continuity to operators T and T ∗ in [H+ , H− ], respectively. It is clear that T and T ∗ are adjoint to each other operators in [H+ , H− ]. Thus, ˙ By Theorem 3.1.6 and Theorem 3.1.8 the operators T and T ∗ are T ∈ E(A). (+, ·)-continuous. This yields (+, ·)-continuity to operators T and T ∗ . Therefore, ˙ T is a regular extension of A. Suppose that A˙ is a regular symmetric operator but is not an O-operator. If ˙ < ∞ and, according to the deﬁciency indices of A are ﬁnite, then codim Dom(A) ˙ are regular. In this case the equalities Theorem 4.1.3, all T from the class Ω(A) Dom(T ) = H+ and Dom(T ∗ ) = H+ can not hold if both semi-deﬁciency numbers ˙ with −i ∈ ρ(T ) be determined by the operator are non-zero. Indeed, let T ∈ Ω(A) M ∈ [Ni ⊕ N, N−i ⊕ N] in accordance with Theorem 4.1.9. Since dim Ran(M + I) = dim Ni + dim N = dim Ni , dim Ran(M ∗ + I) = dim N−i + dim N = dim N−i , and dim M = dim Ni + dim N−i + dim N, we get Dom(T ) = H+ and Dom(T ∗ ) = H+ . If Ni = {0} (respect., N−i = {0}, then Dom(T ∗ ) = H+ and Dom(T ) = H+ (respect., Dom(T ) = H+ and Dom(T ∗ ) = H+ ).

4.2 Quasi-self-adjoint bi-extension In the following we will suppose that closed symmetric operator A˙ is regular. We begin with its deﬁnition. ˙ is called a quasi-self-adjoint bi-extension Deﬁnition 4.2.1. An operator A ∈ E(A) ˙ if (or q.s.-a. bi-extension) of a regular T ∈ Ω(A) A ⊃ T ⊃ A˙

and

˙ A∗ ⊃ T ∗ ⊃ A.

Deﬁnition 4.2.2. We say that a quasi-self-adjoint bi-extension A of an operator ˙ has the range property (R) if T ∈ Ω(A) Ran(A − λ0 I) ⊇ Ran(A − A∗ ),

¯ 0 I) ⊇ Ran(A − A∗ ) Ran(A∗ − λ

for some λ0 ∈ ρ(T ). ˙ belongs to the class Deﬁnition 4.2.3. We say that the operator T of the class Ω(A) ˙ R(A) if T admits q.s.-a. bi-extensions with the range property (R).

82

Chapter 4. Quasi-self-adjoint Extensions

In particular, if T is a regular self-adjoint extension of A˙ having a self-adjoint ˙ and A is a q.s.-a. bi-extension with the range propbi-extension A, then T ∈ R(A) erty (R) of T . In Chapter 3 it is established that a regular self-adjoint extension of A˙ admits self-adjoint bi-extensions and the description of all such bi-extensions is obtained (see Theorem 3.3.8). In this section we are going to ﬁnd necessary ˙ to be in the class R(A). ˙ We will and suﬃcient conditions for a regular T ∈ Ω(A) also give a description of all q.s.-a. bi-extensions A with the range property (R) of ˙ T ∈ R(A). Lemma 4.2.4. If A is a quasi-self-adjoint bi-extension of T , then T and T ∗ are quasi-kernels of A and A∗ , respectively. Proof. If f ∈ Dom(T ), then Af = T f ∈ H. Conversely, let Af ∈ H. Then for any g ∈ Dom(T ∗ ), (T ∗ g, f ) = (A∗ g, f ) = (g, Af ), which implies that f ∈ Dom(T ) and T f = Af .

+ + Let J = (PN − PN ) M be the operator in M (see (3.10)). Clearly J is (1)i

−i

+ + self-adjoint and J2 = (PN + PN ) M. Every bi-extension A of a regular closed i −i symmetric operator A˙ can be represented in the form (3.13)

i −1 + + ∗ ˙ ˙ A = APDom(A) (S − J) PM , ˙ + A +R 2 where S belongs to [M, M], and the adjoint operator A∗ is given by i −1 + + ∗ ∗ ∗ ˙ ˙ A = APDom(A) (S − J) PM , ˙ + A +R 2 with (1)-adjoint operator S ∗ . The next statement immediately follows from Theorem 4.1.9, Proposition 3.1.11, and the equality R−1 N = 2(A˙ A˙ ∗ + I) N (see (2.26) and (2.7)). ˙ −i ∈ ρ(T ), be a regular quasi-self-adjoint extension Theorem 4.2.5. Let T ∈ Ω(A), ˙ An operator of a regular closed symmetric operator A. i −1 + + ∗ ˙ ˙ A = APDom(A) (S − J) PM , ˙ + A +R 2 is a quasi-self-adjoint bi-extension of T if and only if S(M + I) =

i (I − M ), 2

S ∗ (M ∗ + I) =

i (M ∗ − I), 2

(4.14)

where M ∈ [Ni ⊕ N, N−i ⊕ N] determines T by Theorem 4.1.9, and M ∗ ∈ [N−i ⊕ N, Ni ⊕ N] is (1)-adjoint to M operator.

4.2. Quasi-self-adjoint bi-extension

83

Observe that i i ∗ ∗ (I − M )f, (I + M )h = (I + M )f, (M − I)h 2 2 1 1 i ∗ = ((f, g)1 − (M f, M g)1 ) 2

(4.15)

for all f ∈ Ni ⊕ N and all h ∈ N−i ⊕ N. In the sequel we will need the following statement. ˙ Then: Lemma 4.2.6. Let T ∈ Ω(A). 1) (1)-orthogonal complements to Dom(T ) and Dom(T ∗ ) in H+ are given by + H+ Dom(T ) = ϕ ∈ M : A˙ ∗ ϕ ∈ Dom(T ∗ ), (T ∗ A˙ ∗ + I)ϕ = −(A˙ A˙ ∗ + I)PN ϕ , + H+ Dom(T ∗ ) = ψ ∈ M : A˙ ∗ ψ ∈Dom(T ), (T A˙ ∗ + I)ψ = −(A˙ A˙ ∗ + I)PN ψ ; (4.16) −1 + ˙ + ˙∗ 2) if A = AP (S − 2i J) PM is a quasi-self-adjoint bi-exten˙ + A +R Dom(A) sion of T , then + ∗ + PN S Jϕ = iPN ϕ,

ϕ ∈ H+ Dom(T ),

+ + PN SJψ = iPN ψ,

ψ ∈ H+ Dom(T ∗ ).

Proof. 1) Clearly, h ∈ H+ is (+)-orthogonal to Dom(T ) if and only if (T f, A˙ ∗ h) = −(f, h) for all f ∈ Dom(T ) that are equivalent to A˙ ∗ h ∈ Dom(T ∗ ) and (T ∗ A˙ ∗ + I)h = 0. It follows that if ϕ ∈ H+ is (1)-orthogonal to Dom(T ), then + 0 = (f, ϕ)1 = (f, (I + PN )ϕ)+ ,

f ∈ Dom(T ),

+ which is equivalent to h = (I + PN )ϕ being (+)-orthogonal to Dom(T ). Using that ∗ ˙ ˙ A N ⊂ Dom(A), we get (4.16). 2) Let ϕ ∈ H+ be (+)-orthogonal to Dom(T ). Then by (4.16) + A˙ ∗ ϕ = iJϕ + A˙ ∗ PN ϕ ∈ Dom(T ∗ ).

Consequently, Jϕ ∈ Dom(T ∗ ) ∩ (Ni ⊕ N−i ). Since A∗ ⊃ T ∗ , we get i + + ∗ (PN J)Jϕ = 0. + PN )(S − i −i 2 Further

i + + T ∗ A˙ ∗ ϕ = A∗ (iJϕ + A˙ ∗ PN ϕ) = A˙ A˙ ∗ PN ϕ + iA˙ ∗ Jϕ + iR−1 S ∗ − J Jϕ 2 + + + + + ∗ ∗ = −(P + P + P )ϕ + (A˙ A˙ + I)(P ϕ + 2iP S Jϕ). Ni

N−i

N

N

N

84

Chapter 4. Quasi-self-adjoint Extensions

+ ∗ + From (4.16) we obtain PN S Jϕ = iPN ϕ.

In order to establish the existence of quasi-self-adjoint bi-extension for a given ˙ we need the following statement. regular T ∈ Ω(A) Proposition 4.2.7. Let K be a Hilbert space and let C1 and C2 two bounded linear operators deﬁned on the proper subspaces of K such that Dom(C1 ) = K1 and Dom(C2 ) = K2 . Suppose that (C1 φ, ψ)K = (φ, C2 ψ)K Let

for all

K = K⊥ 1 ⊕ K1 ,

φ ∈ K1 , ψ ∈ K2 .

K = K⊥ 2 ⊕ K2 .

(4.17) (4.18)

Then all operators Q ∈ [K, K] satisfying the condition Q ⊃ C1 , Q∗ ⊃ C2 take, according to decomposition (4.18), the operator-matrix form PK2 C1 (PK⊥ C2 )∗ 1 Q= , PK⊥ C1 H 2

(4.19)

(4.20)

⊥ ∗ where H is an arbitrary operator in [K⊥ 1 , K2 ]. The adjoint Q is of the form PK1 C2 (PK⊥ C1 )∗ ∗ 2 Q = , (4.21) PK⊥ C2 H∗ 1 ⊥ where H ∗ ∈ [K⊥ 2 , K1 ] is the adjoint to H.

Proof. In view (4.17) the operator given by (4.21) is the adjoint to the operator Q deﬁned by (4.20). Moreover, Q and Q∗ satisfy (4.19). Conversely, if Q ∈ [K, K] and Q ⊃ C1 , then Q has an operator-matrix representation of the form PK2 C1 X Q= , PK⊥ C1 H 2 ⊥ ⊥ where X ∈ [K⊥ 1 , K2 ] and H ∈ [K1 , K2 ]. Hence, (PK2 C1 )∗ (PK⊥ C1 )∗ 2 Q∗ = . X∗ H∗

From (4.17) it follows that the adjoint (PK2 C1 )∗ to PK2 C1 ∈ [K1 , K2 ] is equal to PK1 C2 ∈ [K2 , K1 ]. Hence, the condition Q∗ ⊃ C2 implies X ∗ = PK⊥ C2 . So, Q is of 1 the form (4.20). This completes the proof. ˙ be a regular quasi-self-adjoint extension of a regular Theorem 4.2.8. Let T ∈ Ω(A) ˙ Then there exists a bi-extension A of A˙ that is a closed symmetric operator A. quasi-self-adjoint bi-extension of T .

4.2. Quasi-self-adjoint bi-extension

85

Proof. It is suﬃcient to prove the statement for the case −i ∈ ρ(T ). Let M ∈ [Ni ⊕ N, N−i ⊕ N] determine T in accordance with Theorem 4.1.9. Let K = M be equipped with a (1)-inner product, K1 = (I + M )(Ni ⊕ N), K2 = (I + M ∗ )(N−i ⊕ N), i C1 (M + I)f = (I − M )f, f ∈ Ni ⊕ N, 2 i ∗ C2 (M + I)g = (M ∗ − I)g, g ∈ N−i ⊕ N. 2 Then it follows from (4.15) that C1 and C2 satisfy (4.17). Applying Proposition 4.2.7 we obtain the existence of an operator S ∈ [M, M] with the property S ⊃ C1 and S ∗ ⊃ C2 . Let S be such an operator and let i −1 + + ∗ ˙ ˙ A = APDom(A) (S − J) PM . ˙ + A +R 2 Since

˙ + ˙ ∗ + R−1 (S ∗ − i J) P + , A∗ = AP + A ˙ M Dom(A) 2

applying Theorem 4.2.5 we get A ⊃ T and A∗ ⊃ T ∗ .

If A˙ is a densely deﬁned closed symmetric operator, then all operators from ˙ with (−i) ∈ ρ(T ) take the form the class Ω(A) ˙ ⊕ (M + I)Ni , T = A˙ ∗ Dom(T ), Dom(T ) = Dom(A) ˙ ⊕ (M ∗ + I)N−i , T ∗ = A˙ ∗ Dom(T ∗ ), Dom(T ∗ ) = Dom(A) ˙ where M ∈ [Ni , N−i ]. In this case all quasi-self-adjoint bi-extensions of T ∈ Ω(A) are given by i + A = A˙ ∗ + R−1 (S − J)PM , 2

i + A∗ = A˙ ∗ + R−1 (S ∗ − J)PM , 2

(4.22)

where S ∈ [M, M] satisﬁes conditions S(M + I) =

i (I − M ), 2

S ∗ (M ∗ + I) =

i (M ∗ − I). 2

In the densely deﬁned case it is not diﬃcult to derive the general block-matrix form of all such S : Ni ⊕ N−i → Ni ⊕ N−i , i I − HM H 2 S= , −(iI − M H)M 2i I − M H where H ∈ [N−i , Ni ] is an arbitrary operator. We are also going to introduce block-operator matrices SA and SA∗ which will be needed later and are of the

86 form

Chapter 4. Quasi-self-adjoint Extensions i −HM H SA = S − J = , M (HM − iI) iI − M H 2 i −M ∗ H ∗ − iI (M ∗ H ∗ − iI)M ∗ ∗ SA∗ = S − J = . H∗ iI − H ∗ M ∗ 2

(4.23)

As usual we stick to the notations Re S = (S + S ∗ )/2 and Im S = (S − S ∗ )/2i for S ∈ [M, M]. Theorem 4.2.9. Let A˙ be a regular closed symmetric operator with non-zero semi˙ Then T ∈ deﬁciency indices. Let T be a regular extension from the class Ω(A). ˙ R(A) if and only if there exists a (·)-isometric operator U mapping Ni onto N−i such that the equalities ˙ + U )Ni = Dom(T ∗ )+(I ˙ + U )Ni = H+ Dom(T )+(I

(4.24)

hold. ˙ Let us set Proof. Necessity. Let T ∈ R(A). ˙ MT = Dom(T ) Dom(A), M⊥ T = M MT , ˙ MT ∗ = Dom(T ∗ ) Dom(A), M⊥ T ∗ = M MT ∗ , where the orthogonal complements to a(1)-inner product in are taken with respect −1 + + i ∗ ˙ ˙ H+ . Suppose A = APDom(A) (S − 2 J) PM is a q.s.-a. bi-extension ˙ + A +R of T with the range property (R). Then for an arbitrary λ ∈ ρ(T ) we have + + ˙∗ A − λI = (T − λI)PDom(T ) + (A − λI)PM⊥ T

+R and

−1

+ + PN SPM ⊥ T

+R

−1

+ (PN i

+

+ PN )(S −i

i + − J)PM ⊥, T 2

+ + + + + Im A = R−1 PN (Im S)PM + R−1 (PN + PN )(Im S)PM . i

−i

By Deﬁnition 4.2.1 the equation (A − λ0 I)x = (Im A)f, λ0 ∈ ρ(T ), has a solution x ∈ H+ for an arbitrary f ∈ M. Since + + R−1 PN (Im S)PM f,

+ + R−1 PN SPM ⊥ x ∈ L ⊂ H, T

it follows from (4.25) that there exists a vector g ∈ M⊥ T such that i + + + + + (PN J)g = (PN + PN )(S − + PN )(Im S)PM f. i −i i −i 2

(4.25)

4.2. Quasi-self-adjoint bi-extension

87

Consequently + + x = g + (T − λ0 I)−1 R−1 PN (Im S)f − R−1 PN Sg − (A˙ ∗ − λ0 I)g . Thus, condition Ran(A − λ0 I) ⊃ Ran(Im A), yields the inclusion i + + + + Ran (PN + P )(S − J) ⊃ Ran (PN . + PN )(Im S) N i −i i −i 2 Therefore, i + + + + ker S ∗ (PN + P ) + J) ⊆ ker (Im S)(PN . + PN N i −i i −i 2 ¯ 0 I) ⊃ Ran(Im A) implies Similarly, the condition Ran(A∗ − λ i + + + + ker S(PN + PN ) + J ⊆ ker (Im S)(PN . + PN ) i −i i −i 2 Now from (4.26) and (4.27) we obtain i i + + + + ker S(PN J = ker S ∗ (PN J . + PN ) + + PN ) + i −i i −i 2 2

(4.26)

(4.27)

(4.28)

Therefore, + + + + i i ker S(PN = ker S ∗ (PN + PN ) + 2 J + PN ) + 2 J −i i −i i + + i ⊆ ker (Re S)(PN J . + PN ) + 2 i

−i

Since Re S is a (1)-self-adjoint operator in M we get i ker(Re S + J) = N ⊕ (U + I)Dom(U ), 2

Dom(U ) ⊆ Ni , Ran(U ) ⊆ N−i ,

and U is a (·) and (1)-isometry. Let i + + F := ker S(PN + PN ) + J N. i −i 2 ˆ , where N ˆ is some subspace in Dom(U ). Then (4.28) yields F = (U + I)N i i Let f ∈ MT ∩ F . Then, since f ∈ MT , we have i + + ((PN J)f = 0. + PN )S − i −i 2 On the other hand f ∈ F and hence, i + + ((PN J)f = −iJf. + PN )S − i −i 2

(4.29)

88

Chapter 4. Quasi-self-adjoint Extensions

˙ . Then f = f1 + f2 , where f1 ∈ MT , Therefore, MT ∩ F = {0}. Let f = MT +F f2 ∈ F . Hence, i + + + + iJ (PN + PN )S − J f = (PN + PN )f2 = f2 . i −i i −i 2 ˙ is (1)-closed. Since It follows that the linear manifold MT +F i + + ∗ ker (PN − J) = MT ∗ , + PN )(S i −i 2 taking closure in (+)-metric, we obtain i + + Ran S(PN + PN ) + J = M⊥ T∗. i −i 2 + + Let ψ ∈ M⊥ T ∗ . Then by Lemma 4.2.6 Jψ ∈ MT ∩ (Ni ⊕ N−i ) and PN SJψ = iPN ψ. Consequently, + + + i S(PN J Jψ = PN SJψ + iJ(Jψ) + (S − 2i J)Jψ = iψ. + PN ) + 2 i

−i

Hence

i i + + + + ⊥ Ran S(PN + PN ) + J = S(PN + PN ) + J JM⊥ T ∗ = MT ∗ . (4.30) i −i i −i 2 2 ˙ Let h ∈ M MT +F . Then (4.30) implies that there exists a vector ψ ∈ M⊥ T∗ such that i i + + + + S(PN + PN ) + J Jψ = S(PN + PN ) + J h. i −i i −i 2 2 It follows from (4.29) that h = Jψ + f + g, where f ∈ F, g ∈ N. Because Jψ ∈ MT and h is (1)-orthogonal to F and MT , we have h = g ∈ N. Since R−1 h ∈ L, we get that R−1 h is (·)-orthogonal to Dom(T ). Therefore, h = 0. Hence, ˙ = M. MT +F

(4.31)

˙ Similarly, one can prove that MT ∗ +F = M. Further, from (4.31) we get that + + i Ran (PN + PN )S − 2 J = JF . Since i −i i i + + + + ker S ∗ (PN + P ) + J ⊕ Ran (P + P )S − J = M, N−i Ni N−i i 2 2 ˆ , then from (4.29) we get JF ⊕ F = Ni ⊕ N−i . Now the equality F = (U + I)N i ˆ is a subspace in Dom(U ), yields N ˆ = N and Ran(U ) = N . Thus, we where N i i i −i have proved the equalities + + + + i i ∗ ker S(PN J = ker S (P J + PN ) + + PN ) + N 2 2 i −i −i i + + i = ker Re S(PN J = N ⊕ (I + U )Ni , + PN ) + 2 i

−i

4.2. Quasi-self-adjoint bi-extension and

89

˙ + U )Ni = MT ∗ +(I ˙ + U )Ni = M, MT +(I

(4.32)

where U is a (1)-isometric operator mappingNi onto N−i . Clearly, equalities (4.32) are equivalent to (4.24). Suﬃciency. Suppose that (4.32) holds for some (·)-isometry U : Ni → N−i , Ran(U ) = N−i . For the sake of simplicity let us assume that −i ∈ ρ(T ). Then (4.7) and (4.8) hold, i.e., MT = (I + M )(Ni ⊕ N), and MT ∗ = (I + M ∗ )(N−i ⊕ N). Let an operator S ∈ [M, M] be deﬁned as S((I + M )f + (I + U )ϕ) =

i ((I − M )f − (I − U )ϕ), f ∈ Ni ⊕ N, ϕ ∈ Ni . (4.33) 2

By direct calculations one veriﬁes that the (1)-adjoint S ∗ ∈ [M, M] is given by S ∗ ((I + M ∗ )h + (I + U )ϕ) =

i ((M ∗ − I)h − (I − U )ϕ), h ∈ N−i ⊕ N, ϕ ∈ Ni . 2

Then the quasi-kernels of i −1 + + ∗ ˙ ˙ A = APDom(A) (S − J) PM ˙ + A +R 2 i −1 + + ∗ ∗ ˙ ˙ A = APDom(A) (S − J) PM ˙ + A +R 2

and

∗

coincide with T and T ∗ , respectively. Observe that by construction, we have + + + + i i ∗ ker S(PN J = ker S (P + P ) + J + PN ) + Ni N−i 2 2 i −i (4.34) + + = N ⊕ (I + U )Ni ⊆ ker((Im S)(PN + PN )), i

−i

and + + + + i i ∗ Ran (PN J) = Ran (PN J) + PN )(S − + PN )(S − 2 2 i

−i

i

−i

(4.35) = (I −

U )Ni

⊇

+ Ran((PN i

+

+ PN )Im S). −i

¯ ⊇ Ran(A − A∗ ) for an Therefore, Ran(A − λI) ⊇ Ran(A − A∗ ) and Ran(A∗ − λI) arbitrary λ ∈ ρ(T ). Thus, A is a q.s.-a. bi-extension of T with the range property ˙ This completes the proof. (R). Consequently, T ∈ R(A). Corollary 4.2.10. Suppose that an m-dissipative extension T of A˙ belongs to the ˙ Then each of its q.s.-a. bi-extensions A with the range property (R) is class R(A). dissipative (Im (Ah, h) ≥ 0 for all h ∈ H+ ).

90

Chapter 4. Quasi-self-adjoint Extensions

˙ + ˙ ∗ + R−1 (S − i J) P + be a q.s.-a. bi-extension of Proof. Let A = AP + A ˙ M 2 Dom(A) T . As proved in Theorem 4.2.9, the operator S is of the form (4.33). Then for f ∈ Ni ⊕ N and ϕ ∈ Ni , using that U : Ni → N−i is a (1)-isometry, we get Im (S((I + M )f + (I + U )ϕ), (I + M )f + (I + U )ϕ)1 =

1 f 21 − M f 21 . 2

Since T is m-dissipative, the operator M is a contraction. So, Im (Ah, h) ≥ 0 for all h ∈ H+ .

4.3 The (∗)-extensions and uniqueness theorems ˙ is called a Deﬁnition 4.3.1. A q.s.-a. bi-extension A of an operator T ∈ Ω(A) ˙ (∗)-extension of T if Re A is a t-self-adjoint bi-extension of A. ˆ then A In particular, if A is a t-self-adjoint bi-extension with quasi-kernel A, ˆ is a (∗)-extension of A. ˙ and U is a (·)Theorem 4.3.2. If T is a regular extension from the class R(A) isometric operator, mapping Ni onto N−i , such that the equalities (4.24) holds, then the operator S given by (4.33) deﬁnes a (∗)-extension A of T . Moreover, 1. the operator ˙ + ˙ ∗ + R−1 (Re S − i J) P + Re A = AP + A ˙ M Dom(A) 2 is a t-self-adjoint bi-extension of A˙ and the quasi-kernel Aˆ of Re A is given by ˆ = Dom(A) ˙ ⊕ (I + V)(N ⊕ N), Dom(A) i where V = −(Re S − 2i I)(Re S + 2i I)−1 (Ni ⊕ N), the equality i i U = −(Re S + I)−1 (Re S − I) Ni , 2 2 holds for the operator U , and ˆ +(I ˙ + U )Ni = H+ ; Dom(A)

(4.36)

¯ do not depend on λ ∈ ρ(T ) 2. the linear manifolds Ran(A−λI) and Ran(A∗ −λI) and ¯ = H+R ˙ −1 (U − I)Ni , Ran(A − λI) = Ran(A∗ − λI) (4.37) ¯ ˙ −1 (U − I)Ni , (A − λI)Nλ = (A∗ − λI)N ¯ = L+R λ for all λ ∈ ρ(T );

(4.38)

4.3. The (∗)-extensions and uniqueness theorems

91

ˆ and 3. the linear manifolds Ran(Re A − μI) do not depend on μ ∈ ρ(A) ˙ −1 (U − I)Ni , Ran(Re A − μI) = H+R

(4.39)

˙ −1 (U − I)Ni (Re A − μI)Nμ = L+R

(4.40)

ˆ for all μ ∈ ρ(A); 4. the inclusion

˙ −1 (U − I)Ni Ran(Im A) ⊆ L+R

is valid and if A˙ is a maximal common symmetric part of T and T ∗ , then the equality (−) ˙ −1 (U − I)Ni Ran(Im A) = L+R (4.41) holds. Proof. The statement concerning the operator S is already established in the course of the proof of Theorem 4.2.9. Let us prove statement 1. It follows from (4.34) that ˜ Re S = ker Re S(P + + P + ) + i J ⊇ (I + U )N ⊕ N. G i Ni N−i 2 By Proposition 3.3.4 the operator ˙ + ˙ ⊕G ˜ Re S ) AP + A˙ ∗ P + (Dom(A) ˙ Dom(A)

M

is a closed symmetric extension of B˙ deﬁned by (2.19). But ˙ + ˙ ∗ P + (Dom(A) ˙ ⊕ (I + U )N ⊕ N) AP + A i ˙ M Dom(A) ˙ Therefore, G ˜ Re S = (I + U )N ⊕ N. Applying is a self-adjoint extension of B. i Theorem 3.3.6, we get that ˙ + ˙ ∗ + R−1 (Re S − i J) P + Re A = AP + A ˙ M Dom(A) 2 ˙ is a t-self-adjoint bi-extension of A. 2. Clearly, H = Ran(T − λI) ⊂ Ran(A − λI),

¯ ⊂ Ran(A∗ − λI) ¯ H = Ran(T ∗ − λI)

for an arbitrary λ ∈ ρ(T ). From (4.25) and (4.35) we get ˙ −1 (U − I)Ni , Ran(A − λI) ⊆ H+R

¯ ⊆ H+R ˙ −1 (U − I)Ni . Ran(A∗ − λI)

92

Chapter 4. Quasi-self-adjoint Extensions

If f ∈ Nλ , then A˙ ∗ f = P f and ˙ + ˙∗ + ˙ + (AP ˙ + A PM )f − λf = PL (APDom(A) ˙ − λI)f. Dom(A) Furthermore, by virtue of (4.35) we obtain −1 + + ˙ + (A − λI)f = PL (AP PN SPM f ˙ − λI)f + R Dom(A)

i + + + ˙ −1 (U − I)Ni . + R−1 (PN J)PM f ∈ L+R + PN )(S − i −i 2 ¯ ˙ −1 (U − I)Ni . On the other hand the vectors Similarly, (A∗ − λI)N ¯ ⊆ L+R λ ϕ = (I + U )φ + (T − λI)−1 g − (A˙ ∗ − λI)(I + U )φ and

¯ −1 g − (A˙ ∗ − λI)(I ¯ ψ = (I + U )φ + (T ∗ − λI) + U )φ

are the solutions of the equations (A − λI)ϕ = R−1 (U − I)φ + g

¯ and (A∗ − λI)ϕ = R−1 (U − I)φ + g,

where φ ∈ Ni , g ∈ L, respectively. In addition, one can verify that ϕ ∈ Nλ and ψ ∈ Nλ¯ . Hence, equalities (4.37) and (4.38) are valid. Statement 3. can be proved similarly with the help of the equality i + + Ran (PN + P )Re S − J = (I − U )Ni . N−i i 2 Observe that the vector

χ = (I + U )φ + (Aˆ − μI)−1 g − (A˙ ∗ − μI)(I + U )φ

belongs to Nμ and is the unique solution of the equation (Re A − μI)χ = R−1 (U − I)φ + g, where φ ∈ Ni and g ∈ L. (+)

4. Since ker(Im S) ⊇ (I + U )Ni , we get Ran(Im S) ⊆ N ⊕ (I − U )Ni . −1 ˙ Consequently, Ran(Im A) ⊆ L+R (U − I)Ni . Suppose that A˙ is a maximal common symmetric part of T and T ∗ . Note that ker(Im A) ∩ MT = {0}. Indeed, if h ∈ MT and (Im A)h = 0, then, since Ah = T h, we get A∗ h = T h ∈ H. It follows that h ∈ Dom(T ∗ ) and A∗ h = T ∗ h. Since A˙ is a maximal common symmetric ˙ Consequently, h = 0. Using decomposition part of T and T ∗ , we get h ∈ Dom(A). (4.32) and relation (4.47) we get now that + + + ker(Im A) = ker(R−1 (Im S)PM ) = ker((Im S)(PN + PN )) = (I + U )Ni . i

−i

Therefore, taking closure in the (−)-metric, we obtain ˙ −1 (U − I)Ni . Ran(Im A) = R−1 Ran(Im S) = R−1 (N ⊕ (U − I)Ni ) = L+R

4.3. The (∗)-extensions and uniqueness theorems

93

Recall that each bi-extension A of a regular operator A˙ can be written in the form (see (3.12)) i −1 + ∗ ˙ A=B +R S − J PM , 2 where B˙ is deﬁned by (2.19). Then i −1 + ∗ ˙ A−B =R S − J PM . 2 Deﬁne the linear manifold ˙ . F0 = ϕ ∈ H− : (ϕ, f ) = 0 for all f ∈ Dom(B)

(4.42)

˙ = Dom(A) ˙ ⊕ N, F0 is a subspace in H− , F0 ∩ H = {0}, and Since Dom(B) F0 = R−1 (Ni ⊕ N−i ). For a (∗)-extension A of T determined by S of the form (4.33) we have F0 ∩ Ran(A − B˙ ∗ ) = R−1 (U − I)Ni . In what follows we will use the set LA , LA = R−1 (U − I)Ni .

(4.43)

LA = F0 ∩ Ran(A − B˙ ∗ ).

(4.44)

Thus, we have In addition, equalities (4.37)–(4.40) can be rewritten as ¯ = H+L ˙ A, Ran(A − λI) = Ran(A∗ − λI) ¯ ˙ (A − λI)Nλ = (A∗ − λI)N = L +L ¯ A , λ ∈ ρ(T ), λ ˆ ˙ ˙ A , μ ∈ ρ(A). Ran(Re A − μI) = H+LA , (Re A − μI)Nμ = L+L

(4.45)

˙ and let A be a Theorem 4.3.3. Let T be a regular extension from the class Ω(A) ˙ (∗)-extension of T . Suppose A is a maximal common symmetric part of T and T ∗ . Then A is a q.s.-a. bi-extension of T with the range property (R) and, therefore, ˙ T ∈ R(A). Proof. For the sake of simplicity we assume that −i ∈ ρ(T ). Let M ∈ [Ni ⊕ N, N−i ⊕N] determine T via Theorem 4.1.9. Then equalities (4.14) hold. It follows that + + i (PN J)(M + I)f = 0, f ∈ Ni ⊕ N, + PN )(S − 2 i

−i

i

−i

+ + i ∗ (PN J)(M ∗ + I)h = 0, h ∈ N−i ⊕ N. + PN )(S − 2

94

Chapter 4. Quasi-self-adjoint Extensions

Since S = Re S + iIm S, we get + + + + i (PN + P )(Re S − N 2 J)(M + I)f = −i(PN + PN )(Im S)(M + I)f, i

−i

i

−i

i

−i

+ + + + i ∗ ∗ (PN + I)h = i(PN + I)h, + PN )(Re S − 2 J)(M + PN )(Im S)(M i

−i

and, therefore, + + i ∗ (PN + PN )(Re S − 2 J)((M + I)f + (M + I)h) i

−i

+ + ∗ = −i(PN + I)h) + PN )(Im S)((M + I)f − (M i

−i

for all f ∈ Ni ⊕ N and all h ∈ N−i ⊕ N. By Theorem 4.1.11 the linear manifold Ran(M + I) + Ran(M ∗ + I) is dense in M = Ni ⊕ N−i ⊕ N. Now, taking closure in the (+)-metric, we get the relation i + + + + Ran((PN J)) = Ran((PN + PN )(Re S − + PN )Im S). i −i i −i 2 Hence, the equality i + + + + ker (Re S)(PN + PN ) + J = ker((Im S)(PN (4.46) + PN )) i −i i −i 2 ˙ + ˙ ∗ + R−1 (S − i J) P + be a quasi-self-adjoint biholds. Let A = AP + A ˙ M 2 Dom(A) extension of T and i −1 + + ∗ ˙ ˙ Re A = APDom(A) (Re S − J) PM ˙ + A +R 2 ˙ We set be a t-self-adjoint bi-extension of A. i + + ˜ GRe S := ker (Re S)(PN + PN ) + J . i −i 2 According to Theorem 3.3.6 the operator ˙ + ˙ ∗ P + (Dom(A) ˙ ⊕G ˜ Re S ) A˜ =: AP + A ˙ M Dom(A) is a self-adjoint extension of the operator ˙ = Dom(A) ˙ ⊕ N, Dom(B)

˙ + ˙∗ + B˙ = AP ˙ + A PN . Dom(A)

It follows that dim Ni = dim N−i and there exists a (·)-isometric operator U mapping Ni onto N−i such that ˜ = Dom(B) ˙ ⊕ (I + U )N , Dom(A) i

4.3. The (∗)-extensions and uniqueness theorems

95

˜ Re S = (I + U )N ⊕ N, and i.e., G i i (Re S)(I + U )ϕ = − (I − U )ϕ, 2

ϕ ∈ Ni .

Relation (4.46) then yields (Im S)(I + U )ϕ = 0, ϕ ∈ Ni . Hence i S(I + U )ϕ = S ∗ (I + U )ϕ = − (I − U )ϕ, 2

(4.47)

for all ϕ ∈ Ni . Let the operator M ∈ [Ni ⊕ N, N−i ⊕ N] determine T in accordance with Theorem 4.1.9. Because T and T ∗ are the quasi-kernels of A and A∗ , respectively, we get S(I + M )f =

i (I − M )f, 2

S ∗ (I + M ∗ )h =

i (M ∗ − I)h 2

for all f ∈ Ni ⊕ N and all h ∈ N−i ⊕ N. Let us show that ˙ + U )Ni = M, (I + M )(Ni ⊕ N)+(I ˙ + U )Ni = M. (I + M ∗ )(N−i ⊕ N)+(I

(4.48)

Suppose ψ ∈ (I + M )(Ni ⊕ N) ∩ (I + U )Ni . Then ψ = (I + U )ϕ = (I + M )f, where ϕ ∈ Ni , f ∈ Ni ⊕ N. Then from (4.47) we obtain i (S − J)(I + U )ϕ = i(U − I)ϕ. 2 On the other hand i + (S − J)(I + M )f = PN (I − M )f. 2 So, (S − 2i J)ψ ∈ Ni ⊕ N−i and (S − 2i J)ψ ∈ N. It follows that (I − U )ψ = 0 if and only if ψ = 0. Hence (I + M )(Ni ⊕ N) ∩ (I + U )Ni = {0}. Similarly it can be shown that (I + M ∗ )(N−i ⊕ N)∩(I +U )Ni = {0}. Now suppose ˙ + U )Ni . Then from that a vector f ∈ M is (1)-orthogonal to (I + M )(Ni ⊕ N)+(I (4.12) we have the equalities + ∗ f = h − PN M h, −i

+ PN (M ∗ + I)h = 0,

and f = (U − I)φ + g,

h ∈ N−i ⊕ N,

φ ∈ Ni , g ∈ N.

96

Chapter 4. Quasi-self-adjoint Extensions

+ + ∗ It follows that φ = −PN M h, U φ = PN h. Consequently, −i

−i

(I + U )φ = (M ∗ + I)h. Because of (I + M ∗ )(N−i ⊕ N) ∩ (I + U )Ni = {0}, we get φ = h = 0. Therefore, ˙ + U )Ni is (1)-dense in M. Similarly, the the linear manifold (I + M )(Ni ⊕ N)+(I ∗ ˙ linear manifold (I + M )(N−i ⊕ N)+(I + U )Ni is (1)-dense in M. Observe that it follows from (4.47) that i S + I ((M + I)f + (I + U )ϕ) = if + iU ϕ (4.49) 2 for all f ∈ Ni ⊕ N and all ϕ ∈ Ni . Consequently, since S is (1)-bounded, we get ( (2 ||f ||21 + ||ϕ||21 = ( S + 2i I ((M + I)f + (I + U )ϕ)(1 ≤ c(M + I)f + (I + U )ϕ)21 for some c > 0 and all f ∈ Ni ⊕ N, ϕ ∈ Ni . This yields that the linear manifold ˙ + U )Ni is (1)-closed. Because it is dense in M, the equality (I + M )(Ni ⊕ N)+(I ˙ + U )Ni = M (I + M )(Ni ⊕ N)+(I holds. Similar arguments show that ˙ + U )Ni = M. (I + M ∗ )(N−i ⊕ N)+(I ˙ Moreover, These equalities are equivalent to (4.24), i.e., T ∈ R(A). i i + + + + ∗ Ran (PN + P )(S − J) = Ran (P + P )(S − J) N−i Ni N−i i 2 2 + + = (I − U )Ni = Ran((PN + PN )Im S), i

−i

where the closure is taken in the (+)-metric. Therefore, Ran(A − λI) ⊃ Ran(Im A) ¯ ⊃ Ran(Im A) for each λ ∈ ρ(T ). This means that A is a (∗)and Ran(A∗ − λI) extension of T . Remark 4.3.4. Let us make a few remarks. 1) As we have proved above (see (4.49)), if ˙ + ˙ ∗ + R−1 (S − i J) P + , A = AP + A ˙ M Dom(A) 2 ˙ and −i ∈ ρ(T ), then −i/2 ∈ ρ(S). Thus i/2 ∈ ρ(S ∗ ) is a (∗)-extension of T ∈ R(A) and if the equality ˙ + U )Ni = M (I + M )(Ni ⊕ N)+(I

4.3. The (∗)-extensions and uniqueness theorems

97

is valid, then it follows from the equalities M = Ni ⊕ N−i ⊕ N and i (S ∗ − I)((I + M ∗ )g + (I + U )ϕ) = −ig − iϕ 2 for all g ∈ N−i ⊕ N and all ϕ ∈ Ni , that (I + M ∗ )(N−i ⊕ N) ∩ (I + U )Ni = {0}. Moreover,

˙ + U )Ni = M. (I + M ∗ )(N−i ⊕ N)+(I

Thus, the conditions ˙ + U )Ni = H+ Dom(T )+(I

˙ + U )Ni = H+ and Dom(T ∗ )+(I

are equivalent. 2) One can easily derive that equalities (4.48) are equivalent to the following conditions: + + + (a) the operators U PN M : ker(PN (M + I)) → N−i − PN −i i + + + and PN (M + I) : ker(U PN − PN M ) → N are bijections, i

−i

+ + + ∗ (b) the operators U PN − PN : ker(PN (M ∗ + I)) → Ni M −i i + + + ∗ ∗ and PN (M + I) : ker(U PN M − PN−i ) → N are bijections. i

When A˙ is a closed and densely deﬁned, then the conditions: (i) the equality

(ii) the equality

˙ + U )Ni = Ni ⊕ N−i , (I + M )Ni +(I

(4.50)

˙ + U )Ni = Ni ⊕ N−i , (I + M ∗ )N−i +(I

(4.51)

(iii) the operator M − U : Ni → N−i is a bijection, (iv) the operator I − U ∗ M : Ni → Ni is a bijection, (v) the operator I − M ∗ U : Ni → Ni is a bijection, are equivalent. 3) By von Neumann’s formula (1.15), a (·)-unitary mapping U from Ni onto N−i deﬁnes a self-adjoint extension AU of the symmetric operator P A˙ in the space H0 via ˙ +(I ˙ + U )Ni . Dom(AU ) = Dom(A) If U satisﬁes (4.24), then ˙ Dom(AU ) ∩ Dom(T ∗ ) = Dom(A), ˙ Dom(AU ) ∩ Dom(T ) = Dom(A),

98

Chapter 4. Quasi-self-adjoint Extensions

and Dom(AU ) + Dom(T ) = Dom(AU ) + Dom(T ∗ ) = Dom(A˙ ∗ ) = H+ .

(4.52)

4) Denote by PT U and PT ∗ U the projectors onto Dom(T ) and Dom(T ∗ ), corresponding to the direct decompositions ˙ + U )Ni , H+ = Dom(T )+(I and

(4.53)

˙ + U )Ni , H+ = Dom(T ∗ )+(I

(4.54) ∗

respectively. Then the (∗)-extension A of T and its adjoint A , corresponding to the choice of the operator U , take the form A = T PT U + (A˙ ∗ − R−1 A˙ ∗ )(I − PT U ), A∗ = T ∗ PT ∗ U + (A˙ ∗ − R−1 A˙ ∗ )(I − PT ∗ U ). In addition, if A˙ is a maximal common symmetric part of T and T ∗ , then ker(Im A) = Dom(AU ). Theorem 4.3.5. Let the deﬁciency indices of A˙ be equal to r < ∞. Suppose that ˙ and that A˙ is a maximal common symmetric part of T and T ∗ . Then T ∈ Ω(A) the quasi-self-adjoint bi-extension A of T has the range property (R) if and only if the conditions Ran(Im A) ⊃ L

and

dim(Ran(Im A)) = r

(4.55)

hold. Proof. If A is a (∗)-extension of T , then the parameter S ∈ [M, M] is of the form (4.33), where U ∈ [Ni , N−i ] is a (1)-isometry. Then dim(U − I)Ni = dim Ni = dim Ni − dim N. Now from (4.41) we get dim(Ran(Im A)) = r and Ran(Im A) ⊃ L. −1 + + ˙ ˙∗ Conversely, let A = APDom(A) (S − 2i J) PM be a quasi-self˙ + A +R adjoint extension of T and let conditions (4.55) hold true. Then ker(Im S) ⊆ Ni ⊕ N−i . It follows from (4.46) that i + + ker Re S(PN + PN ) + J = ker(Im S) ⊕ N. i −i 2 Since

i + + ker Re S(PN + P ) + J = (I + W )Fi ⊕ N, N−i i 2

4.3. The (∗)-extensions and uniqueness theorems

99

where Fi ⊆ Ni and W : Fi → N−i is a (1)-isometry, we get dim ker(Im S) = dim Fi ≤ dim Ni = r − dim N. On the other hand conditions (4.55) yield dim ker(Im S) = r − dim N. Hence, Fi = Ni . From Theorem 3.3.6 we get that Re A is a t-self-adjoint bi˙ Consequently, A has the range property (R). extension of A. Under the conditions of Theorem 4.3.5 the formula (4.55) implies that A is a (∗)-extension. Corollary 4.3.6. Let A˙ be a closed densely deﬁned operator with equal ﬁnite de˙ and that A˙ is a maximal common ﬁciency indices (r, r). Suppose that T ∈ Ω(A) ∗ symmetric part of T and T . Then the quasi-self-adjoint bi-extension A of T has the range property (R) if and only if dim Ran(Im A) = r. Theorem 4.3.7. Let the deﬁciency indices of A˙ be equal to r < ∞. Suppose that ˙ and that A˙ is a maximal common symmetric part of T and T ∗ . Then the T ∈ Ω(A) quasi-self-adjoint bi-extension A of T is a (∗)-extension if and only if conditions (4.55) hold. Proof. If conditions (4.55) hold, then, as we have already mentioned, Theorem 4.3.5 implies that A is a (∗)-extension. Conversely, let A be a (∗)-extension. Then according to Theorem 4.3.3 it satisﬁes the range property (R). Thus we can apply Theorem 4.3.5 and obtain conditions (4.55). Corollary 4.3.8. Let A˙ be a closed densely deﬁned operator with ﬁnite deﬁciency ˙ and that A˙ is a maximal common symmetric indices (r, r). Suppose that T ∈ Ω(A) part of T and T ∗ . Then the quasi-self-adjoint bi-extension A of T is a (∗)-extension of T if and only if dim Ran(Im A) = r. The following theorems will be referred at as the uniqueness theorems. ˙ and let A and A be two (∗)-extensions of T . If Theorem 4.3.9. Let T ∈ R(A) Re A = Re A , then A = A . If A˙ is a maximal common symmetric part of T and T ∗ and Im A = Im A , then A = A . Proof. Let the operators A and A be given by ˙ + ˙ ∗ + R−1 (S − i J) P + A = AP + A ˙ M Dom(A) 2 and

˙ + ˙ ∗ + R−1 (S − i J) P + , A = AP + A ˙ M Dom(A) 2

100

Chapter 4. Quasi-self-adjoint Extensions

respectively, with S , S ∈ [M, M]. Then by Theorem 4.3.2 there are two (1)isometries U , U ∈ [Ni , N−i ], Ran(U ) = Ran(U ) = N−i such that

+ + + + i i ∗ ker(S (PN + P ) − N 2 J) = ker(S (PN + PN ) − 2 J) i

−i

i

−i

+ + i = ker((Re S )(PN + PN ) − 2 J) = N ⊕ (I + U )Ni i

−i

i

−i

+ + ⊆ ker((Im S )(PN + PN )),

and

+ + i ker(S (PN + PN ) − 2 J) = ker(S i

−i

∗

+ + i (PN + PN ) − 2 J) i

−i

+ + i = ker((Re S )(PN + PN ) − 2 J) = N ⊕ (I + U )Ni i

−i

i

−i

+ + ⊆ ker((Im S )(PN + PN ))

(see equalities (4.34)). If Re A = Re A , then Re S = Re S . Hence U = U . By Theorem 4.3.2 A = A . Suppose Im A = Im A , i.e., Im S = Im S . Since A˙ is a maximal common symmetric part of T and T ∗ , then by relation (4.46) i i + + + + ker (Re S )(PN + P ) + J = ker (Re S )(P + P ) + J . N−i Ni N−i i 2 2 Since A and A are (∗)-extensions, we get the equality U = U , which implies A = A . Theorem 4.3.10. If a closed symmetric operator A˙ has equal and ﬁnite deﬁciency ˙ and R(A) ˙ coincide. indices, then the classes Ω(A) Proof. Clearly, by Theorem 2.4.1 and the remark afterwards the operator A˙ is a regular symmetric operator. If A˙ is an O-operator, then the statement is already proved (see Theorem 4.1.12). Therefore, we assume that Ni and N−i are nontrivial. Let dim Ni = dim N−i = r and let dim L = dim N = p < r. Then dim Ni = dim N−i = r − p. ˙ then by Theorem 4.1.3 both operators P T and P T ∗ are closed. If T ∈ Ω(A), Let M ∈ [Ni ⊕ N, N−i ⊕ N] and M ∗ ∈ [N−i ⊕ N, Ni ⊕ N] deﬁne T and T ∗ , respectively, in accordance with Theorem 4.1.9. Because T and T ∗ have dense domains in H, the equalities + + PN Ran(M + I) = PN Ran(M ∗ + I) = N

hold true. We are going to use the following notation: + + + N := ker(PN (M + I)), F := ker(PN G = ker(PN N ), M N ), i −i + ˙ L := N (F +G), K := N−i (PN M (F ⊕ L)), + N∗ = ker(PN (M ∗ + I)).

Observe that

−i

4.3. The (∗)-extensions and uniqueness theorems

101

1. ker(M + I) = {0} implies F ∩ G = {0}, 2. dim N = dim N∗ = dim N−i = r − p, 3. dim K = dim G, + + 4. PN G ∩ PN L = {0}, i i

5.

+ + ˙ N dim(PN G +P L) i i

+ = dim(PN M L ⊕ K). −i

Let U be a (1)-isometry such that Dom(U ) = Ni , Ran(U ) = N−i , and + ˙ + L) = P + M L ⊕ K. U (PN G +P N N i i

−i

Let us consider the following equation with respect to h ∈ N , + + U PN h = αPN M h, i

(4.56)

−i

where α is a complex parameter. If hα is a nontrivial solution of (4.56), then + + + g = αPN M hα ∈ Ran(PN M N ) = PN M (F ⊕ L) −i

−i

−i

+ ˙ + = PN M F +PN M L. −i

−i

On the other hand + + + + ˙ g = U PN hα ∈ U PN N = U PN G +U PN L. i

i

i

i

+ + ˙ + It follows that g ∈ PN ∩ PN . From the deﬁnition ML ⊕ K M F +PN M L i

−i

−i

+ of K we get that g ∈ PN M L. Consequently, hα ∈ L ⊕ G. −i Since the dimensions are ﬁnite, it is only possible that either 1) the equation (4.56) has nontrivial solution h ∈ N for all complex α, or 2) there is a ﬁnite number of α’s, for which (4.56) has nontrivial solutions h ∈ N . If the case 1) takes place, then choose a sequence {αn } such that limn→∞ αn = 0 and let {hn }, hn + = 1, be a corresponding sequence of solutions of (4.56). The sequence {hn} is compact since dim Ni < ∞, and hence, we may assume that {hn } converges. Let limn→∞ hn = h ∈ L ⊕ G. Then h+ = 1. Since + PN M hn 1 ≤ M , −i + we get limn→∞ αn PN M hn = 0. Hence, −i + lim U PN hn = 0. i

n→∞

+ Thus, PN h = 0 ⇒ h ∈ F and we arrive at a contradiction. Therefore, case i 2) holds, and hence there exists a ﬁnite set of complex numbers α s, for which

102

Chapter 4. Quasi-self-adjoint Extensions

equation (4.56) has a nontrivial solution. Let us denote this set by A and assume that α0 ∈ / A and |α0 | = 1. Then the operator + + α ¯ 0 U PN − PN M i −i

is a one-to-one correspondence between N and N−i . Let U := α ¯ 0 U. Suppose also that the equation + ∗ PN−i y − UPN M y = 0,

(4.57)

i

has a nontrivial solution f ∈ N∗ . Then for all h ∈ N , + + ∗ ∗ 0 = (h, U −1 (PN f − UPN M f ))1 = (UPN h, f )1 − (h, PN M f )1 i i −i

= =

+ (UPN h, f )1 i + (UPN h, f )1 i

i

∗

− (h, M f −

+ PN M ∗ f )1

+ − (M h, f )1 − (h, PN f )1

+ + = (UPN h, f )1 − (M h, f )1 − (PN h, f )1 i

+ + + + = (UPN h, f )1 − (M h, f )1 + (PN M h, f )1 = ((UPN − PN M )h, f )1 . i

i

−i

+ ∗ ∗ This implies that f ∈ N and, hence, PN M f = 0, i.e., M f ∈ N. Because i ∗ f + M f = 0, we get f = 0. Therefore, equation (4.57) has only the trivial solution. This is equivalent to the operator + PN − U PNi M ∗ −i

being a one-to-one correspondence between N∗ and N−i . Now by the construction we get that Ran(M + I) ∩ Ran(I + U) = Ran(M ∗ + I) ∩ Ran(I + U) = {0}. Since dim Ran(M + I) = dim Ran(M ∗ + I) = r, dim Ran(I + U) = p, we get ˙ ˙ Ran(M + I)+Ran(I + U) = Ran(M ∗ + I)+Ran(I + U) = M. ˙ We note that any By Theorem 4.2.9 the operator T belongs to the class R(A). ˙ T ∈ R(A) has inﬁnitely many (∗)-extensions. If the semi-deﬁciency indices are both inﬁnite, the situation is diﬀerent. Theorem 4.3.11. Let A˙ be a closed densely deﬁned symmetric operator with equal ˙ inﬁnite deﬁciency indices. Then it admits a quasi-self-adjoint extension T ∈ Ω(A) that does not have a (∗)-extension.

4.3. The (∗)-extensions and uniqueness theorems

103

˙ that does not have Proof. We will construct an example of an operator T ∈ Ω(A) ˙ a (∗)-extension. Let A be closed, densely deﬁned, symmetric operator in H with equal inﬁnite deﬁciency indices. Let M ∈ [Ni , Ni ], where N±i are deﬁciency spaces ˙ By Remark 4.3.4, if U is an isometry of Ni onto N−i such that of A. ˙ + U)Ni = Ni ⊕ N−i , (M + I)Ni +(I then the operator M − U is bijection from Ni onto N−i . Also let U be a partial isometry from Ni into N−i such that dim(Ran(U ))⊥ = dim ker(U ∗ ) = ∞. Also let N be a compact and self-adjoint operator in Ni such that (N ϕ, ϕ)+ ≥ −ε(ϕ, ϕ)+ ,

(0 < ε < 1), ϕ ∈ Ni ,

and 0 is not an eigenvalue of N . Considering the operator M = U (N + I), we have M ∗ M − I = N (N + 2I). Since 0 is not an eigenvalue and (−2) is a regular point for N , then M ∗ M − I is invertible and compact. Moreover, M M ∗ − I = U (N + I)2 U ∗ − I. Now let (M M ∗ − I)h = 0. Then h = U (N + I)2 U ∗ h and hence N (N + 2I)U ∗ h = 0. Consequently, h = 0. That is why M M ∗ − I is invertible but not compact, since the operator M M ∗ − I coincides with −I on the inﬁnite-dimensional subspace ker U ∗ . Suppose there exists a (+)-isometric operator U from Ni on N−i such that M − U is an isomorphism from Ni on N−i . Let G be a subspace of Ni such that (M − U)G = ker U ∗ . Then dim G = dim ker U ∗ = ∞. Furthermore, for all ϕ ∈ G and φ ∈ Ni , ((M − U )ϕ, M φ)+ = 0. Hence M ∗ Uϕ = M ∗ M ϕ. The adjoint operator M ∗ − U ∗ is an bijection from N−i on Ni . Then M ∗ U − I is also an bijection from Ni on Ni . Then there exists a ξ > 0 such that (M ∗ U − I)ϕ+ ≥ ξϕ+ , (∀ϕ ∈ Ni ). In particular, the last inequality takes place for all ϕ ∈ G. But then ξϕ+ ≤ (M ∗ M − I)ϕ+ ,

(ϕ ∈ G).

Since the operator M ∗ M − I is compact and dim G = ∞, then the last inequality leads to a contradiction. Thus, there is no (+)-isometric operator U from Ni on N−i such that M − U is an bijection from Ni on N−i . Applying Theorems 4.2.9, 4.3.3, and Remark 4.3.4 we conclude that (4.50) fails to be true and the quasi-selfadjoint extension T of A˙ given by ˙ ⊕ (I + M )Ni , Dom(T ) = Dom(A) does not admit a (∗)-extension.

104

Chapter 4. Quasi-self-adjoint Extensions

4.4 The (∗)-extensions in the densely-deﬁned case Let A˙ be a closed densely deﬁned symmetric operator in H. In this case the (+)˙ orthogonal decomposition of H+ = Dom(A), ˙ ⊕ Ni ⊕ N−i , H+ = Dom(A) holds. The operator A˙ ∗ maps M = Ni ⊕ N−i onto itself and A˙ ∗2 f = −f for all ˙ These f ∈ M. Let T be its quasi-self-adjoint extension, i.e., T ⊃ A˙ and T ∗ ⊃ A. conditions are equivalent to A˙ ⊂ T ⊂ A˙ ∗ . Recall that ˙ MT = Dom(T ) Dom(A), ⊥ and M⊥ T = H+ MT . Clearly, MT and MT are subspaces of M and

MT ⊕ M⊥ T = M. The equality

∗ ˙∗ M⊥ T = A MT ∗ = T MT ∗ ˙∗ holds. Indeed, f ∈ M⊥ T if and only if (T y, A f ) = (y, −f ) for all y ∈ Dom(T ). The ∗ ˙∗ latter is equivalent to T A f = −f ∈ Dom(T ∗ ) ∩ M = MT ∗ ⇐⇒ f ∈ A˙ ∗ MT ∗ .

Deﬁnition 4.4.1. Let A˙ be a closed densely deﬁned symmetric operator. Two quasiself-adjoint extensions T1 and T2 of the operator A˙ are called relatively prime (disjoint) if ˙ Dom(T1 ) ∩ Dom(T2 ) = Dom(A) (4.58) and are called mutually transversal if they are disjoint and Dom(T1 ) + Dom(T2 ) = H+ .

(4.59)

Note that in the case of ﬁnite deﬁciency indices the conditions (4.58) and (4.59) are equivalent. Clearly, the disjointness of T and T ∗ is equivalent to A˙ being maximal common symmetric part of T and T ∗ . It is also not hard to see that if T1 and T2 are mutually transversal, then Dom(T1 ) MT2 = Dom(T2 ) MT1 = H+ . ˙ then relations (4.4) and (4.5) hold, i.e., If T ∈ Ω(A), ˙ + (I − Mλ ))Nλ¯ , Dom(T ) = Dom(A) ¯ − λMλ ψ, ˙ + λψ T (g + (I − Mλ )ψ) = Ag

˙ ψ ∈ Nλ¯ , g ∈ Dom(A),

where ¯ Mλ = (T − λI)(T − λI)−1 Nλ¯ : Nλ¯ → Nλ ,

λ ∈ ρ(T ), Im λ = 0.

4.4. The (∗)-extensions in the densely-deﬁned case

105

˙ Let ρ(T1 )∩ Theorem 4.4.2. 1) Let operators T1 and T2 belong to the class Ω(A). ρ(T2 ) = ∅. Then ¯ ∈ ρ(T1 ) ∩ ρ(T2 ), Im λ = 0. (T1 − λI)−1 − (T2 − λI)−1 H ⊆ Nλ , λ, λ 2) If T1 and T2 are mutually transversal, then for any λ ∈ ρ(T1 )∩ρ(T2 ), Im λ = 0 the operator (T1 − λI)−1 − (T2 − λI)−1 Nλ¯ is of the spaces Nλ¯ and Nλ . Conversely, if the operator an isomorphism (T1 − λI)−1 − (T2 − λI)−1 Nλ¯ is an isomorphism of Nλ¯ and Nλ for some λ ∈ ρ(T1 ) ∩ ρ(T2 ), Im λ = 0, then T1 and T2 are mutually transversal. ˙ In order for T and T ∗ to be mutually transversal it is necessary 3) Let T ∈ Ω(A). for all, and suﬃcient for at least one, λ ∈ ρ(T ), Im λ = 0 that the operator I − Mλ∗ Mλ is an isomorphism of the deﬁciency subspace Nλ¯ . Proof. Since both sets ρ(T1 ) and ρ(T2 ) are open, one can always ﬁnd a non-real point λ of the set ρ(T1 ) ∩ ρ(T2 ). Since Tk∗ ⊃ A˙ (k = 1, 2), then ¯ −1 (A˙ − λI)g ¯ (Tk∗ − λI) = g,

˙ (k = 1, 2, g ∈ Dom(A)).

Hence for an arbitrary f ∈ H, ¯ ˙ − λg [(T1 − λI)−1 − (T2 − λI)−1 ]f, Ag ¯ −1 − (T ∗ − λI) ¯ −1 ](Ag ¯ ˙ − λg)) = (f, [(T ∗ − λI) = 0. 1

2

Then the following inclusion holds: (T1 − λI)−1 − (T2 − λI)−1 H ⊆ Nλ . Also

˙ (I − M (k) )N ¯ , Dom(Tk ) = Dom(A) λ λ

k = 1, 2.

˙ then the von If Dom(T1 ) + Dom(T2 ) = H+ and Dom(T1 ) ∩ Dom(T2 ) = Dom(A), Neumann formulas imply (1)

(2)

Nλ Nλ¯ = (I − Mλ )Nλ (I − Mλ )Nλ¯ . If fλ ∈ Nλ , then there are two uniquely deﬁned vectors f1 and f2 such that (1)

(2)

f = (I − Mλ )f1 + (I − Mλ )f2 . (1)

(2)

Therefore, one concludes that the operator Mλ − Mλ is an isomorphism of spaces Nλ¯ and Nλ . Furthermore, (1) (2) −1 −1 ¯ ¯ Mλ − Mλ = (T1 − λI)(T − (T2 − λI)(T Nλ¯ 1 − λI) 2 − λI) −1 −1 = 2iIm λ (T1 − λI) − (T2 − λI) Nλ¯ .

106

Chapter 4. Quasi-self-adjoint Extensions

(1) (2) Conversely, if Mλ − Mλ = (T1 − λI)−1 − (T2 − λI)−1 Nλ¯ is an isomorphism of spaces Nλ¯ and Nλ , then (1)

(2)

Nλ Nλ¯ = (I − Mλ )Nλ¯ (I − Mλ )Nλ¯ , which implies the transversality of T1 and T2 . The second statement can be proved similarly.

Using Theorem 4.2.9 and the deﬁnition of transversality we obtain the following result. Theorem 4.4.3. Let A˙ be a closed densely deﬁned symmetric operator and let T ∈ ˙ Then T ∈ R(A) ˙ if and only if A˙ has equal deﬁciency indices and there exists Ω(A). a self-adjoint extension A˜ of A˙ transversal to T. Moreover, the formulas A = A˙ ∗ − R−1 A˙ ∗ (I − PT A˜ ), A∗ = A˙ ∗ − R−1 A˙ ∗ (I − PT ∗ A˜ ),

(4.60)

˙ with the range set a bijection between the set of all q.s.-a. bi-extensions of T ∈ R(A) property (R) and their adjoints and all self-adjoint extensions A˜ of the operator A˙ that are transversal to T . Here PT A˜ and PT ∗ A˜ are the projectors in H+ onto Dom(T ) and Dom(T ∗ ), corresponding to the direct decompositions ˙ A˜ , H+ = Dom(T )+M

˙ A˜ . H+ = Dom(T ∗ )+M

(4.61)

In addition, if A˙ is maximal common symmetric part of T and T ∗ , then ker(Im A) = Dom(A). If a quasi-self-adjoint bi-extension A of T has a form (4.60), we say that A ˜ is generated by A. ˙ with the range property Theorem 4.4.4. If A is a q.s.-a. bi-extension of T ∈ Ω(A) ˜ (R) generated by A via (4.60), then Re A is a t-self-adjoint bi-extension generated ˜ Moreover, if by A. ˜ = Dom(A) ˙ ⊕ (I + U )Ni , Dom(A) ˙ ⊕ (I + M )Ni , then the operator W ∈ [Ni , N−i ] −i ∈ ρ(T ), and Dom(T ) = Dom(A) given by W = −(I − M U ∗ )−1 (I + M M ∗ − 2M U ∗ )(I + M M ∗ − 2U M ∗ )−1 (U − M ) (4.62) (·)-unitarily maps Ni onto N−i and ˆ = Dom(A) ˙ ⊕ (I + W )Ni , Dom(A) where Aˆ is the quasi-kernel of Re A.

4.4. The (∗)-extensions in the densely-deﬁned case

107

+ ˙ with Proof. Let A = A˙ ∗ + R−1 (S − 2i J)PM be a q.s.-a. bi-extension of T ∈ Ω(A) ˜ Then the ﬁrst statement follows from the range property (R) generated by A. Theorem 4.3.2 and equality (4.36). ˙ + U )Ni we have Let h ∈ M. Since M = (I + M ∗ )N−i +(I

h = (I + M ∗ )ψ + (I + U )φ, ψ ∈ N−i , φ ∈ Ni . Due to the decomposition ˙ + U )Ni , M = (I + M )Ni +(I every vector (I + M ∗ )ψ, ψ ∈ N−i , has unique representation (I + M ∗ )ψ = (I + M )ϕ + (I + U )χ, where ϕ, χ ∈ Ni . Therefore, χ = (U − M )−1 (I − M M ∗ )ψ. Since i i (S − J)(I + M ) = 0, (S ∗ − J)(I + M ∗ ) = 0, 2 2 and (see (4.47)) i i i (Re S + J)(I + U ) = (S ∗ + J)(I + U ) = (S + J)(I + U ) = 0, 2 2 2 we obtain (Re S − 2i J)h = (Re S − 2i J)(I + M ∗ )ψ + (Re S − 2i J)(I + U )φ = 12 (S − 2i J)(I + M ∗ )ψ − i(I − U )φ.

Furthermore, i i i (S − J)(I + M ∗ )ψ = (S − J)(I + M )ϕ + (S − J)(I + U )χ = −i(I − U )χ. 2 2 2 Therefore, i i (Re S − J)h = − (I − U )χ − i(I − U )φ, 2 2 and thus (Re S − 2i J)h = 0 if and only if φ = − 12 χ. Hence, if h ∈ ker(Re S − 2i J), then 1 h = (I + M ∗ )ψ − (I + U )(U − M )−1 (I − M M ∗ )ψ. (4.63) 2 Conversely, if ψ ∈ N−i , then one can verify that the vector h ∈ M given by (4.63) belongs to ker(Re S − 2i J). The equality (4.63) yields 1 + PN h = M ∗ ψ − (U − M )−1 (I − M M ∗ )ψ i 2 1 = (U − M )−1 (2U M ∗ − I − M M ∗ )ψ, 2 1 + PN−i h = ψ − U (U − M )−1 (I − M M ∗ )ψ 2 1 = (I − M U ∗ )−1 (I + M M ∗ − 2M U ∗ )ψ, 2

ψ ∈ N−i .

108

Chapter 4. Quasi-self-adjoint Extensions

Observe that if B = I − M U ∗ , then I + M M ∗ − 2M U ∗ = BB ∗ − 2iIm B, I + M M ∗ − 2U M ∗ = BB ∗ + 2iIm B. Because B −1 ∈ [N−i , N−i ], we obtain that (I + M M ∗ − 2M U ∗ )−1 , (I + M M ∗ − 2U M ∗ )−1 ∈ [N−i , N−i ]. Hence, the operator W = −(I − M U ∗ )−1 (I + M M ∗ − 2M U ∗ )(I + M M ∗ − 2U M ∗ )−1 (U − M ) is well deﬁned and Dom(W ) = Ni , Ran(W ) = N−i . By construction we have i ker(Re S − J) = (I + W )Ni . 2 + ˙ the Since A = A˙ ∗ + R−1 (Re S − 2i J)PM is a t-self-adjoint bi-extension of A, ˆ = operator W is (·) and then (+)-unitary mapping Ni onto N−i , and Dom(A) ˙ Dom(A) ⊕ (I + W )Ni (see Theorems 3.3.6 and 3.4.5).

Corollary 4.4.5. If W is given by (4.62), then W − M = −(I − M M ∗ )(I + M M ∗ − 2U M ∗ )−1 (U − M ). Proof. Due to the equality (U − M )(U ∗ − M ∗ ) = I + M M ∗ − U M ∗ − M U ∗ , for W − U we get W − U = −U (U − M )−1 (I + M M ∗ − 2M U ∗ )(I + M M ∗ − 2U M ∗ )−1 ×(U − M ) + I) = −U (U − M )−1 (I + M M ∗ − 2M U ∗ )(I + M M ∗ − 2U M ∗ )−1 + I ×(U − M ) = −2U (U − M )−1 (I + M M ∗ − M U ∗ − U M ∗ )(I + M M ∗ − 2U M ∗ )−1 ×(U − M ) = −2(I − U M ∗ )(I + M M ∗ − 2U M ∗ )−1 (U − M ). Furthermore, since W − M = (W − U ) + (U − M ), we have W − M = − 2(I − U M ∗ )(I + M M ∗ − 2U M ∗ )−1 − I (U − M ) = − (2(I − U M ∗ ) − (I + M M ∗ − 2U M ∗ )) (I + M M ∗ − 2U M ∗ )−1 (U − M ) = −(I − M M ∗ )(I + M M ∗ − 2U M ∗ )−1 (U − M ). The next statement reinforces Theorem 4.3.9.

4.4. The (∗)-extensions in the densely-deﬁned case

109

˙ and A˙ be a maximal common symmetric part of T Theorem 4.4.6. Let T ∈ R(A) ∗ and T . If A1 and A2 are (∗)-extensions of T such that the quasi-kernels of real parts of Re A1 and Re A2 coincide, then A1 = A2 . Proof. Without loss of generality we may suppose that −i ∈ ρ(T ). Then ˙ ⊕ (I + M )Ni , Dom(T ∗ ) = Dom(A) ˙ ⊕ (I + M ∗ )N−i . Dom(T ) = Dom(A) Let A1 and A2 be generated by (+)-unitary mappings U1 , U2 : Ni → N−i (Ran(U1 ) = Ran(U2 ) = N−i ). By assumption the quasi-kernels of Re A1 and Re A2 ˙ Let coincide with some self-adjoint extension Aˆ of A. ˆ = Dom(A) ˙ ⊕ (I + W )Ni . Dom(A) Then by Theorem 4.4.4 and Corollary 4.4.5 we have the equality (I − M M ∗ )(I + M M ∗ − 2U1 M ∗ )−1 (U1 − M ) = (I − M M ∗ )(I + M M ∗ − 2U2 M ∗ )−1 (U2 − M ). Since A˙ is a maximal common symmetric part of T and T ∗ , the operator I −M M ∗ has zero null-space. Hence, (I + M M ∗ − 2U1 M ∗ )−1 (U1 − M ) = (I + M M ∗ − 2U2 M ∗ )−1 (U2 − M ) or equivalently (U1 − M )−1 (I + M M ∗ − 2U1 M ∗ ) = (U2 − M )−1 (I + M M ∗ − 2U2 M ∗ ). Using the relations I + M M ∗ − 2U1 M ∗ = I − M M ∗ − 2(U1 − M )M ∗ , I + M M ∗ − 2U2 M ∗ = I − M M ∗ − 2(U2 − M )M ∗ , we obtain

(U1 − M )−1 (I − M M ∗ ) = (U2 − M )−1 (I − M M ∗ ).

This yields (U1 − M )−1 = (U2 − M )−1 . Therefore, U1 = U2 and A1 = A2 .

˙ generated by Theorem 4.4.7. If A is a quasi-self-adjoint bi-extension of T ∈ Ω(A) ˜ we A˜ via (4.60), then for all φ = h + f ∈ H+ , h ∈ Dom(T ), and f ∈ Dom(A) have ˜ f ) + 2Re (T h, f ). (Aφ, φ) = (T h, h) + (Af, Proof. Let A = A˙ ∗ − R−1 A˙ ∗ (I − PT A˜ ) according to (4.60). Note that for any ˜ we have that P ˜ f ∈ Dom(A) ˙ and hence f ∈ Dom(A) TA ˙ (PT A˜ f, T h) = (AP ˜ f, h), TA

110

Chapter 4. Quasi-self-adjoint Extensions

for any h ∈ Dom(T ). Besides, R−1 A˙ ∗ (I − PT A˜ )f, g = 0,

˜ ∀f, g ∈ Dom(A).

˜ = Dom(A) ˙ ⊕ (U + I)Ni , where U is a unitary operator from Indeed, since Dom(A) Ni onto N−i , we have (I − PT A˜ )f = (I + U )ϕ, for ϕ ∈ Ni , and A˙ ∗ (I − PT A˜ )f = i(I − U )ϕ. Moreover, from (+)-orthogonality of (U +I)Ni and (U −I)Ni we obtain the desired equation. Further, ˜ − R−1 A(I ˜ − P ˜ )f, h + f ) (Aφ, φ) = (T h + Af TA ˜ ˜ −P = (T h, h) + (Af, f ) + (T h, f ) − (A(I

˜ )f, h)+ TA

˜ h), + (Af,

and ˜ − P ˜ )f, h)+ = (A(I ˜ − P ˜ )f, h) − ((I − P ˜ )f, T h) (A(I TA TA TA ˜ h) − (AP ˙ = (Af, f, h) − (f, T h) + (PT A˜ f, T h) ˜ TA ˜ h) − (f, T h). = (Af, ˜ f ) + 2Re (T h, f ). Consequently, (Aφ, φ) = (T h, h) + (Af,

4.5 Resolvents of quasi-self-adjoint extensions Let C be a linear operator in H. Following Chapter 1, we denote by ρ(C) the resolvent set of C. For every λ ∈ ρ(C), the resolvent of C is the operator Rλ (C) = (C − λI)−1 ˙ that is deﬁned on entire H and (·)-continuous. We also recall that we write Ω(A) for the class of all quasi-self-adjoint extensions T (of a closed symmetric operator ˙ with nonempty resolvent sets. A) ˙ Then for every λ ∈ ρ(T ) the operator Rλ (T ) is Theorem 4.5.1. Let T ∈ Ω(A). (·, +)-continuous and is a holomorphic function of λ with values in [H, H+ ]. Proof. Based on Theorems 4.1.2 and 4.1.3 we have Rλ (T )H = Dom(T ) ⊂ H+ and A˙ ∗ Rλ (T ) = P T (T − λI)−1 = P [I + λRλ (T )],

(4.64)

˙ Hence where P is a (·)-orthogonal projection operator in H onto H0 = Dom(A). Rλ (T )f 2+ = Rλ (T )f 2 + A˙ ∗ Rλ (T )f 2 ≤ Rλ (T )f 2 + f + λRλ (T )f 2 ≤ γf 2

4.5. Resolvents of quasi-self-adjoint extensions

111

for some γ > 0, i.e., Rλ (T ) ∈ [H, H+ ]. Furthermore, if f ∈ H and g ∈ H+ , then the function (Rλ (T )f, g)+ = (Rλ (T )f, g) + (A˙ ∗ Rλ (T )f, A˙ ∗ g) = (Rλ (T )f, g) + (f + λRλ (T )f, A˙ ∗ g) is holomorphic on ρ(T ). Therefore Rλ (T ) is a holomorphic function with values in [H, H+ ]. ˙ and λ ∈ ρ(T1 ) ∩ ρ(T2 ), then Proposition 4.5.2. If T1 and T2 belong to Ω(A) Ran(Rλ (T1 ) − Rλ (T2 )) ⊂ Nλ . Proof. Since Tk∗ ⊃ A˙ (k = 1, 2), then ¯ Rλ (Tk∗ )(A˙ − λI)g = g,

˙ (k = 1, 2, g ∈ Dom(A)).

Hence for an arbitrary f ∈ H, ¯ = (f, [R ¯ (T ∗ ) − R ¯ (T ∗ )](Ag ¯ ˙ − λg ˙ − λg)) [Rλ (T1 ) − Rλ (T2 )]f, Ag = 0, λ 1 λ 2

and this completes the proof.

For a rigged Hilbert space H+ ⊂ H ⊂ H− we call an operator C bi-continuous if C ∈ [H− , H] and C H ∈ [H, H+ ]. Now we will show that the resolvents of ˙ can be extended to H− by continuity and study the properties operators in Ω(A) of such resolvents. ˙ and λ ∈ ρ(T ). Then: Theorem 4.5.3. Let T ∈ Ω(A) 1. Rλ (T ) can be extended to H− by (−, ·)-continuity; ˆ λ (T ) is bi-continuous and is a holomorphic function 2. the extended operator R of λ ∈ ρ(T ) that belongs to [H− , H]; 3. for all h ∈ H0 , ˆ λ (T )(Ah ˙ − λh) = h, R

(4.65)

˙ by (·, −)-continuity. where A˙ is the extension of A˙ to H0 = Dom(A) ¯ ∈ ρ(T ∗ ), then the operator R ¯ (T ∗ ) belongs to ˙ and λ Proof. 1. Since T ∗ ∈ Ω(A) λ ∗ ∗ [H, H+ ]. Its adjoint [Rλ¯ (T )] then belongs to [H− , H]. If f, g ∈ H, then ([Rλ¯ (T ∗ )]∗ f, g) = (f, Rλ¯ (T ∗ )g) = (Rλ (T )f, g), ˆ λ (T ) = [Rλ¯ (T ∗ )]∗ is an i.e., [Rλ¯ (T ∗ )]∗ f = Rλ (T )f for all f ∈ H. Therefore R extension of the operator Rλ (T ) on H− by (−, ·)-continuity. Note that it follows ˆ λ (T ) that from the deﬁnition of R ˆ λ (T )f, g) = (f, R ¯ (T ∗ )g) (R λ

(∀f ∈ H− , g ∈ H).

(4.66)

112

Chapter 4. Quasi-self-adjoint Extensions

ˆ λ (T ) is clearly bi-continuous. Since Rλ¯ (T ∗ ) is a holomor2. The operator R ¯ ˆ λ (T ) is a holomorphic on ρ(T ) phic function of λ that belongs to [H, H+ ], then R function that belongs to [H− , H]. 3. According to (4.64) (A˙ ∗ − λP )Rλ¯ (T ∗ ) = P . Passing to the adjoint we get (4.65). ˆ λ (T )) = H if and only if T ∗ is (+, ·)-continuous. Theorem 4.5.4. Ran(R Proof. The (+, ·)-continuity of T ∗ means that the operator Rλ¯ (T ∗ ) ∈ [H, H+ ] has (+, ·)-continuous inverse. The latter is equivalent to the fact that ˆ λ (T ) [Rλ¯ (T ∗ )]∗ = R maps H− onto H.

˙ and λ ∈ ρ(T1 ), μ ∈ ρ(T2 ). Then Theorem 4.5.5. Let T1 and T2 belong to Ω(A) ˆ ˆ ˆ λ (T1 )− R ˆ μ (T2 )] [Rλ (T1 )− Rμ (T2 )] ∈ [H− , H+ ] and the [H− , H+ ]-valued function [R is holomorphic in λ and μ on ρ(T1 ) × ρ(T2 ). ˙ and g ∈ H− . Then [R ˆ λ (T1 ) − R ˆ μ (T2 )]g ∈ H and Proof. Indeed, let f ∈ Dom(A) ˙ [R ˆ λ (T1 ) − R ˆ μ (T2 )]g = [R ¯ (T ∗ ) − Rμ¯ (T ∗ )]Af, ˙ g . Af, λ

1

2

˙ then Since T1∗ ⊃ A, ¯ ¯ (T ∗ )f, ˙ = Rλ¯ (T1∗ )T1∗ f = f + λR Rλ¯ (T1∗ )Af λ 1 and similarly Hence

˙ = Rμ¯ (T ∗ )T ∗ f = f + μ Rμ¯ (T2∗ )Af ¯Rμ¯ (T2∗ )f. 1 1 ¯ ¯ (T ∗ )f − μ ˙ [R ˆ λ¯ (T1∗ ) − R ˆ μ¯ (T2∗ )]g = λR Af, ¯ Rμ¯ (T2∗ )f, g λ 1 ˆ μ (T2 )]g . = f, [λRλ (T1 ) − μR

ˆ μ (T2 )]g ∈ H+ and Furthermore, [Rλ (T1 ) − R ˆ λ (T1 ) − R ˆ μ (T2 ) g = P λR ˆ λ (T1 ) − μR ˆ μ (T2 ) g. A˙ ∗ R

(4.67)

ˆ λ (T1 ) − R ˆ μ (T2 ) ∈ [H− , H+ ]. The analyticity of the function R ˆ λ (T1 ) − Thus, R ˆ ˆ ˆ μ (T2 )]g Rμ (T2 ) follows from the fact that for any g ∈ H− both functions [Rλ (T1 )−R ∗ ˆ λ (T1 ) − R ˆ μ (T2 )]g are holomorphic. and A˙ [R We note that it follows from the above theorem and (4.66) that ˆ λ (T1 ) − R ˆ μ (T2 )]∗ = R ˆ λ¯ (T ∗ ) − R ˆ μ¯ (T ∗ ) [R 1 2 in [H− , H+ ]. In the case when λ = μ Theorem 4.5.5 can be reﬁned as follows.

(4.68)

4.5. Resolvents of quasi-self-adjoint extensions

113

˙ and λ ∈ ρ(T1 ) ∩ ρ(T2 ). Then Theorem 4.5.6. Let T1 and T2 belong to Ω(A) ˆ ˆ [Rλ (T1 ) − Rλ (T2 )]H− ⊆ Nλ . Proof. Using (4.66) with λ = μ we have ¯ [R ˙ − λg, ˆ λ (T1 ) − R ˆ λ (T2 )]f ) = 0, (Ag ˙ and f ∈ H− . for all g ∈ Dom(A)

˙ (⊂ H+ ) is We recall (see (3.1)) that the orthogonal complement of Dom(A) denoted by F (⊂ H− ), the linear manifold B is given by B = H + F = H0 F,

(4.69)

and the projection onto H0 parallel to F is denoted by Π. ˙ and λ ∈ ρ(T ). A vector f ∈ H− belongs to F Lemma 4.5.7. Let T belong to Ω(A) ˆ λ (T )f ∈ Nλ . if and only if R Proof. The proof immediately follows from the relation ¯ R ˙ − λg, ˆ λ (T )f ) = (g, f ), (Ag

˙ f ∈ H− ). (g ∈ Dom(A),

Corollary 4.5.8. The following statements hold true: ˆ λ (T )F ⊂ Nλ ; 1. R ˆ λ (T )) = H, then R ˆ λ (T )F = Nλ ; 2. If Ran(R ˆ λ (T )f is (+)-holomorphic on ρ(T ). 3. If f ∈ F, then the vector function R The last condition is true because for every h ∈ H+ the scalar function ˆ λ (T )f, h)+ = (R ˆ λ (T )f, h) + (A˙ ∗ R ˆ λ (T )f, A˙ ∗ h) (R ˆ λ (T )f, h) + λ(P R ˆ λ (T )f, A˙ ∗ h), = (R is holomorphic on ρ(T ). ˙ and λ ∈ ρ(T ). Then: Theorem 4.5.9. Let T ∈ Ω(A) ˆ λ (T )f belongs to H+ if and only if f ∈ B; 1. R ˆλ (T ) B coincides with Π. 2. the operator (A˙ ∗ − λP )R ˆ λ (T )B ⊂ H+ . Conversely, Proof. According to Lemma 4.5.7 and Theorem 4.5.1 R ˆ λ (T )B ⊂ H+ . We set let f ∈ H− and R ˆ λ (T )f, ϕ = (A˙ ∗ − λP )R

ψ = f − ϕ.

˙ we have For an arbitrary g ∈ Dom(A) ˆ λ (T )f, g) (ψ, g) = (f, g) − ((A˙ ∗ − λP )R ˙ − λg)) = 0, = (f, g) − (f, Rλ¯ (T ∗ )(Ag

114

Chapter 4. Quasi-self-adjoint Extensions

˙ Therefore, f = ϕ + ψ ∈ H0 + F = B and since T ∗ ⊃ A. ˆ λ (T )f. Πf = ϕ = (A˙ ∗ − λP )R

ˆ λ (T )g is independent of the choice It follows from the proof that (A˙ ∗ − λP )R ˙ and λ ∈ ρ(T ). of T ∈ Ω(A) ˙ and λ ∈ ρ(T ). By Lemma 4.5.7 Let F be a (−)-subspace of F , T ∈ Ω(A), ˆ λ (T ) maps H0 + F in H+ . Let the operator R ˆ λ (T ) (H0 + F ). Q=R Then the following theorem holds. Theorem 4.5.10. The operator Q is (−, +)-closed. (−)

(−)

Proof. Let {hn } ⊂ Dom(Q), hn −−→ h(∈ H), and Qhn −−→ f (∈ H+ ). It follows from (+, ·)-continuity of the operator (A˙ ∗ − λP ) H+ that (A˙ ∗ − λP )Qhn → (A˙ ∗ − λP )f ∈ H0 . Using the proof of Theorem 4.5.9 we have (A˙ ∗ − λP )Qhn = Phn and hence hn = hn − Phn ∈ F. Since F is (−)-closed we have (−) hn −−→ h − (A˙ ∗ − λP )f ∈ F.

˙ Furthermore, the (−, ·)-continuity of R ˆ λ (T ) yields Thus, h ∈ H0 + F = Dom(A). ˆ ˆ ˆ Rλ (T )hn → Rλ (T )h, i.e., f = Rλ (T )h = Qh. ˙ and λk ∈ ρ(Tk ) (k = 1, 2). It follows from (4.68) that for Let T1 , T2 ∈ Ω(A) arbitrary f, g ∈ H− , ˆ λ1 (T1 ) − R ˆ λ2 (T2 )]f, g) = (f, [R ˆ λ¯ (T1∗ ) − R ˆ λ¯ (T2∗ )]g). ([R 1 2 If f, g ∈ B, then the last equation can be re-written as ˆ λ1 (T1 )f, g) − (f, R ˆ λ¯ (T1∗ )g) = (R ˆ λ2 (T2 )f, g) − (f, R ˆ λ¯ (T2∗ )g). (R 1 2 Therefore it is clear that the sesquilinear functional Ω(f, g) =

1 ˆ ˆ λ¯ (T ∗ )g) , (Rλ (T )f, g) − (f, R 2i

(4.70)

˙ and λ ∈ ρ(T ). which is deﬁned on B × B, is independent of the choice of T ∈ Ω(A) This functional is also symmetric. Indeed, taking into account that T ∗ belongs to

4.5. Resolvents of quasi-self-adjoint extensions

115

¯ in the right-hand side ˙ together with T , we replace T with T ∗ and λ with λ Ω(A) of (4.70) 1 ˆ ˆ λ (T )g) . Ω(f, g) = (Rλ¯ (T ∗ )f, g) − (f, R 2i Comparing with (4.70) we get Ω(f, g) = Ω(g, f ). It follows from (4.66) that Ω(f, g) = 0 if either f or g belongs to H. Therefore, it is suﬃcient to consider Ω(f, g) for f, g ∈ F. Theorem 4.5.11. The functional Ω(f, g) (F × F ) is (−)-continuous in both arguments. (−) (−) (·) ˆ λ (T )fn − Proof. Let {fn }, {gn } ⊂ F, fn −−→ 0, and gn −−→ 0. Then R → 0. ˆ λ (T )fn ∈ Nλ and Nλ is poly-closed, then Since, according to Lemma 4.5.7, R (+) ˆ λ (T )fn − ˆ λ (T )fn , gn ) → 0. Similarly, (fn , R ˆ λ¯ (T ∗ )gn ) → 0, R −→ 0 and hence, (R and thus Ω(fn , gn ) → 0 as n → ∞.

Let us now derive an explicit formula for Ω(f, g) (F × F ) using the (−)orthogonal decomposition (2.18). It follows from Theorem 4.5.11 that Ω(f, g) = 0 (−)

(−)

if either f ∈ L or g ∈ L . Thus we only need to consider the case when f, g ∈ R−1 N±i . First we will show that Ni is (+)-orthogonal to N−i . Indeed, if ϕ ∈ Ni and ψ ∈ N−i , then A˙ ∗ ϕ = iϕ, A˙ ∗ ψ = −iP ψ, and (ϕ, ψ)+ = (ϕ, ψ) + (A˙ ∗ ϕ, A˙ ∗ ψ) = (ϕ, ψ) + (iϕ, −iP ψ) = 0. Similarly one shows that N−i is (+)-orthogonal to Ni . Let ϕ ∈ N±i . Applying (2.7) we get R−1 ϕ = ϕ + AA˙ ∗ ϕ = ϕ + A(±iϕ) = ±i(Aϕ ∓ iϕ). Let T be such a quasi-self-adjoint extension of operator A˙ that i is its regular ˙ Then point (for example one can take the maximal symmetric extension of A). ˆ i (T )R−1 ϕ = iϕ for all ϕ ∈ N . If ψ ∈ H+ , using (4.65) and (2.1) we have that R i then ˆ i (T )R−1 ϕ, R−1 ψ) = i(ϕ, R−1 ψ) = i(R−1 ϕ, R−1 ψ)− = i(ϕ, ψ)+ . (R On the other hand, ˆ −i (T ∗ )R−1 ψ) = (ϕ, R ˆ−i (T ∗ )R−1 ψ)+ . (R−1 ϕ, R ˆ −i (T ∗ )R−1 ψ ∈ Ni and hence If ψ ∈ M, then according to Corollary 4.5.8, R ˆ−i (T ∗ )R−1 ψ)+ = 0. (ϕ, R Therefore, Ω(R−1 ϕ, R−1 ψ) =

1 (ϕ, ψ)+ , 2

(∀ϕ ∈ Ni , ψ ∈ M).

116

Chapter 4. Quasi-self-adjoint Extensions

In particular, −1

Ω(R

−1

ϕ, R

ψ) =

1 (ϕ, ψ)+ , 2

0,

∀ϕ, ψ ∈ Ni ; ∀ϕ ∈ Ni , ψ ∈ N−i .

(4.71)

Similarly, if T is a quasi-self-adjoint extension of A˙ with a regular point (−i), we have 0, ∀ϕ ∈ N−i , ψ ∈ Ni ; −1 −1 Ω(R ϕ, R ψ) = (4.72) − 12 (ϕ, ψ)+ , ∀ϕ, ψ ∈ N−i . Formulas (4.71) and (4.72) provide the desired representation for Ω R−1 (Ni ⊕ N−i ). We note that the functional Ω is identically zero if and only if A˙ is an O-operator. ˆ λ (T ) and (A − λI)−1 for T ∈ R(A) ˙ Now we establish a connection between R and its quasi-self-adjoint extension A with the range property (R). Recall that relations (4.45) hold, where LA = F0 ∩ Ran(A − B˙ ∗ ) (see (4.44)). Theorem 4.5.12. Let A˙ be a regular symmetric operator with equal deﬁciency ˙ and A is a quasi-self-adjoint bi-extension of T with the indices. If T ∈ R(A) range property (R), then the operator A − λI is invertible for all λ ∈ ρ(T ) and (A − λI)−1 can be extended to H− by (−, ·)-continuity. The extended operator coˆ λ (T ) and R ˆ λ (T )H− = H. Moreover, incides with R ¯ −1 , ˆ λ (T ) (H+L ˆ λ¯ (T ∗ ) (H+L ˙ A ) = (A − λI)−1 , R ˙ A ) = (A∗ − λI) R −1 ˆ ˆ ˙ A ) = (Re A − μI) Rμ (A) (H+L ˆ (Aˆ is the quasi-kernel of Re A). In addition, the for λ ∈ ρ(T ) and μ ∈ ρ(A) −1 ∗ ¯ −1 , and (Re A − μI)−1 map (−, +)-continuously resolvents (A − λI) , (A − λI) ˙ A onto Nλ , Nλ¯ , and Nμ , correspondingly. the linear manifold L+L ¯ ⊃ T ∗ − λI ¯ and λ ¯ ∈ ρ(T ∗ ), then Proof. Since A∗ − λI ∗ ∗ ¯ ¯ (A∗ − λI)H + ⊃ (T − λI)Dom(T ) = H.

It follows that ker(A − λI) = {0} and by Theorem 2.1.5 the operator (A − λI)−1 admits an extension on H− by (−, ·)-continuity. Since (A − λI)−1 ⊃ (T − λI)−1 , then the extension of (A − λI)−1 by (−, ·)-continuity coincides with the extension ˆ λ (T ). It follows from Theorems 3.1.8 of (T − λI)−1 by (−, ·)-continuity, i.e., with R and 4.1.13 that operator T ∗ is (+, ·)-continuous. By Theorem 4.5.4, ˆ λ (T )H− = H. R The remaining statements follow from Theorem 4.3.2 (see (4.37), (4.39), (4.38), (4.40)) and the equivalence of (+) and (·) norms on the deﬁciency subspaces.

4.5. Resolvents of quasi-self-adjoint extensions

117

ˆ λ (T )(A − λI)h = h for all h ∈ H+ . Corollary 4.5.13. R In the following theorem we establish a connection between extensions of operator (A − λI)−1 by (−, −)-continuity and by (−, ·)-continuity. ˙ and the operator (A − λI)−1 is (−, −)-continuous, Theorem 4.5.14. If A ∈ E(A) then it is (−, ·)-continuous. Proof. By Theorem 2.1.6 if C ∈ [H+ , H− ], then C −1 is (−, −)-continuous if and ¯ ⊃ H+ and thus Ran(A∗ − λI) ¯ ⊃ Nλ . only if Ran(C ∗ ) ⊃ H+ . Hence, Ran(A∗ − λI) ∗ ¯ ¯ ˙ On the other hand, A − λI ⊃ A − λI which implies ¯ ⊃ Mλ . Ran(A∗ − λI) ¯ ⊃ H. Therefore, (A − λI)−1 is (−, ·)Since H = Mλ¯ ⊕ Nλ , then Ran(A∗ − λI) continuous. The following theorem reﬁnes Theorem 4.5.14. ˙ The Theorem 4.5.15. Let A be a self-adjoint bi-extension of symmetric operator A. −1 necessary and suﬃcient condition for operator (A − λI) to be (−, −)-continuous is to be (−, ·)-continuous. Proof. Taking into account Theorem 4.5.14 we only need to prove that (−, ·)continuity implies (−, −)-continuity. Let (A − λI)−1 be (−, ·)-continuous. Then (A − λI)−1 ≤ k f − ,

k > 0, ∀f ∈ Ran(A − λI).

Since for all g ∈ H, g− ≤ g, then it follows that (A − λI)−1 − ≤ k f − ,

k > 0, ∀f ∈ Ran(A − λI),

i.e., (A − λI)−1 is (−, −)-continuous.

˙ where A˙ has ﬁnite and equal semi-deﬁciency Corollary 4.5.16. Let A = A∗ ∈ E(A) indices. If for some λ operator (A − λI)−1 is (−, −)-continuous, then A is a self˙ adjoint bi-extension of operator A. The proof of the corollary easily follows from Theorem 4.5.15. ˙ then for every λ we have (A − λI)Nλ ⊂ F . Lemma 4.5.17. If A ∈ E(A), ˙ Then Proof. Let g ∈ Nλ and f ∈ Dom(A). ¯ g) = (A − λf, ¯ g) = (f, A∗ g − λg), ˙ − λf, 0 = (Af and hence A∗ g − λg ∈ F.

˙ and the operator (A − λI)−1 can be extended by Theorem 4.5.18. If A ∈ E(A) (−, −)-continuity to the operator Rλ , then Rλ F = Nλ .

118

Chapter 4. Quasi-self-adjoint Extensions

Proof. It follows from Lemma 4.5.17 that Rλ F ⊃ Nλ . In order to prove the inverse inclusion we note that according to Theorem 4.5.14 the operator Rλ is (−, ·)continuous. Let f ∈ F , then there exists such a sequence {hn } ⊂ H+ that f = (−) lim (A − λI)hn , n→∞

hn → Rλ f.

˙ we have For an arbitrary g ∈ Dom(A) ¯ ¯ (hn , (A˙ − λI)g) = (hn , (A∗ − λI)g) = ((A − λI)hn , g). Passing to the limit as n → ∞ we obtain ¯ (Rλ f, (A˙ − λI)g) = (f, g) = 0, and hence Rλ f is orthogonal to Mλ and Rλ f ∈ Nλ .

Chapter 5

The Livˇsic Canonical Systems with Bounded Operators In this chapter we present the foundations of the theory of the Livˇsic canonical open systems with bounded state-space (main) operators. We provide an analysis of such a type of systems in terms of transfer functions and their linear-fractional transformations. We also consider couplings of these systems and present multiplication and factorization theorems of the transfer functions.

5.1 The Livˇsic canonical system and the Brodski˘i theorem Let A be a bounded linear operator in a Hilbert space H, K ∈ [E, H], and J be a bounded, self-adjoint, and unitary operator in E, where E is another Hilbert space with dim E < ∞.1 Let also Im A = KJK ∗ and L2[0,τ0 ] (E) be the Hilbert space of E-valued functions equipped with an inner product τ0 (ϕ, ψ)L2[0,τ ] (E) = (ϕ, ψ)E dt, ϕ(t), ψ(t) ∈ L2[0,τ0 ] (E) . 0

0

Consider the system of equations ⎧ dχ ⎪ ⎨ i dt + Aχ(t) = KJψ− (t), χ(0) = x ∈ H, ⎪ ⎩ ψ+ = ψ− − 2iK ∗ χ(t).

(5.1)

The following lemma holds. 1 Here

and in the proof of Lemma 5.1.1 we assume that dim E < ∞. However, Lemma 5.1.1 also holds true for the case of a separable inﬁnite-dimensional Hilbert space E.

Y. Arlinskii et al., Conservative Realizations of Herglotz–Nevanlinna Functions, Operator Theory: Advances and Applications, DOI 10.1007/978-3-7643-9996-2_5, © Springer Basel AG 2011

119

120

Chapter 5. The Livˇsic Canonical Systems with Bounded Operators

Lemma 5.1.1. If for a given continuous in E function ψ− (t) ∈ L2[0,τ0 ] (E) we have that χ(t) ∈ H and ψ+ (t) ∈ L2[0,τ0 ] (E) satisfy (5.1), then a system of the form (5.1) satisﬁes the metric conservation law τ τ 2 2 2χ(τ ) − 2χ(0) = (Jψ− , ψ− )E dt − (Jψ+ , ψ+ )E dt, τ ∈ [0, τ0 ]. (5.2) 0

0

Proof. Using the ﬁrst equation of system (5.1) we obtain d (χ(t), χ(t)) = (χ (t), χ(t)) + (χ(t), χ (t)) = ((−iKJψ− (t) − Aχ(t)), χ(t)) dt + (χ(t), (−iKJψ− (t) − Aχ(t))) = −(iKJψ− (t), χ(t)) − (χ(t), iKJψ− (t)) − (iA∗ χ(t), χ(t)) + (iAχ(t), χ(t)) = −(iKJψ− (t), χ(t)) − (χ(t), iKJψ− (t)) − (2KJK ∗ χ(t), χ(t)). Now we apply the second equation and continue d (χ(t), χ(t)) = −(iKJ(ψ+(t) + 2iK ∗ χ(t)), χ(t)) dt − (χ(t), iKJ(ψ+ (t) + 2iK ∗χ(t))) − (2KJK ∗ χ(t), χ(t)) = −(iKJψ+ (t), χ(t)) − (iKJψ− (t), χ(t)) = −(iJψ+ (t), K ∗ χ(t))E − (iJψ− (t), K ∗ χ(t))E = −(iJψ+ (t), (ψ− (t) − ψ+ (t))/2i)E − (iJψ− (t), (ψ− (t) − ψ+ (t))/2i)E 1 1 = (Jψ− (t), ψ− (t))E − (Jψ+ (t), ψ+ (t))E . 2 2 Taking into account that ψ± (t) ∈ L2[0,τ0 ] (E) and χ(t) is continuously diﬀerentiable, we integrate both sides from 0 to τ ∈ [0, τ0 ], and multiply by 2 to obtain (5.2). Given an input vector ψ− = ϕ− eizt ∈ E, we seek solutions to the system (5.1) as an output vector ψ+ = ϕ+ eizt ∈ E and a state-space vector χ(t) = xeizt ∈ H. Substituting the expressions for ψ± (t) and χ(t) allows us to cancel exponential terms and convert the system (5.1) to the stationary form (A − zI)x = KJϕ− , z ∈ ρ(A), (5.3) ϕ+ = ϕ− − 2iK ∗x, which is called the Livˇsic canonical system. Here ϕ− ∈ E is an input vector, ϕ+ ∈ E is an output vector, and x is a state-space vector in H. The spaces H and E are called state and input-output, and the operators A, K, J are state-space, channel, and directing, respectively. The relation KJK ∗ = Im A

(5.4)

Ran(Im A) ⊆ Ran(K).

(5.5)

implies

5.1. The Livˇsic canonical system and the Brodski˘i theorem

121

The subspace Ran(K) is called the channel subspace. Brieﬂy the Livˇsic canonical system (5.3) can be written as an array2 A K J Θ= , (5.6) H E which we will sometimes refer to as a canonical system. The following theorem is due to Brodski˘i [89]. Theorem 5.1.2. If A is a bounded linear operator acting in a separable Hilbert space H, and G is any subspace containing Ran(Im A), then there exists a canonical system of the form (5.6) for which A is a state-space operator and G is the channel subspace. Proof. The operator Im A maps Ran(Im A) into itself and annihilates its orthogonal complement. Consider the spectral decomposition b Im A Ran(Im A) = t dΣ(t). a

Putting

b |t|

K0 =

1/2

b dΣ(t),

J0 =

a

we get

J0 = J0∗ ,

J02 = I,

sign t dΣ(t), a

K0 J0 K0∗ = Im A Ran(Im A).

Next, we construct the orthogonal sum E = Ran(Im A) ⊕ H1 ⊕ H2 , where H1 and H2 are Hilbert spaces whose dimensions coincide with the dimension of the space H0 = G Ran(Im A). Suppose that U is some isometric mapping of H2 onto H1 and that K1 is a bounded linear mapping of H1 into H2 with dense range in H0 . Then K2 = K1 U ∈ [H2 , H1 ] and K2 K2∗ h = K1 K1∗ h, h ∈ H0 . Let K ∈ [E, H] and J ∈ [E, E] be deﬁned as follows: ⎧ ⎧ ⎨ K0 g, g ∈ Ran(Im A), ⎨ J0 g, g ∈ Ran(Im A), Kg = Jg = K1 g, g ∈ H1 , g, g ∈ H1 , ⎩ ⎩ K2 g, g ∈ H2 , −g, g ∈ H2 . It is easy to see that J = J ∗ , J 2 = IE , and ⎧ ⎨ K0∗ h, ∗ K h= (K1∗ + K2∗ )h, ⎩ 0, 2 In

Ran(K) is dense in G. Since h ∈ Ran(Im A), h ∈ H0 , h ⊥ G,

operator theory the array (5.6) is also referred as an operator colligation.

122

Chapter 5. The Livˇsic Canonical Systems with Bounded Operators

we have KJK ∗ . Thus

Θ=

A H

K

J E

is the desired system.

A construction of a system based upon a given linear operator A is called an inclusion of A into a system. It is clear that this operation is not unique. It is also not hard to see that if dim Ran(Im A) < ∞, then A can be included into a system Θ with operator K such that Ran(K) = dim Ran(Im A) or into a system with an invertible operator K. The systems with these conditions will play an important role in the next chapters.

5.2 Minimal canonical systems A bounded linear operator A on a Hilbert space H is called prime if H cannot be represented as an orthogonal sum of two subspaces G and G ⊥ ( = 0) with the following properties: 1. G and G ⊥ are invariant to A; 2. A induces a self-adjoint operator in G ⊥ . Theorem 5.2.1. The closed linear span G of the form G = c. l. s.{An (Im A)h, n = 0, 1, . . . ; h ∈ H}

(5.7)

and its orthogonal complement G ⊥ = H G are invariant with respect to A. The operator A induces a prime operator in G, and a self-adjoint operator in G ⊥ . Moreover, the operator A is prime if and only if H = G. Proof. By deﬁnition AG ⊆ G. Therefore A∗ G ⊥ ⊆ G ⊥ . Moreover, since Ran(Im A) ⊂ G, we have G ⊥ ⊆ ker(Im A). Thus Ah = A∗ h for all h ∈ G ⊥ and AG ⊥ ⊆ G ⊥ and A G ⊥ is a self-adjoint operator in G ⊥ . Let A1 := A G. We suppose that G = G1 ⊕ G0 , where: 1) G1 and G0 are invariant with respect to A1 ; 2) A1 induces in G0 a self-adjoint operator. Then A induces in G0 ⊕ G ⊥ a self-adjoint operator, Ran(Im A) ⊂ G1 , and therefore An (Im A) h ∈ G1 , It follows that G0 = {0}.

(n = 0, 1, . . . ; h ∈ H).

Corollary 5.2.2. The space H can be represented in one and only one way in the form of an orthogonal sum of subspaces G and G ⊥ which are invariant with respect to A and in which A induces a prime and a self-adjoint operator, respectively.

5.2. Minimal canonical systems

123

Let us consider a canonical system Θ of the form (5.6) and denote by F the closed linear span of vectors An Kg, i.e., F = c. l. s.{An Kg, n = 0, 1, . . . ; g ∈ E}.

(5.8)

⊥

The subspaces F and F = H F are called principal and excess subspaces, respectively. It easily follows from Theorem 5.2.1 and relation (5.5) that each of the subspaces F and F ⊥ is invariant with respect to A and A∗ , and that A F ⊥ is a self-adjoint in ∈ F ⊥ . A canonical system Θ of the form (5.6) is said to be minimal if F = H and excess or non-minimal otherwise. For a canonical system to be minimal it is suﬃcient that its state-space operator is prime. The converse statement is generally speaking not true. Indeed, letting G = H in Theorem 5.1.2, we ﬁnd that every bounded linear operator may be included in a minimal canonical system. Having a system A K J Θ= , H E we may construct a new canonical system ∗ A K ∗ Θ = H

−J E

,

which is called adjoint to Θ. The state subspaces of the systems Θ and Θ∗ coincide. Indeed, since the subspace F is invariant with respect to A∗ and Ran K ⊆ F , we have A∗ n Kg ∈ F, (n = 0, 1, . . . ; g ∈ H), or F∗ ⊆ F , where F∗ is the principle subspace of Θ∗ of the form (5.8) where A is replaced with A∗ . Similarly one shows that F ⊆ F∗ . Therefore F = F∗ . Lemma 5.2.3. Suppose that

Θ=

A H

K

J E

,

is a canonical system. If the subspace H0 ⊂ H is invariant with respect to A and orthogonal to Ran (K), then it belongs to the excess subspace F ⊥ . Proof. For any vector h ∈ H0 the equation (h, A∗ n Kg) = (An h, Kg) = 0,

(n = 0, 1, . . . ; g ∈ H),

holds, and hence h ⊥ F∗ . Since F = F∗ , we have h ⊥ F, i.e., h ∈ F ⊥ .

Theorem 5.2.4. The Livˇsic canonical system A K J Θ= , H E is non-minimal if and only if there exists a nontrivial subspace H0 ⊂ H which is invariant with respect to A and orthogonal to Ran(K).

124

Chapter 5. The Livˇsic Canonical Systems with Bounded Operators

Proof. If Θ is non-minimal, then F ⊥ is diﬀerent from zero, invariant with respect to A, and orthogonal to Ran (K). The suﬃciency follows from Lemma 5.2.3. Suppose that the system Θ of the form (5.6) is not minimal. Deﬁne AF = A F and AF ⊥ = A F ⊥ . We obtain the systems AF K J AF ⊥ 0 J ΘF = and ΘF ⊥ = , F E F⊥ E which are called respectively the principal and excess parts of the canonical system Θ. It is easy to see that ΘF is a minimal canonical system.

5.3 Couplings of canonical systems First of all we give the deﬁnition of equal canonical systems. Deﬁnition 5.3.1. We say that two canonical systems A K J A Θ= and Θ = H E H

K

J E

are equal, and write Θ = Θ , if H = H , E = E , A = A , K = K , and J = J . Consider two canonical systems A1 K1 J Θ1 = and H1 E

Θ2 =

A2 H2

K2

J E

,

for which the input-output spaces and directing operators coincide, and denote by P1 and P2 the orthoprojections onto H1 and H2 acting in the space H = H1 ⊕ H2 . Introduce the operators A = A1 P1 + A2 P2 + 2iK1 JK2∗ P2 ,

K = K1 + K2 ,

operating respectively in H and from E into H. Since A1 − A∗1 , 2i ∗ ∗ ∗ ∗ A = A1 P1 + A2 P2 − 2iK2 JK1 P1 ,

A2 − A∗2 2i K ∗ = K1∗ P1 + K2∗ P2 ,

K1 JK1∗ =

K2 JK2∗ =

we have A − A∗ = K1 JK1∗ P1 + K2 JK2∗ P2 + K1 JK2∗ P2 + K2 JK1∗ P1 2i = (K1 + K2 )J(K1∗ P1 + K2∗ P2 ) = KJK ∗ .

It follows that Θ=

A H

K

J E

(5.9)

5.3. Couplings of canonical systems

125

is a canonical system. This system is called the coupling of the systems Θ1 and Θ2 . We will write Θ = Θ1 Θ2 . One shows by a direct veriﬁcation that the following formulas hold: (Θ1 Θ2 )Θ3 = Θ1 (Θ2 Θ3 ),

(Θ1 Θ2 )∗ = Θ∗2 Θ∗1 .

(5.10)

If Θ1 Θ2 = Θ1 Θ3 or Θ2 Θ1 = Θ3 Θ1 , then Θ2 = Θ3 . Now we will introduce a notion of a projection of a canonical system. Let us select in the state space H of the system A K J Θ= H E a subspace H0 , and deﬁne in it an operator A0 h = P0 Ah (h ∈ H0 ), where P0 is the orthoprojection onto H0 . Moreover, we construct a mapping K0 = P0 K of the space E into H0 . Since for any h ∈ H0 we have A∗0 h = P0 A∗ h,

K0∗ h = K ∗ h, A − A∗ A0 − A∗0 K0 JK0∗ h = P0 KJK ∗ h = P0 h= h, 2i 2i then the array

Θ0 =

A0 H0

K0

J E

is a canonical system of the form (5.6). The system Θ0 is called the projection of the system Θ onto the subspace H0 and is denoted by Θ0 = prH0 Θ. Observe that the relations prH0 Θ∗ = (prH0 Θ)∗ ,

prH1 Θ = prH1 (prH2 Θ),

(H1 ⊆ H2 ),

follow directly from the deﬁnition of the system projection. If A1 K 1 J A2 K2 J A Θ1 = , Θ2 = , Θ = Θ1 Θ2 = H1 E H2 E H

(5.11)

K

J E

,

then, as is shown by formulas (5.9), Θ1 and Θ2 are projections of the system Θ onto H1 and H2 respectively, and H1 is invariant with respect to A. Conversely, every canonical system Θ is the coupling of its projections A1 K1 J A2 K2 J Θ1 = and Θ2 = , H1 E H2 E onto an arbitrary subspace H1 invariant with respect to A and its orthogonal complement H2 . Indeed, denoting by Pj the orthoprojection onto Hj (j = 1, 2), we get P2 AP1 = 0, P1 A∗ P2 = 0, A − A∗ P1 AP2 = 2iP1 P2 = 2iP1 KJK ∗ P2 = 2iK1 JK2∗ P2 , 2i A = (P1 + P2 )A(P1 + P2 ) = A1 P1 + A2 P2 + 2iK1 JK2∗ P2 .

126

Chapter 5. The Livˇsic Canonical Systems with Bounded Operators

In particular, every canonical system is equal to the coupling of its principal and excess parts. A canonical system Θ0 is called a left (right) divisor of a system Θ if it is a projection of Θ onto a subspace H0 ⊆ H which is invariant with respect to A (A∗ ). Suppose that Θ is a canonical system and that the subspaces 0 = H0 ⊂ H1 ⊂ H2 ⊂ · · · ⊂ Hn = H are invariant with respect to A. Since the subspace Hk−1 is invariant with respect to Ak = A Hk (k = 2, 3, . . . , n − 1), from formula (5.11) we obtain Θ = prH1 Θ prH2 H1 Θ . . . prHn Hn−1 Θ.

(5.12)

Below we will discuss the coupling of minimal canonical systems. Lemma 5.3.2. If a canonical system A Θ= H is the coupling of the systems A1 K1 J Θ1 = H1 E then

K

J E

,

and

Fj⊥ = F ⊥ ∩ Hj ,

Θ2 =

A2 H2

K2

J E

(j = 1, 2).

,

(5.13)

Proof. The subspace F1⊥ is invariant with respect to A and orthogonal to Ran(K1 ). It is also orthogonal to Ran(K), since Ran(K1 ) is the projection of Ran(K) onto H1 . By Lemma 5.2.3, F1⊥ ⊆ F ⊥ ∩ H1 . On the other hand, the subspace F ⊥ ∩ H1 is invariant with respect to A1 and orthogonal to Ran(K1 ). Applying Lemma 5.2.3 to the system Θ1 , we ﬁnd that F ⊥ ∩ H1 ⊆ F1⊥ . Thus equation (5.13) is proved for j = 1. It remains to note that Θ∗ = Θ∗2 Θ∗1 , so that, from what has already ⊥ ⊥ been proved, F∗,2 = F∗⊥ ∩ H2 . Inasmuch as F∗,2 = F2⊥ and F∗⊥ = F ⊥ , we have ⊥ ⊥ F2 = F ∩ H2 . Theorem 5.3.3. If Θ = Θ1 Θ2 · · · Θn , then Fj⊥ = F ⊥ ∩ Hj ,

(j = 1, 2, . . . , n),

where Hj is the state space of the canonical system Θj . ⊥ Proof. In view of Lemma 5.3.2, Fj⊥ = F1,2,...,j ∩ Hj , and ⊥ F1,2,...,j = F ⊥ ∩ (H1 ⊕ H2 ⊕ · · · ⊕ Hj ), ⊥ where F1,2,...,j is the excess subspace of the coupling Θ1 Θ2 · · · Θj . Accordingly,

Fj⊥ = F ⊥ ∩ (H1 ⊕ H2 ⊕ · · · ⊕ Hj ) ∩ Hj = F ⊥ ∩ Hj .

5.3. Couplings of canonical systems

127

As a consequence of Theorem 5.3.3 we obtain the following statement. Theorem 5.3.4. If Θ = Θ1 Θ2 · · · Θn is a minimal canonical system, then all the systems Θj (j = 1, 2, . . . , n) are minimal. One can see that a coupling of minimal canonical systems may turn out to be a non-minimal system. Moreover, the following theorem holds. Theorem 5.3.5. Let A0 be a self-adjoint bounded operator in a separable Hilbert space H0 . Then there exist minimal canonical systems Θ1 and Θ2 such that A0 is a state-space operator of the excess portion of the coupling Θ1 Θ2 . ˜ = H0 ⊕ H, where H is some Hilbert space Proof. Consider the orthogonal sum H whose dimension is equal to the dimension of the space H0 . Suppose that we are given an isometric mapping U of H0 onto H and that the operator A = U A0 U −1 is included in the minimal canonical system A K J Θ= . H E It is not hard to see that the system A˜ K J A0 ˜ Θ = ΘΘ0 = , where Θ0 = ˜ H0 H E

K

J E

,

˜ has the principal part Θ and the excess part Θ0 . The subspace H1 of the space H ˜ consisting of vectors of the form f + U f (f ∈ H0 ) is invariant with respect to A, since ˜ + U f ) = A0 f + AU f = A0 f + U A0 f, (f ∈ H0 ). A(f Accordingly, ˜ = Θ1 Θ 2 , Θ

˜ Θ2 = pr Θ, ˜ H2 = H ˜ H1 ). (Θ1 = prH1 Θ, H2

˜ Then, since H0 = F˜ ⊥ and Hj ∩ H0 = 0 Let F˜ ⊥ be the excess subspace of Θ. (j = 1, 2), by Lemma 5.3.2 Θ1 and Θ2 are minimal canonical systems. Now we will deﬁne and discuss the spectrum of the coupling of canonical systems. Theorem 5.3.6. Suppose that the Livˇsic canonical system A K J Θ= , H E is the coupling of the systems A1 K1 J Θ1 = H1 E

and

Θ2 =

A2 H2

K2

J E

.

128

Chapter 5. The Livˇsic Canonical Systems with Bounded Operators

If λ is a regular point for the operators A1 and A2 , then it is regular for the operator A as well, while (A − λI)−1 = (A1 − λI)−1 P1 + (A2 − λI)−1 P2 − 2i(A1 − λI)−1 K1 JK2∗ (A2 − λI)−1 P2 ,

(5.14)

where P1 and P2 are orthoprojections onto H1 and H2 . Proof. From the ﬁrst of equalities (5.9) we have A − λI = (A1 − λI)P1 + (A2 − λI)P2 + 2iK1 JK2∗ P2 .

(5.15)

Now it is easy to verify that the right side of (5.14) is an operator which is both left and right invertible for the operator (5.15). The theorem is proved. We agree that by the spectrum of a canonical system we will mean the spectrum of its state-space operator. Theorem 5.3.6 means that the spectrum of the coupling of two canonical systems is contained in the union of the spectra of the factors. Lemma 5.3.7. Suppose that H0 is an invariant subspace of the bounded linear operator A acting in the space H, and O is a bounded open connected piece of the complex plane C, all of whose points are regular for A. If at least one point λ0 ∈ O is regular for the operator A0 induced by A in H0 , then all the points of O have the same property. Proof. We note in preparation that the point λ ∈ O is regular for the operator A0 if and only if (A − λI)−1 H0 ⊆ H0 . Let Oλ0 be a suﬃciently small neighborhood of λ0 contained in O and such that the Taylor expansion Rλ = Rλ0 + (λ − λ0 )Rλ2 0 + (λ − λ0 )2 Rλ3 0 + · · · ,

λ ∈ Oλ0 , Rλ = (A − λI)−1

uniformly converges. Then Rλ0 H0 ⊆ H0 implies Rλ H0 ⊆ H0 for all λ ∈ Oλ0 . Hence (Rλ h, f ) = 0 for all h ∈ H0 , all f ∈ H H0 , and all λ ∈ Oλ0 . Because the set O is connected and open, and the function (Rλ h, f ) is holomorphic on O, we get (Rλ h, f ) = 0 for all λ ∈ O. So, Rλ H0 ⊆ H0 for all λ ∈ O.

The following is an easy consequence of Theorem 5.3.6, Lemma 5.3.7, and equation (5.10). Theorem 5.3.8. Suppose that Θ is the coupling of the canonical systems Θ1 and Θ2 . If the set of regular points of the operator A is connected, then the spectrum of Θ is equal to the union of the spectra of Θ1 and Θ2 .

5.4. Transfer functions of canonical systems

129

Now we introduce a notion of unitarily equivalent canonical systems. We recall that an operator A1 acting in a space H1 is said to be unitarily equivalent to the operator A2 in H2 if there exists an isometric mapping U of H1 onto H2 such that U A1 = A2 U . We say that the canonical system A1 K1 J Θ1 = , H1 E is unitarily equivalent to the system A2 Θ2 = H2

K2

J E

,

if there exists an isometric mapping U of the space H1 onto H2 such that U A1 = A2 U,

U K 1 = K2 .

(5.16)

Obviously, the relation of unitary equivalence is reﬂexive, symmetric, and transitive. It is also easy to see that if one of two unitarily equivalent canonical systems is minimal, then so is the other. If A K J Θ= H E is a minimal canonical system, and if for some unitary operator U in H the equations U A = AU, U K = K are satisﬁed, then U An Kg = An Kg,

(n = 0, 1, . . . ; g ∈ H),

which means that U = IH . Using this remark, we arrive at the following conclusion: if A1 K 1 J A2 K 2 J Θ1 = and Θ2 = H1 E H2 E are unitarily equivalent minimal canonical systems, then the isometric mapping satisfying conditions (5.16) is deﬁned uniquely.

5.4 Transfer functions of canonical systems Consider the Livsi˘c canonical system A Θ= H

K

J E

.

130

Chapter 5. The Livˇsic Canonical Systems with Bounded Operators

Taking into account (5.3), we call the function of complex variable z, WΘ (z) = I − 2iK ∗ (A − zI)−1 KJ,

(5.17)

the transfer function of the canonical system Θ. It is easy to see that if ϕ− ∈ E is an input vector and ϕ+ ∈ E is an output vector of the system Θ, then ϕ+ = WΘ (z)ϕ− . The function WΘ (z) is obviously deﬁned and holomorphic on the set ρ(A) of regular points of the operator A, and its values are bounded linear operators acting in the input-output space E. Theorem 5.4.1. Suppose that Θ=

A H

K

J E

is the coupling of the Livˇsic canonical systems A1 K1 J A2 Θ1 = and Θ2 = H1 E H2

K2

J E

.

If z is a regular point for the operators A1 and A2 , then WΘ1 Θ2 (z) = WΘ1 (z)WΘ2 (z).

(5.18)

Proof. Denote by Pj (j = 1, 2) the orthoprojection onto Hj acting in the space H = H1 ⊕ H2 . Applying Theorem 5.3.6, we obtain WΘ1 Θ2 (z) = I − 2iK ∗ (A − zI)−1 KJ = I − 2iK ∗ P1 (A1 − zI)−1 P1 KJ − 2iK ∗P2 (A2 − zI)−1 P2 KJ + (2i)2 K ∗ P1 (A1 − zI)−1 K1 JK2∗ (A2 − zI)−1 P2 KJ = I − 2iK1∗ (A1 − zI)−1 K1 J − 2iK2∗(A2 − zI)−1 K2 J + (2i)2 K1∗ (A1 − zI)−1 K1 JK2∗ (A2 − zI)−1 K2 J = [I − 2iK1∗ (A1 − zI)−1 K1 J][I − 2iK2∗ (A2 − zI)−1 K2 J] = WΘ1 (z)WΘ2 (z).

Corollary 5.4.2. If the state-space operator of the canonical system A K J Θ= H E has invariant subspaces 0 = H0 ⊂ H1 ⊂ · · · ⊂ Hn = H and z is a regular point for the state-space operators of the canonical systems Θj = prHj Hj−1 Θ (j = 1, 2, . . . , n), then WΘ (z) = WΘ1 (z)WΘ2 (z) . . . WΘn (z).

(5.19)

5.4. Transfer functions of canonical systems

131

The proof follows from formula (5.12). We will now state the criteria for unitary equivalence of canonical systems. First we note that if the canonical systems A1 K 1 J A2 K 2 J Θ1 = and Θ2 = H1 E H2 E are unitarily equivalent, then the set ρ(A1 ) of regular points of A1 coincides with the set ρ(A2 ) of regular points of A2 , and WΘ1 (z) = WΘ2 (z) (z ∈ ρ(A1 )). Indeed, in view of (5.16), WΘ2 (z) = I − 2iK2∗ (A2 − zI)−1 K2 J = I − 2iK1∗ U −1 [U (A1 − zI)−1 U −1 ]U K1 J = I − 2iK1∗ (A1 − zI)−1 K1 J = WΘ1 (z). Theorem 5.4.3. Suppose that A1 K 1 Θ1 = H1

J E

and

Θ2 =

A2 H2

K2

J E

are minimal canonical systems. If in some neighborhood G of inﬁnity WΘ1 (z) = WΘ2 (z), then Θ1 and Θ2 are unitarily equivalent. Proof. It is given in the statement of the theorem that K1∗ (A1 − zI)−1 K1 = K2∗ (A2 − zI)−1 K2 ,

(z ∈ G).

Then for z, ζ ∈ G and j = 1, 2 we have ¯ −1 = (A∗ − ζI) ¯ −1 [(A∗ − ζI) ¯ − (Aj − zI)] × (Aj − zI)−1 (Aj − zI)−1 − (A∗j − ζI) j j ¯ ∗ − ζI) ¯ −1 (Aj − zI)−1 − 2i(A∗ − ζI) ¯ −1 Kj JK ∗ (Aj − zI)−1 , = (z − ζ)(A j

j

j

which means that ¯ ∗ (A∗ − ζI) ¯ −1 (A1 − zI)−1 K1 = K ∗ (A1 − zI)−1 K1 (z − ζ)K 1 1 1 ∗ −1 ∗ ¯ ¯ −1 K1 JK ∗ (A1 − zI)−1 K1 − K (A1 − ζI) K1 + 2iK (A1 − ζI) 1 1 ∗ −1 ∗ ¯ −1 K2 = K2 (A2 − zI) K2 − K2 (A2 − ζI) ¯ −1 K2 JK ∗ (A2 − zI)−1 K2 + 2iK2∗ (A2 − ζI) 2 ∗ ∗ −1 ¯ ¯ = (z − ζ)K2 (A2 − ζI) (A2 − zI)−1 K2 .

1

This yields ¯ −1 (A1 − zI)−1 K1 = K ∗ (A∗ − ζI) ¯ −1 (A2 − zI)−1 K2 , K1∗ (A∗1 − ζI) 2 2 Using the expansion A2j I Aj (Aj − zI)−1 = − − 2 − 3 − . . . , z z z

(|z| > Aj ),

(z, ζ ∈ G).

132

Chapter 5. The Livˇsic Canonical Systems with Bounded Operators

we arrive at the equation n m n (Am 1 K1 g, A1 K1 g ) = (A1 K1 g, A2 K2 g ), (m, n = 0, 1, . . . ; g, g ∈ E).

(5.20)

Denote by Gj (j = 1, 2) the linear span of vectors of the form Am j Kj g (m = 0, 1, . . .; g ∈ E) and consider mapping U of the set G1 onto G2 which assigns to )l )l each vector of the form n=0 An1 K1 gn the vector n=0 An2 K2 gn . In view of (5.20) the mapping U is isometric. Since G1 and G2 are dense in H1 and H2 respectively, U can be extended by continuity to an isometric mapping of the space H1 onto H2 . Denoting the extended mapping again by U , we obtain the equation U An1 K1 = An2 K2 ,

(n = 0, 1, . . .).

Thus U K1 = K2 , and, moreover, U A1 An1 K1 = A2 An2 K2 = A2 U An1 K1

(n = 0, 1, . . .),

i.e., U A1 = A2 U . Corollary 5.4.4. Suppose that A1 K 1 Θ1 = H1

J E

and

Θ2 =

A2 H2

K2

J E

are minimal canonical systems. If in a neighborhood G of inﬁnity WΘ1 (z) = WΘ2 (z), then ρ(A1 ) = ρ(A2 ) and WΘ1 (z) ≡ WΘ2 (z), (z ∈ ρ(A1 )). Note that in Theorem 5.4.3 we cannot get along without the requirement of minimality of the systems. For example, let us consider a non-minimal system Θ with principal part ΘF and excess part ΘF ⊥ . Obviously Θ and ΘF cannot be unitarily equivalent. At the same time WΘF ⊥ (z) ≡ I, and, by Theorem 5.4.1, there exists a neighborhood of inﬁnity in which WΘ (z) = WΘF (z)WΘF ⊥ (z) = WΘF (z). Now we will discuss analytic properties of a transfer function. Suppose that Θ is some canonical system. Then ¯ −1 K, z, ζ ∈ ρ(A). (5.21) WΘ (z)JWΘ∗ (ζ) − J = 2i(ζ¯ − z)K ∗ (A − zI)−1 (A∗ − ζ) Indeed, since ¯ −1 = (A − zI)−1 [(A∗ − ζI) ¯ − (A − zI)](A∗ − ζ) ¯ −1 (A − zI)−1 − (A∗ − ζ) ¯ ¯ −1 − 2i(A − zI)−1 KJK ∗ (A∗ − ζ) ¯ −1 , = (z − ζ)(A − zI)−1 (A∗ − ζ)

5.4. Transfer functions of canonical systems

133

we have WΘ (z)JWΘ∗ (ζ) − J ¯ −1 K] − J = [I − 2iK ∗ (A − zI)−1 KJ ]J[I − 2iJK ∗ (A∗ − ζI) ¯ −1 + 2i(A − zI)−1 KJK ∗ (A∗ − ζI) ¯ −1 ]K = −2iK ∗[(A − zI)−1 − (A∗ − ζI) ¯ −1 K. = 2i(ζ¯ − z)K ∗ (A − zI)−1 (A∗ − ζ) In particular, if z, z¯ ∈ ρ(A), then WΘ (z)JWΘ∗ (¯ z ) − J = 0.

(5.22)

Moreover, at each point z ∈ ρ(A), WΘ (z)JWΘ∗ (z) − J = 4Im zK ∗ (A − zI)−1 (A∗ − z¯I)−1 K, which means that WΘ (z)JWΘ∗ (z) − J

≥ 0 (Im z > 0, z ∈ ρ(A)),

(5.23)

−J

≤ 0 (Im z < 0, z ∈ ρ(A)).

(5.24)

WΘ (z)JWΘ∗ (z) Similarly, using the identity

¯ −1 = (z − ζ)(A ¯ ¯ −1 (A − zI)−1 − (A∗ − ζ) − zI)−1 (A∗ − ζ) ¯ −1 , − 2i(A − zI)−1 KJK ∗ (A∗ − ζ) we obtain ¯ −1 (A − zI)−1 KJ, WΘ∗ (ζ)JWΘ (z) − J = 2i(ζ¯ − z)JK ∗ (A∗ − ζI)

(5.25)

which shows that WΘ∗ (¯ z )JWΘ (z) − J = 0,

z, z¯ ∈ ρ(A),

(5.26)

and WΘ∗ (z)JWΘ (z) − J

≥ 0 (Im z > 0, z ∈ ρ(A)),

−J

≤ 0 (Im z < 0, z ∈ ρ(A)).

WΘ∗ (z)JWΘ (z)

If z, z¯ ∈ ρ(A), then by (5.22) and (5.26) the operator WΘ (z) has a bounded inverse WΘ−1 (z) = JWΘ∗ (¯ z )J.

(5.27)

Since there exists a neighborhood of the point at inﬁnity in which the resolvent of A decomposes into a series I A (A − zI)−1 = − − 2 − · · · , z z

134

Chapter 5. The Livˇsic Canonical Systems with Bounded Operators

which converges in norm, it follows that in this same neighborhood 2i ∗ K KJ + · · · . z Now we will introduce a linear-fractional transformation of the transfer function. We assign to a canonical system Θ the operator-function WΘ (z) = I +

VΘ (z) = K ∗ (Re A − zI)−1 K,

Re A = (1/2)(A + A∗ ).

(5.28)

The function VΘ (z) is holomorphic on the set ρ(Re A), and its values, as those of the function WΘ (z), are operators acting in E. We note that ρ(Re A) contains all non-real points. From the equality VΘ (z) − VΘ∗ (z) = 2iIm zK ∗ (Re A − z¯I)−1 (Re A − zI)−1 K, we obtain that VΘ (z) − VΘ∗ (z) ≥0 2i and

(Im z > 0),

VΘ (z) = VΘ∗ (z),

VΘ (z) − VΘ∗ (z) ≤ 0 (Im z < 0), 2i

(Im z = 0, z ∈ ρ(Re A)).

(5.29)

At each point of the set ρ(A) ∩ ρ(Re A) there exist the operators (WΘ (z) + I)−1 and (I + iVΘ (z)J)−1 , while VΘ (z) = i(WΘ (z) + I)−1 (WΘ (z) − I)J = i(WΘ (z) − I)(WΘ (z) + I)−1 J, (5.30) WΘ (z) = (I + iVΘ (z)J)−1 (I − iVΘ (z)J) = (I − iVΘ (z)J)(I + iVΘ (z)J)−1 . (5.31) Indeed, since (Re A − zI)−1 − (A − zI)−1 = i(A − zI)−1 Im A(Re A − zI)−1 , we have K ∗ (Re A − zI)−1 K − K ∗ (A − zI)−1 K = iK ∗ (A − zI)−1 KJK ∗ (Re A − zI)−1 K. Thus

i 1 VΘ (z) + (I − WΘ (z))J = (I − WΘ (z))VΘ (z), 2 2

so that (WΘ (z) + I)(I + iVΘ (z)J) = 2I.

(5.32)

Similarly, starting from the relations (Re A − zI)−1 − (A − zI)−1 = i(Re A − zI)−1 Im A(A − zI)−1 , we get (I + iVΘ (z)J)(WΘ (z) + I) = 2I.

(5.33)

In view of (5.32) and (5.33) each of the operators WΘ (z) + I and I + iVΘ (z)J has a bounded inverse for z ∈ ρ(A) ∩ ρ(Re A) . Formulas (5.30) and (5.31) follow easily from (5.32).

5.5. Class ΩJ and its realization

135

5.5 Class ΩJ and its realization In what follows all the matrices will be considered as linear operators in the space Cn when it deems necessary. The following theorem holds. Theorem 5.5.1. Let V (z) be an (n × n) matrix-valued function in a Hilbert space Cn that has an integral representation3 b 1 V (z) = dσ(t), (5.34) a t−z where σ(t) is a non-negative, non-decreasing (n × n) matrix-function in Cn deﬁned on a ﬁnite interval [a, b]. Then V (z) can be realized in the form V (z) = i(WΘ (z) − I)(WΘ (z) + I)−1 J,

(5.35)

where WΘ (z) is a transfer function of a minimal canonical system of the form (5.6), z, (Im z = 0) is such that WΘ (z) is deﬁned, and J = J ∗ = J −1 is an arbitrary pre-assigned directing operator. Proof. Let C[a,b] (Cn ) be the the set of all continuous on [a, b], Cn -valued functions 2 (Cn , dσ) be the completion of C[a,b] (Cn ) with respect to the semi-inner and let L [a,b] product b (f , g ) = f(t)dσ(t)g ∗ (t). a

Then the Hilbert space

L2[a,b] (Cn , dσ)

is the quotient space

2 (Cn , dσ)/ ker(p), L [a,b] where

= f = 0}. 2 (Cn , dσ) : p(f) ker(p) = {f ∈ L [a,b]

Consider in L2[a,b] (Cn , dσ) the operator

b

(Af)(t) = tf(t) + 2i

f(t)dσ(t)J,

(5.36)

a

where f(t) = (f1 (t), . . . , fn (t)) is a row-vector function L2[a,b] (Cn , dσ). Here the directing operator J = [jαβ ] is ((n × n)) signature matrix in Cn with the property that J = J ∗ = J −1 . Obviously, (Im A) f =

n # α,β=1

3 Functions

(f, hα )jαβhβ =

b

f(t)dσ(t)J = KJK ∗ f.

a

of this type belong to the class of Herglotz-Nevanlinna functions that will be studied in detail in Chapter 6 and further on.

136

Chapter 5. The Livˇsic Canonical Systems with Bounded Operators

Also, Kg =

n #

K ∗ f = (f, h1 ), . . . , (f, h1 ) ,

(g, hα )Cn hα (t), g ∈ Cn ,

α=1

(·, ·)Cn is the dot product in Cn , and hk = hk (t) (k = 1, . . . , n) is a vector whose k-th component is 1 and the rest are zeros. Consider the system Θ with the state-space operator A of the form (5.36) A K J Θ= , (5.37) L2[a,b] (Cn , dσ) Cn Since the real part of the operator A is an operator of multiplication by independent variable (Re Af )(t) = tf (t), then b 1 VΘ (z) = K ∗ (Re A − zI)−1 K = dσ(t) = V (z). t − z a Applying (5.30), we get b 1 V (z) = dσ(t) = VΘ (z) = i(WΘ (z) − I)(WΘ (z) + I)−1 J, a t−z where WΘ (z) is a transfer function of a system Θ of the form (5.37). We can also show that this system Θ is minimal because its state-space operator A is prime. In order to do that we will use Corollary 5.2.1 and show that c. l. s.{Akhα , k = 0, 1, . . . ; α = 1, . . . , n} = L2[a,b] (Cn , dσ). Consider the vector function hα (z, t) = hα (WΘ (z) + I) , 2i(t − z)

|z| > A, Im z = 0.

Then b a

hα (WΘ (z) + I) dσ(t) hα (z, t)dσ(t)J = hα (WΘ (z) + I) J= V (z)J 2i t−z 2i b

a

1 = hα (WΘ (z) − I). 2 Hence, Ahα (z, t) = zhα (z, t) + (t − z)hα (z, t) + i

b

hα (z, t)dσ(t)J a

hα (WΘ (z) + I) hα (WΘ (z) − I) = zhα (z, t) + − 2i 2i = z hα (z, t) − ihα ,

5.5. Class ΩJ and its realization

137

or (A − zI)−1hα = ihα (z, t) =

hα (WΘ (z) + I) . 2(t − z)

(5.38)

= 0, k = 0, 1, . . . ; α = 1, . . . , n. Then ((A − zI)−1hα , f) = 0, for Let (Akhα , f) α = 1, . . . , n and (5.38) implies that b dσ(t)f∗ (t) hα (WΘ (z) + I) = 0. t−z a Since det(WΘ (z) + I) ≡ 0, then the integral term in the above equation is zero. We will show that this implies that f(t) = 0. First we observe that ∞ k ∞ # 1 1 1 1 # t tk = − = − =− , t−z z 1 − t/z z z z k+1 k=0

k=0

for large enough |z|. Thus, b b ∞ # dσ(t)f∗ (t) 1 0= =− tk dσ(t)f∗ (t), t−z z k+1 a a k=0

which yields

b

tkhα dσ(t)f∗ (t) = 0,

∀k = 0, 1, . . . ; α = 1, . . . , n.

(5.39)

a

According to the Weierstrass approximation theorem, every continuous function on a ﬁnite interval [a, b] can be uniformly approximated by a polynomial function. Consequently, the linear span of monomials {tk }, k = 0, 1, . . . is dense in L2[a,b] (C, dσ). Therefore, the linear span of vectors {tkhα , k = 0, 1, . . .; α = 1, . . . , n} is dense in L2 (Cn , dσ) and hence (5.39) yields f(t) = 0. [a,b]

Remark 5.5.2. It is not hard to see that Theorem 5.5.1 can also be proved for an operator-valued function V (z) of the form (5.34) whose values are bounded linear operators in an arbitrary ﬁnite-dimensional Hilbert space E. The following theorem is well known [89]. Theorem 5.5.3. For the function V (z), whose values are bounded linear operators in a ﬁnite-dimensional Hilbert space E, to admit the representation b V (z) = a

dF (s) , s−z

outside the ﬁnite interval [a, b] of the real axis and F (s) (a ≤ x ≤ b) is a nonnegative, non-decreasing and bounded operator-function, it is necessary and suﬃcient that V (z) satisﬁes the following conditions:

138

Chapter 5. The Livˇsic Canonical Systems with Bounded Operators

1. V (z) is holomorphic outside [a, b]; 2. V (∞) = 0; 3. in the upper half-plane V (z) has a non-negative imaginary part; 4. V (z) takes self-adjoint values on the intervals (−∞, a) and (b, +∞) of the real axis. Suppose that the linear operator J, acting in a Hilbert space E, satisﬁes the conditions J = J ∗ and J 2 = I. We will say that the function of a complex variable W (z), whose values are bounded linear operators in E, belongs to the class ΩJ if it has the following properties: 1. W (z) is holomorphic in some neighborhood GW of the point at inﬁnity; 2. limz→∞ W (z) − I = 0; 3. for all z ∈ GW the operator W (z) + I has a bounded inverse, while the operator-function V (z) = i(W (z) + I)−1 (W (z) − I)J = i(W (z) − I)(W (z) + I)−1 J,

(5.40)

satisﬁes the conditions of Theorem 5.5.3. In view of(5.40) we have (W (z) + I)(I + iV (z)J) = (I + iV (z)J)(W (z) + I) = 2I,

(z ∈ GW ).

Thus at each z ∈ GW the operator (I + iV (z)J) has a bounded inverse, and W (z) = (I + iV (z)J)−1 (I − iV (z)J) = (I − iV (z)J)(I + iV (z)J)−1 .

(5.41)

It can be easily shown that a transfer function of any canonical system Θ lies in the class ΩJ . Let W (z) be a function in a ﬁnite-dimensional Hilbert space E. Then the following theorem holds. Theorem 5.5.4. If the operator-function W (z) belongs to the class ΩJ , then there exists the Livˇsic canonical system Θ with directing operator J such that WΘ (z) ≡ W (z) in some neighborhood of the point at inﬁnity. Proof. It follows from conditions (1)-(3) that the function V (z) satisﬁes the requirements of Theorem 5.5.3 and hence there exists a non-negative non-decreasing function F (t) (−∞ < a ≤ t ≤ b < +∞) in the ﬁnite-dimensional Hilbert space E such that b dF (t) V (z) = , (z ∈ / [a, b]). t−z a

Consequently, we can apply Remark 5.5.2 in conjunction with Theorem 5.5.1 to obtain the system Θ of the form (5.6) such that (5.35) holds. Then, since V (z) =

5.6. Finite-dimensional state-space case

139

VΘ (z), by formulas (5.41) and (5.31) there exists a neighborhood of the point at inﬁnity in which W (z) = (I +iV (z)J)−1 (I −iV (z)J) = (I −iV (z)J)(I +iV (z)J)−1 = WΘ (z).

Corollary 5.5.5. If the function W (z) belongs to the class ΩJ , then in some neighborhood of inﬁnity it satisﬁes relations (5.22), (5.23), (5.24), and (5.26)–(5.27). Corollary 5.5.6. There exists a neighborhood of inﬁnity in which the function W (z) decomposes into a series of the form W (z) = I +

2i HJ + · · · , z

where H ≥ 0. If H = 0, then W (z) ≡ I. Corollary 5.5.7. Along with each two operator-functions the class ΩJ contains their coupling. If W1 (z) ∈ ΩJ , W2 (z) ∈ ΩJ , and W1 (z)W2 (z) = I (z ∈ GW1 ∩ GW2 ), then W1 (z) ≡ W2 (z) ≡ I. The proof of the ﬁrst statement follows from Theorem 5.4.1, and that of the second from Corollary 5.5.6. We have already mentioned that the transfer operator-function of a nonminimal canonical system and its principal part coincide in some neighborhood of the point at inﬁnity. This leads to the following result. Theorem 5.5.8. If W (z) ∈ ΩJ , then there exists a minimal canonical system Θ with a direction operator J such that, in some neighborhood of the point at inﬁnity, WΘ (z) ≡ W (z).

5.6 Finite-dimensional state-space case Suppose the system

Θ=

A H

K

J E

is such that both state-space H and input-output space E are ﬁnite-dimensional with dim H = n and dim E = m. We will refer to this type of systems as the Livˇsic canonical ﬁnite-dimensional systems. Clearly, any operator A in a ﬁnitedimensional space H can be included in a ﬁnite-dimensional system system as a state-space operator. Suppose that J is a linear operator in E, (dim E < ∞) such that J = J ∗ = −1 J . We say that the function W (z) with values in E belongs to the class Ωm J if: 1. W (z) is holomorphic in a region GW that is a complex plane with a ﬁnite number of points (poles of W (z)) removed; 2. limz→∞ W (z) − I = 0;

140

Chapter 5. The Livˇsic Canonical Systems with Bounded Operators

3. W ∗ (z)JW (z) − J ≥ 0, (Im z > 0, z ∈ GW ); 4. W ∗ (z)JW (z) − J = 0, (Im z = 0, z ∈ GW ). The class Ωm J contains the product of any two if its elements. It is not hard to see that the transfer function of a ﬁnite-dimensional system Θ belongs to this class. Theorem 5.6.1. Every function W (z) ∈ Ωm sic J is a transfer function of the Livˇ minimal canonical ﬁnite-dimensional system. Proof. By Theorem 5.5.4 there exists a minimal system Θ for which ρ(A) = GW , and WΘ (z) = W (z), (z ∈ GW ). It remains to prove that H is ﬁnite-dimensional. It (0) was proved earlier in this chapter that at the points of the set GA = ρ(A)∩ρ(Re A) the operator WΘ (z) + I has an inverse, and VΘ (z) = K ∗ (Re A − zI)−1 K = i(WΘ (z) + I)−1 (WΘ (z) − I)J. At the same time, since all scalar products of the form (WΘ (z)f, g) are rational functions, the operator WΘ (z)+I can fail to have an inverse only at a ﬁnite number of points, and only those points may be in the spectrum of the operator Re A. Accordingly, the orthogonal resolution of identity Σ(s) corresponding to Re A is a piece-wise constant function with a ﬁnite number of jumps, and therefore the closure G0 of the linear span of vectors if the form Σ(s)Kg, (−∞ < s < ∞, g ∈ E) is ﬁnite dimensional. On the other hand, G0 is invariant with respect to A, since s AΣ(s)Kg = (Re A + iIm A)Σ(s)Kg = t dΣ(t)Kg + iKJK ∗ Σ(s)Kg ∈ G0 , −∞

which means that A Kg ∈ G0 , (n = 0, 1, . . .; g ∈ E). In view of the minimality of Θ, G0 = H. n

Now we consider decomposition of functions of the class Ωm J into factors. Lemma 5.6.2. Suppose that H is an n-dimensional space and that A is a linear operator in H. Then there exist subspaces 0 =H0 ⊂ H1 ⊂ H2 ⊂ · · · ⊂ Hn−1 ⊂ Hn = H, (dim Hk = k, k = 1, 2, . . . , n − 1),

(5.42)

invariant with respect to A. Proof. Construct in H an orthonormal basis e1 , . . . , en such that the vector e1 is an eigenvector for A, and the vector ek+1 (k = 1, . . . , n − 1) is an eigenvector for the operator Pk A, where Pk is the orthoprojection operator onto the orthogonal complement to the linear span Hk of vectors e1 , . . . , ek . Then Ae1 = z1 e1 , Ae2 = α21 e1 + z2 e2 , ·················· Aen = αn1 e1 + αn2 e2 + · · · + zn en ,

5.6. Finite-dimensional state-space case

141

which means that the subspaces Hk (k = 1, . . . , n − 1) are invariant with respect to A. Theorem 5.6.3. Suppose that the function W (z) ∈ Ωm J . Then it may be represented in the form 2iσ1 2iσ2 2iσn W (z) = I + P1 J I+ P2 J · · · I + Pn J , (5.43) z − z1 z − z2 z − zn where zj are complex numbers, σj are positive numbers, and the Pj are onedimensional orthogonal projection operators in E satisfying the condition Pj JPj =

Im zj Pj , σj

j = 1, . . . , n.

(5.44)

Proof. It was proved in Theorem 5.6.1 that the function W (z) is a transfer function for some minimal ﬁnite-dimensional system Θ. By Lemma 5.6.2 the space H contains a subspace (5.42) invariant with respect to A. Using formula (5.19) we have W (z) = WΘ1 (z)WΘ2 (z) · · · WΘn (z),

Θj = prHj Hj−1 Θ, (j = 1, . . . , n).

Suppose that Aj and Kj are the state-space and channel operators of the system Θj . Since the operator Aj acts in the one-dimensional space Hj Hj−1 , we have WΘj (z) = I − 2iKj∗ (Aj − zI)−1 Kj J = I +

2i K ∗ Kj J. z − zj j

In view of Theorem 5.3.4 the operator Kj∗ Kj is not zero. Accordingly, Kj∗ Kj = σj Pj , where σj > 0 and Pj is the orthoprojection onto the one-dimensional subspace. Moreover, Pj JPj =

1 ∗ 1 Im zj ∗ Im zj K Kj JKj∗ Kj = 2 Kj∗ Im Aj Kj = Kj Kj = Pj . σj2 j σj σj2 σj

The theorem is proved. We note that each function of the form 2iσ0 Im z0 W0 (z) = I + P0 J, σ0 > 0, P0 JP0 = P0 , z − z0 σ0

(5.45)

where P0 is an orthoprojection operator onto a one-dimensional subspace, belongs to the class Ωm J . Indeed, from the easily veriﬁed equation W0∗ (z)JW0 (z) − J =

4σ0 Im z JP0 J, |z − z0 |2

142

Chapter 5. The Livˇsic Canonical Systems with Bounded Operators

it follows that the function W0∗ (z)JW0 (z) − J is positive in the upper half-plane and equal to zero on the real axis. Thus, the class Ωm J coincides with the collection of all possible products of simplest factors of the form (5.45). It also directly follows from Theorem 5.6.3 and relations (5.43) and (5.44) that if dim E = 1 and J = 1, then any function W (z) from the class Ω11 takes the form n * z − z¯k W (z) = , Im zk > 0, k = 1, . . . , n, (5.46) z − zk k=1

that is applied to a vector of a one-dimensional space E. Let us consider the matrix ⎛ ⎞ α1 + 2i β12 iβ1 β2 · · · iβ1 βn ⎜ 0 α2 + 2i β22 · · · iβ2 βn ⎟ =⎜ ⎟, A ⎝ ⎠ · · · · · · i 2 0 0 · · · αn + 2 βn where {αk }nk=1 are real numbers and {βk }nk=1 calculations one ﬁnds that ⎛ 2 β1 β1 β2 · 1⎜ β1 β2 β22 · ⎜ Im A = ⎝ · · · 2 β1 βn β2 βn ·

(5.47)

are positive numbers. By direct · · · ·

· · · ·

⎞ β1 βn β2 βn ⎟ ⎟. · ⎠ βn2

as an operator A : Cn → Cn that applies to column-vectors We can consider A n is a prime operator. Then by Theorem 5.1.2, A can be included in C . Clearly A into a system K 1 A ΘA = , Cn C where K c = c g and

⎛

⎞ β1 ⎜ β2 ⎟ ⎜ ⎟ 1 ⎜ · ⎟ ⎟. g = √ ⎜ ⎜ 2⎜ · ⎟ ⎟ ⎝ · ⎠ βn f = KK ∗ f = f, g g , f ∈ Cn . Applying Theorem 5.6.3 and (5.46) Clearly, Im A we obtain the following formula for the transfer function of ΘA : WΘA (z) =

n * z − αk + iβk2 /2 . z − αk − iβk2 /2

k=1

The following theorem by Livsi˘c holds.

5.7. Examples

143

Theorem 5.6.4. Let A be a prime dissipative operator in H, (dim H = n) with a rank-one imaginary part Im A. Then A is unitarily equivalent to the operator : Cn → Cn , where A is deﬁned by (5.47). A Proof. Let zk = αk +iβk2 /2, where {αk }nk=1 are real numbers, be the eigenvalues of the operator A. Since A is prime then {βk }nk=1 can be chosen as positive numbers. According to Theorem 5.1.2, A can be included into a system A K 1 Θ= . H C Then we can apply (5.46) and conclude that the transfer function WΘ (z) = WΘA (z). Thus we can apply Theorem 5.3.4 that yields the unitary equivalence of systems ΘA and Θ and hence the operators A and A.

5.7 Examples We conclude this chapter with several simple illustrations. Example. We consider an operator A in H = C2 deﬁned as a matrix 1 0 1 0 1+i 0 A = I + iI = +i = . 0 1 0 1 0 1+i Then its adjoint

1 0 1 0 1−i 0 A = I − iI = −i = , 0 1 0 1 0 1−i 1 0 f1 and Im A = I. If h1 = and h2 = , and f = ∈ C2 , then 0 1 f2 clearly Im A f = If = f = (f, h1 )h1 + (f, h2 )h2 . ∗

Thus we can introduce an operator K : E = C2 → H = C2 such that (f, h1 ) c1 f1 K c = c1 h1 + c2 h2 , K ∗ f = , c= ∈ E, f = ∈ H. (f, h2 ) c2 f2 (5.48) At this point we are ready to form a system Θ of the form (5.6) where A K I Θ= , C2 C2 with all the components deﬁned above and J = I. It is clear that Im A = KK ∗ . Taking into account that Re A = I we calculate ⎛ ⎞ 1 0 ⎟ ⎜ 1−z ⎟, VΘ (z) = K ∗ (Re A − zI)−1 K = ⎜ (5.49) ⎝ 1 ⎠ 0 1−z

144

Chapter 5. The Livˇsic Canonical Systems with Bounded Operators

and ⎛

1−z−i ⎜ 1−z+i WΘ (z) = I − 2iK ∗ (A − zI)−1 K = ⎜ ⎝ 0

⎞ ⎟ ⎟. 1−z−i ⎠ 1−z+i 0

(5.50)

Example. We are going to slightly modify the previous example. Now let A be deﬁned as 1 0 1 0 1+i 0 A= +i = . 0 1 0 −1 0 1−i Then its adjoint A∗ = and Im A =

1 0 0 −1

1 0

0 1

−i

1 0

0 −1

=

1−i 0

0 1+i

,

. Let h1 , h2 , and f be the same as in Example 5.7, then Im A f = (f, h1 )h1 − (f, h2 )h2 .

Let K : E = C2 → H = C2 be deﬁned by (5.48) and set 1 0 J= . 0 −1 We are forming a system Θ=

A C2

K

J C2

,

with all the components deﬁned above. It is clear that Im A = KJK ∗ . Taking into account that both Re A = I and K are the same as in the previous example we note that VΘ (z) is again deﬁned by (5.49) but evaluating WΘ (z) yields a diﬀerent expression, ⎛ ⎞ 1−z−i 0 ⎜ ⎟ 1−z+i ⎟. WΘ (z) = ⎜ (5.51) ⎝ 1−z+i ⎠ 0 1−z−i We note that in Example 5.7 the transfer function WΘ (z) given in (5.50) belongs to the class ΩI while WΘ (z) from Example 5.7 deﬁned by (5.51) is a member of ΩJ class. Moreover, both functions (5.50) and (5.51) are linear-fractional transformations (5.30) of the same function VΘ (z) deﬁned in (5.49).

5.7. Examples

145

Example. Let us consider the following operator in the space L2[0,l] :

l

Af = 2i

f ∈ L2[0,l] .

f (t)dt, x

Its adjoint is A∗ f = −2i and

x

f (t) dt, 0

A − A∗ f= 2i

Im Af =

l

f (t) dt. 0

Let E = C, H = L2[0,l] , and J = 1. We introduce an operator K : E → H as K c = c,

K ∗ f = (f, 1) =

l

f (t) dt. 0

We include A in the system Θ=

A L2[0,l]

K

1 . C

By direct calculations one ﬁnds that 1 2i (A − zI)−1 f (x) = − f (x) − 2 z z

l

2i

e z (t−x) f (t) dt.

x

Using the formula above we obtain WΘ (z) = e2il/z , and VΘ (z) = i

e2il/z − 1 = i tanh(il/z). e2il/z + 1

Chapter 6

The Herglotz-Nevanlinna Functions and Rigged Canonical Systems In this chapter we focus on Herglotz-Nevanlinna functions. After providing all the preliminary results, we introduce the Liv˘sic rigged canonical system (L-system) and its impedance function. We ﬁnd the necessary and suﬃcient conditions for a given Herglotz-Nevanlinna function to be realized as the impedance function of an L-system. The properties of the state-space operator of an L-system based on a given impedance function are determined.

6.1 The Herglotz-Nevanlinna functions and their representations A scalar function φ(z) that is holomorphic in the upper and lower half-planes is called a Herglotz-Nevanlinna function 1 if Im φ(z) ≥ 0 (Im z > 0) and φ(¯ z ) = φ(z). Below we will use some well-known results about the integral representation of a scalar and operator-valued Herglotz-Nevanlinna function. We start oﬀ with the following theorem [3].

1 In addition to the presently used name of Herglotz-Nevanlinna functions one can also ﬁnd the names Pick, Nevanlinna, Herglotz, Nevanlinna-Pick, and R-functions (sometimes depending on the geographical origin of authors and occasionally whether the open upper half-plane C+ or the conformaly equivalent open unit disk D is involved).

Y. Arlinskii et al., Conservative Realizations of Herglotz–Nevanlinna Functions, Operator Theory: Advances and Applications, DOI 10.1007/978-3-7643-9996-2_6, © Springer Basel AG 2011

147

148

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

Theorem 6.1.1. A ﬁnite in the upper half-plane function φ(z) admits the representation ∞ 1 + tz φ(z) = a + b z + dτ (t), (6.1) t−z −∞

where b ≥ 0 and a are two real constants, and τ (t) is a non-decreasing function with bounded variation, if and only if φ(z) is holomorphic and has non-negative imaginary part in the upper half-plane Im z > 0. If, in addition, one applies the normalization conditions τ (t − 0) = τ (t),

τ(−∞) = 0,

then the function τ (t) is uniquely determined. As it turns out, the integral representation (6.1) becomes much more convenient if one replaces the function τ (t) with the function σ(t) such that dσ(t) = (1 + t2 )dτ(t). Then the integral representation (6.1) takes form ∞ φ(z) = a + bz + −∞

1 t − t−z 1 + t2

dσ(t),

(6.2)

where Im z = 0 and the integral is absolutely convergent while σ(t) is such that ∞ −∞

dσ(t) < ∞. 1 + t2

The function σ(t) can be determined using a given function φ(z). We have 1 y→0 π

t2

lim

Im φ(x + iy) dx =

σ(t2 − 0) + σ(t2 + 0) σ(t1 − 0) + σ(t1 + 0) − . (6.3) 2 2

t1

The constants a and b can also be uniquely determined. In particular, φ(iy) Im φ(iy) = lim . y→∞ iy y→∞ y

b = lim

In what follows, the function σ(t) will be normalized by the conditions σ(t) =

σ(t + 0) + σ(t − 0) , 2

σ(0) = 0.

(6.4)

6.1. The Herglotz-Nevanlinna functions and their representations

149

After (6.4) the function σ(t) in the representation (6.2) is determined uniquely. In particular, the inversion formula (6.3) becomes 1 σ(t1 ) − σ(t2 ) = lim y→0 π

t2 Im φ(x + iy) dx, t1

and for any real t, 1 σ(t) = lim y→0 π

t Im φ(x + iy) dx. 0

It follows from (6.2) that for any real y, ∞

2

yIm φ(iy) = b y + −∞

y2 dσ(t). t2 + y 2

(6.5)

We will need the following proposition [159] that can be obtained from (6.5). Theorem 6.1.2. For any Herglotz-Nevanlinna function φ(z) the following statements are true: 1. the function yIm φ(iy) is a non-decreasing function on the interval (0, +∞); 2. the equality lim yIm φ(iy) = sup yIm φ(iy),

y→+∞

y>0

holds. Moreover, both sides of this relation can be ﬁnite or inﬁnite; 3. if b = 0 in (6.1), then ∞ lim yIm φ(iy) = sup yIm φ(iy) =

y→+∞

y>0

dσ(t). −∞

Using standard methods for operator theory we can re-write integral representation (6.2) for the case of an operator-valued Herglotz-Nevanlinna function V (z) whose values are bounded linear operators in a Hilbert space E. We have ∞ V (z) = Q + zX + −∞

1 t − t−z 1 + t2

dG(t),

(6.6)

where Q = Q∗ and X ≥ 0 are linear operators in [E, E], and G(t) is a nondecreasing operator-function on (−∞, +∞) for which +∞

−∞

(dG(t)f, f )E < ∞, 1 + t2

∀f ∈ E.

150

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

We also note that if {zk } (k = 1, . . . , n) is an arbitrary sequence of non-real complex numbers and hk is any sequence of vectors in ﬁnite-dimensional E, then (6.6) is equivalent to n # V (zk ) − V (¯ zl ) h k , hl ≥ 0, zk − z¯l E

(6.7)

k,l=1

and hence V (z) is a Herglotz-Nevanlinna function if and only if (6.7) holds for an arbitrary choice of {zk } and {hk }. In the sequel, if V (z) (Im z = 0) is a function with values in [E, E], then we say that V (z) belongs to a certain function class whenever the function (V (z)f, f ) belongs to the corresponding scalar function class for all f ∈ E.

6.2 Extended resolvents and resolution of identity Let A˙ be a closed symmetric operator with equal defect numbers in H and let A be a self-adjoint extension of A˙ in H. The operator function Rz = (A − zI)−1 is called a canonical resolvent of A˙ and ∞ dE(t) Rz = , (Im z = 0), (6.8) −∞ t − z where E(t) is the resolution of identity or the spectral function of A. In this case ˙ we call E(t) the corresponding canonical spectral function of A. Let H+ ⊂ H ⊂ H− be a rigged Hilbert space generated by the operator A˙ ˆ z the (−, ·)-continuous extension of the operator (see Section 2.2). We denote by R Rz from H− into H which is adjoint to Rz¯, i.e., ˆ z f, g) = (f, Rz¯g), (R

(f ∈ H− , g ∈ H).

ˆ z f = Rz f for f ∈ H, so that R ˆ z is an extension of Rz from H to Obviously, R ˆ z , (Im z = 0) is called the H− with respect to (−, ·)-continuity. The function R ˙ The properties of the resolvents extended canonical resolvent of the operator A. ˆ z were established in Section 4.5. We note that R ˆz − R ˆ ζ ∈ [H− , H+ ], Rz and R ∗ ˆ ˙ ˆ ˆ ˆ A (Rz − Rζ ) = P (z Rz − ζ Rζ ) (see (4.67)), and the Hilbert identity ˆz − R ˆ ζ = (z − ζ)Rz R ˆ ζ = (z − ζ)Rζ R ˆz , R holds. A complex scalar function σ(t) deﬁned on the axis −∞ < t < +∞ is assigned to the class S(k) , (k = 0, 1, 2), if it is of bounded variation on every ﬁnite interval and +∞ |dσ(t)| < ∞. 1 + |t|k −∞

6.2. Extended resolvents and resolution of identity

151

In particular, S(0) is the class of complex functions of bounded variation on the real axis. With each function σ(t) we can associate an interval function σ(Δ) in the usual manner. It was mentioned in Section 6.1 that a scalar Herglotz-Nevanlinna function admits an integral representation of the form (6.2). Suppose φ(z) is such a scalar Herglotz-Nevanlinna function. Then φ(z) can be written in the form (6.2) where a is a real number, b ≥ 0, and σ(t) is a non-decreasing function in S(2) . We will also be interested in a subclass of Herglotz-Nevanlinna functions that has a representation +∞ dσ(t) φ(z) = , (6.9) −∞ t − z where σ(t) is a non-decreasing function in S(0) . ˆ z the Let Rz be the canonical resolvent of a closed symmetric operator and R corresponding extended canonical resolvent. It follows from the integral representation (6.8) that (Rz f, f ) is a Herglotz-Nevanlinna function of the form (6.9) for all f ∈ H. In particular, the operator 1 (Rz − Rz¯) 2i(Im z) is a non-negative operator for all z with Im z = 0. This property is passed to the ˆ z since the operator corresponding extended canonical resolvent R 1 ˆz − R ˆ z¯) (R 2i(Im z) is (−, +)-continuous (see Theorem 4.5.5). However, the quadratic functional ˆ z f, f ) is not a Herglotz-Nevanlinna function for all f ∈ B, where B is deﬁned (R by (4.69). The following theorem clariﬁes the role of the functional Ω deﬁned by ˆz = R ˆ z (A), equality (4.70) written for R Ω(f, g) =

1 ˆ ˆ z¯g) , f, g ∈ B, (Rz f, g) − (f, R 2i

where B is deﬁned by (4.69). ˆ z be an extended canonical resolvent of a closed symmetric Theorem 6.2.1. Let R ˙ operator A. Then for any f ∈ B the function ˆ z f, f ) − iΩ(f, f ) φf (z) = (R is a Herglotz-Nevanlinna function. Proof. The analyticity of the function φf (z) is obvious for all z with Im z = 0. We have 1 ˆ ˆ φf (z) = [(R (Im z = 0), z f, f ) + (f, Rz¯f )], 2

152

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

that implies φf (¯ z ) = φf (z). Furthermore, ˆz f, f ) − Ω(f, f ) Im φf (z) = Im (R 1 ˆ z f, f ) − (R ˆ z f, f ) − (R ˆ z f, f ) − (f, R ˆz f = (R 2i 1 ˆ ˆ z¯f, f ) = 1 (R ˆz − R ˆ z¯)f, f = Im z||R ˆ z f ||2 . = (Rz f, f ) − (R 2i 2i

Thus the number iΩ(f, f ) characterizes the “measure of deviation” of ˆ z f, f ) from the class of Herglotz-Nevanlinna functions. This measure of de(R ˆ z , but viation does not depend on the choice of the extended canonical resolvent R only upon the vector f ∈ B. Let, as above, A˙ be a closed symmetric operator with equal defect numbers in H and let A be a self-adjoint extension of A˙ in H. Let us denote by ℵ the family of all ﬁnite intervals Δ of the real axis. We consider the extension of the interval canonical spectral function E(Δ) of A˙ to the space H− . Theorem 6.2.2. If Δ ∈ ℵ, then E(Δ)H ⊂ H+ and the operators E(Δ) and E(Δ)A˙ are (·, +)-continuous. ˙ We have Proof. Let h ∈ H and f ∈ Dom(A). ˙ E(Δ)h) = (Af, E(Δ)h) = (f, AE(Δ)h) = (f, P AE(Δ)h). (Af, This implies that E(Δ)h ∈ H+

and

A˙ ∗ E(Δ)h = P AE(Δ)h.

(6.10)

˙ Moreover, for every g ∈ Dom(A), ˙ = E(Δ)Ag = AE(Δ)g, E(Δ)Ag

(6.11)

˙ = P AE(Δ)Ag = P A2 E(Δ)g. A˙ ∗ E(Δ)Ag

(6.12)

and, according to (6.8),

For Δ ∈ ℵ the operators AE(Δ) and A2 E(Δ) are bounded, i.e., for all h ∈ H, AE(Δ)h ≤ αh,

A2 E(Δ)h ≤ βh,

α, β > 0.

Thus using (6.10), (6.11), and (6.12) we have E(Δ)h2+ = E(Δ)h2 + A˙ ∗ E(Δ)h2 ≤ (1 + α2 )h2 , ˙ 2 = AE(Δ)g2 + P A2 E(Δ)g2 ≤ (α2 + β 2 )g2 , E(Δ)Ag + which implies the statement of the theorem.

(6.13)

6.2. Extended resolvents and resolution of identity

153

ˆ We denote by E(Δ) the (−, ·)-continuous operator from H− to H that is adjoint to E(Δ) ∈ [H, H+ ]. Thus, ˆ (E(Δ)f, g) = (f, E(Δ)g)

(f ∈ H− , g ∈ H).

ˆ ˆ One can easily see that E(Δ)f = E(Δ)f , ∀f ∈ H, so that E(Δ) is the extension of E(Δ) by continuity. We note that (6.13) implies that ˆ E(Δ)f 2 ≤ (1 + α2 )f 2− ,

(f ∈ H− ).

(6.14)

ˆ We say that E(Δ), as a function of Δ ∈ ℵ, is the extended canonical spectral function of A˙ (or the extended resolution of identity) corresponding to the selfadjoint extension A (or to the original spectral function E(Δ)). Notice that from the equalities E(Δ1 )E(Δ2 ) = E(Δ1 ∩ Δ2 ), E(Δ)Rz = Rz E(Δ) we get ˆ 2 ) = E(Δ ˆ 1 ∩ Δ2 ), E(Δ1 )E(Δ

Δ1 , Δ2 ∈ ℵ

(6.15)

and ˆ z = Rz E(Δ). ˆ E(Δ)R Theorem 6.2.3.

(6.16)

ˆ 1. E(Δ) ∈ [H− , H+ ] for all Δ ∈ ℵ and all f ∈ H− we have ˆ ˆ (E(Δ)f, f ) = ||E(Δ)f ||2 ≥ 0,

and

ˆ 1 )f ||2 ≤ ||E(Δ ˆ 2 )f ||2 . Δ1 ⊂ Δ2 ⇒ ||E(Δ

2. Let FA = {ϕ ∈ H− : (ϕ, g) = 0

for all

g ∈ Dom(A)} ,

then ˆ (E(Δ)ϕ, ϕ) = 0 and

for all

Δ ∈ ℵ ⇐⇒ ϕ ∈ FA ,

ˆ ˙ A. sup (E(Δ)f, f ), Δ ∈ ℵ = +∞ ⇐⇒ f ∈ / H+F

(6.17) (6.18)

˙ and f ∈ H− . By Theorem 6.2.2, Proof. 1. Let φ ∈ Dom(A) + ˙ E(Δ)f ˆ ˙ f )| ≤ E(Δ)Aφ ˙ + · f − ≤ α2 + β 2 φ · f − . |(Aφ, )| = |(E(Δ)Aφ, + ˆ ˆ This yields that E(Δ)f ∈ H+ and A˙ ∗ E(Δ)f ≤ α2 + β 2 · f − . Further, using (6.14) we get ˆ ˆ ˆ E(Δ)f 2+ = E(Δ)f 2 + A˙ ∗ E(Δ)f 2 ≤ (1 + 2α2 + β 2 )f 2− , ˆ which implies that E(Δ) ∈ [H− , H+ ]. Since (E(Δ)f, f ) ≥ 0 for all f ∈ H, then ˆ ˆ (E(Δ)f, f ) ≥ 0 for all f ∈ H− and thus E(Δ) is a non-negative operator. Moreˆ ˆ over, the equality E(Δ)E(Δ) = E(Δ) yields ˆ ˆ (E(Δ)f, f ) = ||E(Δ)f ||2 ,

154

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

for all Δ ∈ ℵ and all f ∈ H− . If Δ1 ⊂ Δ2 , then the inequality ||E(Δ1 )h||2 ≤ ||E(Δ2 )h||2 for all, h ∈ H yields ˆ 1 )f ||2 ≤ ||E(Δ ˆ 2 )f ||2 , f ∈ H− . ||E(Δ ˆ ˆ 2. Since E(Δ) ∈ [H+ , H− ], E(Δ) is the extension of E(Δ), E(Δ)H ⊂ ˆ Dom(A), and H is dense in H− , we get that Ran(E(Δ)) ⊂ Dom(A) for all Δ ∈ ℵ, where Dom(A) is the closure of Dom(A) in H+ . Hence, if ϕ ∈ FA , then ˆ (E(Δ)ϕ, ϕ) = 0 for all Δ ∈ ℵ. Observe that the linear manifold LA = {E(Δ)g, g ∈ Dom(A), Δ ∈ ℵ}, is dense in Dom(A) with respect to the (+)-norm. Indeed, if h ∈ Dom(A) and (h, E(Δ)g)+ = 0 for all g ∈ Dom(A) and all Δ ∈ ℵ, then (h, E(Δ)g) + (A˙ ∗ h, P E(Δ)Ag) = 0. Letting Δ → R, one has (h, g) + (A˙ ∗ h, P Ag) = (h, g)+ = 0, g ∈ Dom(A). 2 ˆ ˆ Hence h = 0. Suppose that (E(Δ)ϕ, ϕ) = 0 for all Δ ∈ ℵ. Since ||E(Δ)ϕ|| = ˆ ˆ (E(Δ)ϕ, ϕ) = 0, we get that E(Δ)ϕ = 0. Therefore,

(ϕ, E(Δ)g) = (Rϕ, E(Δ)g)+ = 0, for all Δ ∈ ℵ and all h ∈ H. Taking g ∈ Dom(A), we obtain that ϕ ∈ FA . Thus, (6.17) holds true. ˆ ˆ If h ∈ H and ϕ ∈ FA , then E(Δ)(h + ϕ) = E(Δ)h and ˆ sup (E(Δ)(h + ϕ), h + ϕ)) < ∞. Δ∈ℵ

2 ˆ ˆ Let f ∈ H− . Assume sup (E(Δ)f, f ) < ∞. Using the equality ||E(Δ)ϕ|| = Δ∈ℵ

ˆ (E(Δ)ϕ, ϕ), we get

ˆ sup ||E(Δ)f || < ∞. Δ∈ℵ

ˆ Therefore, the set {E(Δ)f, Δ ∈ ℵ} is weakly compact, i.e., there exists h ∈ H such that ˆ n )f, g) = (h, g), lim (E(Δ n→∞

for all g ∈ H and for some sequence Δ1 ⊂ Δ2 ⊂ · · · ⊂ Δn ⊂ · · · , lim Δn = R. If n→∞ Δ ∈ ℵ, then there exists n0 such that Δ ⊂ Δn for all n > n0 . Due to the equalities ˆ n ) = E(Δ) ˆ E(Δ)E(Δ for n > n0 and ˆ n )f, g) = (E(Δ ˆ n )f, E(Δ)g), (E(Δ)E(Δ

6.2. Extended resolvents and resolution of identity

155

ˆ we have (E(Δ)f, g) = (h, E(Δ)g). Thus (f − h, E(Δ)g) = 0, for all Δ ∈ ℵ and all g ∈ H. Taking into account that the set LA is dense in Dom(A) with respect to (+)-norm, we obtain that f − h ∈ FA . Corollary 6.2.4. +∞ ˆ d(E(t)f, f ) = +∞

˙ A. f∈ / H+F

if and only if

−∞

Proof. The statement follows from the equality ˆ )f, f ) = (E(Δ)f, ˆ d(E(τ f) Δ

and relation (6.18).

It is well known that the complex scalar measure (E(Δ)f, g) is a complex ˆ function of bounded variation on the real axis. However, (E(Δ)f, g) may be unbounded for f, g ∈ H− . ˆ Theorem 6.2.5. If f ∈ H− and g ∈ H, then (E(Δ)f, g) ∈ S(1) . Proof. Based on (2.1) and (6.10) we have ˆ (E(Δ)f, g) = (f, E(Δ)g) = (Rf, E(Δ)g)+ = (Rf, E(Δ)g) + (A˙ ∗ Rf, A˙ ∗ E(Δ)g) = (Rf, E(Δ)g) + (A˙ ∗ Rf, AE(Δ)g). All that remains is to show that (AE(Δ)g, f ) ∈ S(1) for any h ∈ H. We have +∞

−∞

|d(AE(Δ)g, h)| = 1 + |t|

+∞

−∞

|t| |(dE(Δ)g, h)| ≤ 1 + |t|

+∞ |(dE(Δ)g, h)|.

−∞

Note that we have also proved that (E(Δ)g, x)+ ∈ S(1) for all g ∈ H and ˆ x ∈ H+ . Also applying Theorem 6.2.3 we conclude that (E(Δ)f, g) makes sense for all f, g ∈ H− . Now we will see that equation (6.8) remains true for extended canonical resolvents and extended canonical spectral functions. It follows from (4.64) and

156

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

(6.9) that A˙ ∗ Rz = P (I + zRz ), where P is an orthogonal projection of H onto ˙ This implies Dom(A). A˙ ∗ Rz = P A(A − zI)−1 H +∞ +∞ +∞ dE(t) d(P AE(t)) d(A˙ ∗ E(t)) = PA = = . t−z t−z t−z −∞

−∞

−∞

Hence equality (6.8) takes place in [H, H+ ]. That is why for any f ∈ H− and g ∈ H we have +∞

ˆ z f, g) = (f, R ˆ z¯g) = (f, (R −∞

dE(t)g )= t − z¯

+∞

−∞

d(f, E(t)g) = t−z

+∞

−∞

ˆ d(E(t)f, g) . t−z

Therefore, the representation ˆz = R

+∞ ˆ dE(t) , t−z

−∞

takes place in [H− , H]. In addition, from (6.15) we get ˆ z = Rz E(Δ) ˆ E(Δ)R = Δ

ˆ dE(t) . t−z

(6.19)

Proposition 6.2.6. For each f ∈ H− the equality ˆz − R ˆ ζ )f = (R ˆz − R ˆ ζ )f, lim E(Δ)(R

Δ→R

holds in H+ . Proof. Using (4.67) and (6.16) we have ˆz − R ˆ ζ ) − (R ˆz − R ˆ ζ )) = A˙ ∗ (Rz − Rζ )E(Δ) ˆ ˆz − R ˆζ ) A˙ ∗ (E(Δ)(R − A˙ ∗ (R ˆ ˆ ˆ ˆ ˆ = P (zRz − ζRζ )E(Δ) − P (z Rz − ζ Rζ ) = P (E(Δ) − IH )(z Rz − ζ Rζ ). ˆz − R ˆ ζ )f = A˙ ∗ (R ˆz − R ˆ ζ )f in H. Since It follows that lim A˙ ∗ E(Δ)(R Δ→R

ˆz − R ˆ ζ )f = (R ˆz − R ˆ ζ )f lim E(Δ)(R

Δ→R

in H,

and ||g||2+ = ||g||2 + ||A∗ g||2 , g ∈ H+ , we obtain ˆz − R ˆ ζ )f = (R ˆz − R ˆ ζ )f lim E(Δ)(R

Δ→R

in H+ .

6.2. Extended resolvents and resolution of identity

157

From (6.15), (6.19), and Proposition 6.2.6 it follows that for each f ∈ H− , +∞

ˆz − R ˆ ζ )f = (R −∞

1 1 − t−z t−ζ

ˆ dE(t)f,

where the integral in the right-hand side converges in H+ . In particular, +∞

ˆz − R ˆ ζ )f, g) = (R −∞

1 1 − t−z t−ζ

ˆ d(E(t)f, g).

(6.20)

ˆ z f, g) may not be deﬁned, for all f, g ∈ H− . If f, g ∈ H− then the expression (R and the integral +∞ ˆ d(E(t)f, g) t−z −∞

may diverge. However, a certain regularization both of the extended canonical ˆ z to obtain an integral repreresolvent and of the integral allows us to change R sentation in a form similar to (6.8). Theorem 6.2.7. There exists a bi-continuous self-adjoint operator H such that for ˆ z the operator R ˆ z,H = R ˆ z − H is an operatorany extended canonical resolvent R valued Herglotz-Nevanlinna function with values in [H− , H+ ]. ˆ z(0) be an extended canonical resolvent and z0 be a non-real number. Proof. Let R Set 1 ˆ (0) ˆ (0) H= Rz0 + Rz¯0 , (6.21) 2 and observe that H is bi-continuous and self-adjoint. For an arbitrary extended ˆ z the operator-function canonical resolvent R ˆ z,H = R ˆ z − H = (R ˆz − R ˆ z(0) ) + 1 R ˆ z(0) − R ˆ z(0) + R ˆ z(0) − R ˆ z(0) R , ¯0 0 2 belongs to the class [H− , H+ ] while being holomorphic in the upper and lower ˆ z,H )∗ = R ˆ z¯,H . half-planes (see Theorem 4.5.5). We also note that (R For an f ∈ H− we have ˆ z,H f, f ) ˆ z,H − R ˆ z¯,H )f, f ) ˆz − R ˆ z¯)f, f ) Im (R ((R ((R = = . Im z z − z¯ z − z¯ Referring to our discussion before Theorem 6.2.1, we conclude that the last exˆ z,H f, f ) is a pression is non-negative for all f ∈ H− and Im z = 0. Thus, (R Herglotz-Nevanlinna function for all f ∈ H− .

158

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

The operator H in Theorem 6.2.7 is called a regularizing operator. As it was shown in the proof of the last theorem we can always choose a regularizing operator to be deﬁned by (6.21) where z0 is an arbitrary non-real number and ˆ z(0) R 0 is an extended canonical resolvent. ˆ z,H admits a weak integral representation Theorem 6.2.8. The function R +∞

ˆ z,H = QH + R −∞

1 t − t−z 1 + t2

ˆ dE(t),

(6.22)

where QH is some self-adjoint operator in [H− , H+ ] (depending on the choice of ˆ H) and (E(t)f, g) belongs to the class S(2) . ˆ z,H = R ˆ z − H is [H− , H+ ]-valued HerglotzProof. According to Theorem 6.2.7, R Nevanlinna. Let 1 ˆ ˆ −i , H0 = Ri + R 2 ˆ ˆ and let Rz,H0 = Rz − H0 = Rz,H + (H − H0 ). Using (6.20) we have that

1 ˆ ˆ ˆ Rz − Ri + R−i f, g 2 +∞ 1 t ˆ = − (dE(t)f, g), t−z 1 + t2

ˆ z,H f, g) = (R 0

−∞

for all f, g ∈ H− . This yields (6.22) with Q = H0 − H. Since +∞

−∞

ˆ d(E(t)f, g) 1 ˆ ˆ −i f, g , = R − R i 1 + t2 2i

ˆ for all f, g ∈ H− . Hence d(E(t)f, g) belongs to the class S(2) .

Now we can interpret the preceding results of this section in the following ˆ z be an extended canonical resolvent of a closed symmetric operator way. Let R ˆ A˙ and let E(Δ) be the corresponding extended canonical spectral function. Then for any f, g ∈ H− , +∞ ˆ |d(E(t)f, g)| < ∞, 1 + t2 −∞

and the following integral representation holds: +∞ ˆi + R ˆ −i R 1 t ˆ ˆ Rz − = − dE(t). 2 t − z 1 + t2 −∞

(6.23)

6.2. Extended resolvents and resolution of identity

159

We will reﬁne integral representation (6.23) for the case of the resolvent of a t-self˙ After that we will establish adjoint bi-extension of a regular symmetric operator A. some additional properties of the corresponding extended canonical spectral function. Lemma 6.2.9. Let A be a t-self-adjoint bi-extension of a regular symmetric operator ˆ A˙ with the quasi-kernel Aˆ and let E(Δ) be the extended canonical spectral function ˆ ˙ of A. Then for every f ∈ H+LA , f = 0, and for every g ∈ H− there is an integral representation +∞

(Rz f, g) = −∞

1 t − t−z 1 + t2

1 ˆ ˆ ˆ d(E(t)f, g) + ((R i + R−i )f, g), 2

(6.24)

where LA is deﬁned by (4.44) and Rz = (A − zI)−1 . ˆz = R ˆ z (A) ˆ we have Proof. Due to Theorem 4.5.12 for R ˆ z (H LA ) = (A − zI)−1 . R ˆ z (H+L ˙ A ) ⊆ H+ . Hence for all g ∈ H− and f ∈ H LA , (Rz f, g) is Therefore, R ˆ deﬁned. Consequently, using (6.23) we obtain (6.24) where E(t) is the extended ˆ canonical spectral function of A. Theorem 6.2.10. Let A be a t-self-adjoint bi-extension of a regular symmetric opˆ erator A˙ with the quasi-kernel Aˆ and let E(Δ) be the canonical spectral function ˆ Then for any f ∈ LA L, f = 0, of A. +∞ ˆ d(E(t)f, f ) = ∞,

if

f∈ / L,

(6.25)

if

f ∈ L.

(6.26)

−∞

and

+∞ ˆ d(E(t)f, f ) < ∞, −∞

Moreover, there exist real constants β > 0 and α > 0 such that αf 2−

+∞

≤ −∞

ˆ d(E(t)f, f) ≤ βf 2− , 2 1+t

(6.27)

for all f ∈ LA L, where LA and L are deﬁned by (4.44) and (2.12), respectively. ˙ Proof. Since A is a t-self-adjoint bi-extension of A, then according to (3.13) it −1 + + i ∗ ˙ ˙ takes a form A = AP (S − 2 J) PM . Let us choose a point z ˙ + A +R Dom(A)

160

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

with Im z = 0 to be a regular point of the operator A and consider function φ(z) deﬁned for all f ∈ LA L ⊂ Ran(A − zI) (see (4.44) and Theorem 4.3.2) by the formula: φ(z) = (A − zI)−1 f, f . ( (2 It can be seen that φ(z) = φ(¯ z ) and Imφ(z) = Imz ((A − z¯I)−1 f ( , which means that φ(z) is a Herglotz-Nevanlinna function and according to Lemma 6.2.9 has the integral representation +∞

φ(z) = −∞

1 t − t−z 1 + t2

1 ˆ ˆi + R ˆ −i )f, f ) . d E(t)f, f + (R 2

This representation implies that lim

η→∞

Imφ(iη) = 0. η

ˆ be given by Let Dom(A) ˆ = Dom(A) ˙ ⊕ (I + M )(Ni ⊕ N). Dom(A) According to Theorem 3.3.8, Theorem 4.3.2 and formulas (3.25), (4.33), (4.43) we have ˙ + U )Ni = M, (I + M )(Ni ⊕ N)+(I

i S(I + U )ϕi = − (I − U )ϕi , 2

and LA = R−1 (U − I)Ni . The subspace (U − I)Ni is (1)-orthogonal to (I + U )Ni . Let ˆ . FAˆ = ϕ ∈ H− : (ϕ, g) = 0 for all g ∈ Dom(A) ˆ and If ϕ ∈ LA ∩ FAˆ , then the vector Rϕ ∈ (I − U )Ni is (1)-orthogonal to Dom(A) therefore, Rϕ is (1)-orthogonal to H+ . Hence ϕ = 0, i.e., LA ∩ FAˆ = {0}. Due to Theorem 6.2.3 and Corollary 6.2.4 we get (6.25) and (6.25). Since (A + iI)−1 f ∈ N−i for all f ∈ LA L, the norms · and · + are equivalent on N±i and so are the norms · and · − . Therefore αf 2− ≤ Imφ(i) ≤ βf 2− ,

β > 0, α > 0 – const.

Combining this with 1 ˆ ˆ −i )f, f = Imφ(i) = (Ri − R 2i we obtain the relation (6.27).

+∞

−∞

ˆ d(E(t)f, f) , 1 + t2

6.3. Deﬁnition of an L-system

161

Corollary 6.2.11. In the settings of Theorem 6.2.10 for all f, g ∈ LA L, , , +∞ ˆ +∞ ˆ , R , ˆ ˆ + R d( E(t)f, f ) d(E(t)g, g) , , i −i f, g , ≤ k · , , 2 , , 2 1+t 1 + t2 −∞ −∞

(6.28)

where k > 0 is a constant. Proof. If f ∈ LA L, then (A − iI)−1 f + (A + iI)−1 f = 2(A − iI)−1 A(A + iI)−1 f. From Theorem 4.5.12 we get ((A − zI)−1 f, g) = (f, (A − z¯I)−1 g), and, since A(A + iI)−1 f ∈ H LA , we have , , , R , , ˆ −i , , ˆi + R , f, g , = ,(A(A + iI)−1 f, (A + iI)−1 g), , , , 2 ≤ A(A + iI)−1 f − · (A + iI)−1 g+ ≤ cf − · g− . It follows from Theorem 6.2.10 that there is a constant α > 0 such that f 2−

+∞

≤α −∞

ˆ d(E(t)f, f) . 1 + t2

A similar inequality holds for g2−. This completes the proof.

6.3 Deﬁnition of an L-system In this section we introduce the Liv˘sic rigged canonical system or L-system. In order to do that we ﬁrst deﬁne a class of state-space operators. Deﬁnition 6.3.1. An unbounded operator T acting in the Hilbert space H belongs to the class Λ if ρ(T ) = ∅ and the maximal common symmetric part of T and T ∗ has ﬁnite and equal deﬁciency indices. Deﬁnition 6.3.2. Let A˙ be a symmetric operator with ﬁnite and equal deﬁciency ˙ belongs to the class Λ(A) ˙ if A˙ is the indices. An operator T of the class Ω(A) ∗ maximal common symmetric part of T and T . ˙ where Thus, if T ∈ Λ, then the operators T and T ∗ belong to the class Λ(A), ∗ ˙ A is a maximal common symmetric part of T and T . On the other hand for a ﬁxed ˙ is a subclass of the class Λ. In the sequel, given T ∈ Λ, operator A˙ the class Λ(A) ˙ by A we will denote the maximal common symmetric part of T and T ∗ . Since A˙

162

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

has ﬁnite deﬁciency indices, the operator A˙ is regular. Therefore the operator T from the class Λ is quasi-self-adjoint and a regular extension of A˙ (see Theorem 4.1.3). By Theorem 4.3.10 each operator T from the class Λ admits a (∗)-extension in rigged Hilbert space H+ ⊂ H ⊂ H− constructed by the means of the operator A˙ ∗ (see Section 2.2). Any (∗)-extension A of T ∈ Λ together with A∗ are bounded linear operators from H+ into H− and hence we can call the operators Re A =

1 1 (A + A∗ ) and Im A = (A − A∗ ), 2 2i

(6.29)

real and imaginary parts of T as well as real and imaginary parts of A. Since T ∈ Λ can have many diﬀerent (∗)-extensions, then T can have many diﬀerent real and imaginary parts. Let T ∈ Λ, K be a bounded linear operator from a ﬁnite-dimensional Hilbert space E into H− , K ∗ ∈ [H+ , E], and J = J ∗ = J −1 ∈ [E, E]. Consider the following singular system of equations: ⎧ dχ ⎪ ⎨ i dt + T χ(t) = KJψ− (t), (6.30) χ(0) = x ∈ Dom(T ), ⎪ ⎩ ∗ ψ+ = ψ− − 2iK χ(t). Given an input vector ψ− = ϕ− eizt ∈ E, we seek solutions to the system (6.30) as an output vector ψ+ = ϕ+ eizt ∈ E, and a state-space vector χ(t) = xeizt ∈ Dom(T ). Substituting the expressions for ψ± (t) and χ(t) allows us to cancel exponential terms and convert the system (6.30) to the form (T − zI)x = KJϕ− , z ∈ ρ(T ). (6.31) ϕ+ = ϕ− − 2iK ∗ x, The choice of the operator K in the above system is such that KJϕ− ∈ B ⊂ H− , where B is deﬁned by (4.69). Therefore the ﬁrst equation of (6.31) does not, in general, have a regular solution x ∈ Dom(T ). It has, however, a generalized solution x ∈ H+ that can be obtained in the following way. If z ∈ ρ(T ), then we can use the density of H in H− and therefore there is a sequence of vectors {αn } ∈ H that approximates KJϕ− in the (−)-metric. In this case the state space ˆ z (T )KJϕ− ∈ H is understood as limn→∞ (T −zI)−1 αn , where R ˆ z (T ) vector x = R −1 is the extended to H− by (−, ·)-continuity resolvent (T − zI) . But then we can apply Theorem 4.5.9 to conclude that x ∈ H+ . This explains the expression K ∗ x in the second line of (6.31). In order to satisfy the condition Im T = KJK ∗ we perform the regularization of system (6.31) and use A ∈ [H+ , H− ], a (∗)-extension of T such that Im A = KJK ∗ . This leads to the system (A − zI)x = KJϕ− , z ∈ ρ(T ), (6.32) ϕ+ = ϕ− − 2iK ∗ x,

6.3. Deﬁnition of an L-system

163

where ϕ− is an input vector, ϕ+ is an output vector, and x is a state-space vector of the system. System (6.32) is the stationary version of the system ⎧ dχ ⎨ i dt + Aχ(t) = KJψ− (t), (6.33) χ(0) = x ∈ H+ , ⎩ ψ+ = ψ− − 2iK ∗ χ(t). Let L2[0,τ0 ] (E) be the Hilbert space of E-valued functions equipped with an inner product τ0 (ϕ, ψ)L2[0,τ ] (E) = (ϕ, ψ)E dt, ϕ(t), ψ(t) ∈ L2[0,τ0 ] (E) . 0

0

The following lemma proves the metric conservation law for systems of the form (6.33). Lemma 6.3.3. If for a given continuous in E function ψ− (t) ∈ L2[0,τ0 ] (E) we have that a (+)-continuous and strongly (·)-diﬀerentiable function χ(t) ∈ H+ and ψ+ (t) ∈ L2[0,τ0 ] (E) satisfy (6.33), then a system of the form (6.33) satisﬁes the metric conservation law τ τ 2 2 2χ(τ ) − 2χ(0) = (Jψ− , ψ− )E dt − (Jψ+ , ψ+ )E dt, τ ∈ [0, τ0 ]. 0

0

(6.34)

Proof. Taking into account that A, A∗ ∈ [H+ , H− ], Im A = KJK ∗ , and K ∈ [E, H− ], K ∗ ∈ [H+ , E], we rely on the proof of Lemma 5.1.1. It follows from the statement of our Lemma that the function dχ is both (−)- and (·)-continuous. We dt d will show that the function dt (χ(t), χ(t)) is continuous. We have d (χ(t), χ(t)) = (χ (t), χ(t)) + (χ(t), χ (t)). dt Consequently, for t, t0 ∈ [0, τ0 ] we have |(χ(t), χ (t)) − (χ(t0 ), χ (t0 ))| = |(χ(t), χ (t)) − (χ(t0 ), χ (t)) + (χ(t0 ), χ (t)) − (χ(t0 ), χ (t0 ))| = |(χ(t) − χ(t0 ), χ (t)) + (χ(t0 ), χ (t) − χ (t0 ))| ≤ (χ(t) − χ(t0 )+ χ (t)− + χ(t0 )+ χ (t) − χ (t0 ))− ≤ C1 (χ(t) − χ(t0 )+ + C2 χ (t) − χ (t0 ))− , where C1 > 0 and C2 > 0 are constants. Applying (+)-continuity of χ(t) and d (−)-continuity of χ (t), we obtain that (χ(t), χ (t)) and hence dt (χ(t), χ(t)) is continuous. Now we can apply the algebraic steps described in detail in the proof of Lemma 5.1.1 (replacing A with A) to obtain (6.34). We will refer to systems (6.31)-(6.32) as rigged canonical systems.

164

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

Deﬁnition 6.3.4. A rigged canonical system of the form (6.31)-(6.32) with T ∈ Λ is called the Liv˘sic rigged canonical system or L-system if there exists an imaginary part Im A of T with Im A = KJK ∗ and Ran(ImA) = Ran(K). According to Theorem 4.3.9, any (∗)-extension A of a given operator T ∈ Λ with a ﬁxed imaginary part Im A = KJK ∗ is deﬁned uniquely. Thus, any L-system is well-deﬁned. In the case when T is a bounded operator, its imaginary part is deﬁned naturally and uniquely, and as a result we obtain a canonical system of the Liv˘sic type described in Chapter 5. It is more convenient to write an L-system in the form of one of two arrays: T K J Θ= , (6.35) H+ ⊂ H ⊂ H− E

or Θ=

A H+ ⊂ H ⊂ H−

K

J , E

(6.36)

where (1) A is a (∗)-extension of an operator T of the class Λ; (2) J = J ∗ = J −1 ∈ [E, E],

dim E < ∞;

∗

(3) Im A = KJK , where K ∈ [E, H− ], K ∗ ∈ [H+ , E], and Ran(K) = Ran(Im A).

(6.37)

This is justiﬁed by the following statement. Proposition 6.3.5. Let A˙ be a closed symmetric operator with ﬁnite and equal ˙ and A be an arbitrary (∗)deﬁciency numbers in the Hilbert space H, T ∈ Λ(A), extension of T . Then there exists an L-system of the form (6.36) for which A is the state-space operator. Proof. Let A = RA. Then A ∈[H+ , H+ ] and Im A = R Im A. Set G = Ran(Im A). By Theorem 5.1.2 there exist a Livsi˘c canonical system A K J Θ= , H+ E for which A is a state-space operator and G is the channel subspace, i.e., K ∈ [E, H+ ], K∗ ∈ [H+ , E], Ran(K) = G = Ran(Im A), Im A = KJK∗ . Set K = R−1 K ∈ [E, H− ]. Then Ran(K) = Ran(Im A) and A K J Θ= H+ ⊂ H ⊂ H− E is an L-system.

6.3. Deﬁnition of an L-system

165

As in the case of canonical systems with bounded operators in Chapter 5, the operator K is called a channel operator and J is called a directing operator. A system Θ of the form (6.35)-(6.36) is called a scattering L-system if J = I. Remark 6.3.6. Clearly, for a scattering L-systems or for an L-system with the invertible operator K the condition (6.37) is satisﬁed automatically. The following theorem shows that for the case of the densely deﬁned operator A˙ there is a set of conditions on operator K that guarantee that a system of the form (6.31) becomes an L-system. Theorem 6.3.7. Let Θ be a system of the form (6.31) with T ∈ Λ, (−i) ∈ ρ(T ), ˙ = H, ker(K) = {0}, Ran(K) ⊂ F , where F is deﬁned by (4.69), and Dom(A) Dom(T ) parameterized by an operator M via Theorem 4.1.9. Then Θ is an Lsystem if and only if K satisﬁes −HM + M ∗ H ∗ + iI H − (M ∗ H ∗ + iI)M ∗ + 2iKJK ∗ = R−1 PM , (6.38) M (HM − iI) − H ∗ iI − M H + H ∗ M ∗ where H = i(I − M ∗ M )−1 [(I − M ∗ U )(I − U ∗ M )−1 U ∗ − M ∗ ],

(6.39)

R is a Riesz-Berezansky operator, and U ∈ [Ni , N−i ] is any isometry such that U ∗ M − I is a homeomorphism. Proof. Let Θ be an L-system. Then there is a (∗)-extension A of T whose real part contains a quasi-kernel Aˆ parameterized by von Neumann’s formula (1.13) with an isometry U . Consequently there exists an operator H ∈ [Ni , N−i ] such that A and A∗ are determined by (4.22), (4.23). Furthermore, 1 1 −HM + M ∗ H ∗ − iI H + (M ∗ H ∗ + iI)M ∗ (SA + SA∗ ) = (6.40) M (HM − iI) + H ∗ iI − M H − H ∗ M ∗ 2 2 Applying Theorem 3.4.10 and formula (3.37) to the operator matrix (6.40) we obtain the following equations: −HM + M ∗ H ∗ − iI = −[H + (M ∗ H ∗ + iI)M ∗ ]U, M (HM − iI) + H ∗ = −[iI − M H − H ∗ M ∗ ]U.

(6.41)

˜ = U ∗ M and H ˜ = HU , then we obtain a system of operator equations If we set M ˜ ∗ (I − M ˜ ∗) + M ˜ H( ˜ M ˜ − I) = i(M ˜ ∗ − I), H ˜ ∗H ˜ ∗ (M ˜ ∗ − I) + H(I ˜ − M) ˜ = i(I − M ˜ ∗ ), M

(6.42)

˜ is its solution. Reversing the argument we conclude that Re A has and operator H a self-adjoint quasi-kernel if and only if a system of operator equations (6.42) has

166

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

a solution. Using simple algebraic manipulations we solve system (6.42) to obtain the solution of the form ˜ ∗M ˜ )−1 [(I − M ˜ ∗ )(I − M ˜ )−1 − M ˜ ∗ )]. Y = i(I − M

(6.43)

˜ , H, ˜ (6.43), and (6.41) we obtain (6.39). In order to Combining deﬁnitions of M conﬁrm that U ∗ M − I is a homeomorphism we apply Theorem 4.4.4, Corollary 4.4.5, and Remark 4.3.4 in that order. Since A˙ has ﬁnite and equal deﬁciency indices and is a maximal symmetric part of T and T ∗ , we can use the fact that ˜ ∗M ˜ and I − M ˜ have bounded U ∗ M − I is a homeomorphism to show that I − M inverses. Conversely, let K satisfy (6.38)-(6.39) and let U ∗ M −I be a homeomorphism. Then we have the solvability of the systems (6.42) and (6.42) which implies the existence of a self-adjoint quasi-kernel for Re A, where A is a (∗)-extension we can construct using (4.22) and (4.23). Applying the uniqueness Theorem 4.3.9 we see that our operator T has a unique (∗)-extension A whose real part has a self-adjoint ˆ Because ker(K) = {0}, relation (6.37) is satisﬁed. Formulas (6.38), quasi-kernel A. (4.22), and (4.23) will provide us with the condition A − A∗ = 2iKJK ∗ necessary to complete the Deﬁnition 6.3.4 of an L-system. We note that the operator U in the statement of Theorem 6.3.7 above is the parameter in the von Neumann formula (1.13) describing the self-adjoint quasikernel Aˆ of the real part of the (∗)-extension A that deﬁnes the imaginary part of the operator T . Taking into account Theorem 4.3.2, relations (4.45), (6.37), and following Chapter 5, we associate with an L-system Θ the operator-valued function WΘ (z) = I − 2iK ∗(A − zI)−1 KJ,

z ∈ ρ(T ),

(6.44)

which is called a transfer operator-valued function of the L-system Θ. Following Section 5.4 one can easily show that for the transfer operator-function of the system Θ of the form (6.36) the identities similar to (5.21), (5.25) are valid, i.e., for all z, ζ ∈ ρ(T ), ¯ −1 K, WΘ (z)JWΘ∗ (ζ) − J = 2i(ζ¯ − z)K ∗ (A − zI)−1 (A∗ − ζI) ∗ ∗ ∗ −1 ¯ WΘ (ζ)JWΘ (z) − J = 2i(ζ¯ − z)JK (A − ζI) (A − zI)−1 KJ. Therefore,

WΘ∗ (z)JWΘ (z) − J ≥ 0, WΘ∗ (z)JWΘ (z) − J = 0,

(Im z > 0, z ∈ ρ(T )), (Im z = 0, z ∈ ρ(T )),

− J ≤ 0,

(Im z < 0, z ∈ ρ(T )).

WΘ∗ (z)JWΘ (z)

(6.45)

(6.46)

Similar relations take place if we change WΘ (z) to WΘ∗ (z) in (6.46). Thus, the transfer operator-valued function of the system Θ of the form (6.36) is J-contractive in the lower half-plane on the set of regular points of an operator T and J-unitary on real regular points of an operator T . In addition WΘ−1 (z) = JWΘ∗ (¯ z )J,

if z, z¯ ∈ ρ(T ).

6.3. Deﬁnition of an L-system

167

Let Θ be an L-system of the form (6.36). We consider the operator-valued function VΘ (z) = K ∗ (Re A − zI)−1 K. (6.47) We note that both (6.44) and (6.47) are well deﬁned due to Theorem 4.3.2. The transfer operator-function WΘ (z) of the system Θ and an operator-function VΘ (z) of the form (6.47) are connected by the relations valid for Im z = 0, z ∈ ρ(T ), VΘ (z) = i[WΘ (z) + I]−1 [WΘ (z) − I]J, WΘ (z) = [I + iVΘ (z)J]−1 [I − iVΘ (z)J],

(6.48)

that can be easily derived following the algebraic steps of Section 5.4. The function VΘ (z) deﬁned by (6.47) is called the impedance function of an L-system Θ of the form (6.36). At this point we would like to extend our deﬁnition of a scalar HerglotzNevanlinna function that we gave in Section 6.1. An operator-function V (z) in a ﬁnite-dimensional Hilbert space E is called an operator-valued Herglotz-Nevanlinna function if it is holomorphic in the upper and lower half-planes, Im V (z) ≥ 0 when Im z > 0, and V (¯ z ) = V ∗ (z). ˙ Let A be a symmetric operator with equal ﬁnite deﬁciency indices and A be its self-adjoint extension in a Hilbert space H. Consider a system Δ of the form (A − zI)x = Kψ− , (6.49) ψ+ = K ∗ x, where K is a bounded linear operator from a ﬁnite-dimensional Hilbert space E into H− , K ∗ ∈ [H+ , E], ψ− is an input vector, ψ+ is an output vector, and x is a state-space vector of the system. The rigged Hilbert triplet H+ ⊂ H ⊂ H− ˙ The choice of in the above system is generated by the symmetric operator A. the operator K in the above deﬁnition is such that Kψ− ∈ B ⊂ H− , where B is deﬁned by (4.69). If z ∈ ρ(A), then we can use the density of H in H− and therefore there is a sequence of vectors {βn } ⊂ H that approximates Kψ− in the ˆ z (A)Kψ− ∈ H is understood (−)-metric. In this case the state-space vector x = R ˆ z (A) is the extended to H− by (−, ·)-continuity as limn→∞ (A − zI)−1 βn , where R resolvent (A−zI)−1 . Apply Theorem 4.5.9 to conclude that x ∈ H+ . This explains the expression K ∗ x in the second line of (6.31). If z ∈ ρ(A), then we can solve the ﬁrst equation of the system (6.49) for x and substitute it into the second equation. This leads to the deﬁnition of the corresponding transfer operator-valued function associated with the system Δ, that is ˆ z (A)K, VΔ (z) = K ∗ R z ∈ ρ(A). (6.50) A system Δ of the form (6.49) is called an impedance system if its transfer function is Herglotz-Nevanlinna.

168

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

If Θ is an L-system of the form (6.31)-(6.36) with the state-space operator A, then the system ΔΘ of the form (Aˆ − zI)x = Kψ− , (6.51) ψ+ = K ∗ x, is called an impedance system associated with the L-system Θ if Aˆ is the quasikernel of Re A and both operators A and K are elements of the system Θ. In the next section we will show that ΔΘ is indeed an impedance system. To justify the deﬁnitions of impedance system we present an example of a system of the form (6.49) that is not impedance. Example. Let H = L2[0,1] and let df A˙ f = i , , dt , ˙ Dom(A) = f (t) ∈ L2[0,l] , f (t) − abs. cont., f (t) ∈ L2[0,l] , f (0) = f (1) = 0 , be a symmetric operator in H. Its adjoint df A˙ ∗ f = i , dt is without boundary conditions and H+ = Dom(A˙ ∗ ) = W21 is a Sobolev space with the scalar product 1 1 (f, g)+ = f (t)g(t) dt + f (t)g (t) dt. 0

0

We construct the rigged Hilbert space W21 ⊂ L2[0,1] ⊂ (W21 )− . It is easy to see ˙ = H and its orthogonal complement F is 2-dimensional. We can that Dom(A) choose a basis for F by picking elements δ(t) and δ(t − 1) in (W21 )− that generate (+)-continuous linear functionals (f (t), δ(t)) = f (0)

and

(f (t), δ(t − 1)) = f (1).

Let also df Af = i , , dt , Dom(A) = f (t) ∈ L2[0,l] , f (t) − abs. cont., f (t) ∈ L2[0,l] , f (0) = −f (1) , ˙ Now we set E = C, Kc = cg(ξ), where g(ξ) = be a self-adjoint extension of A. 1 −iξ √ [e δ(t − 1) − δ(t)], and form a system Δ of the form (6.49). Then 2 Rz (A)f = ie−izt

1 0

f (s)eizs ds − 1 + eiz

0

t

f (s)eizs ds ,

f ∈ H,

6.3. Deﬁnition of an L-system

169

and for f ∈ H we have , ˆ z (A)δ(t), f ) = (δ(t), Rz (A)f ) = Rz¯(A)f ,, (R ieiz =− 1 + eiz This implies

t=0

1

−i = 1 + e−iz

1

f (s)e−izs ds

0

e−izs f (s)ds.

0

iz ˆ z (A)δ(t) = −ie e−izt . R 1 + eiz

Similarly, one ﬁnds that ˆ z (A)δ(t − 1) = R

ieiz −izt e . 1 + eiz

Moreover, −ieiz , 1 + eiz ˆ z (A)δ(t), δ(t − 1)) = −i , (R 1 + eiz ˆ z (A)δ(t), δ(t)) = (R

ˆ z (A)δ(t − 1), δ(t − 1)) = (R ˆ z (A)δ(t − 1), δ(t)) = (R

Now let us set ξ = π, then we have g(π) = calculations using the above relations we obtain

√1 [−δ(t 2

ˆ z (A)Kc = (R ˆ z (A)g(π), g(π))c = VΔ (z)c = K ∗ R

i , 1 + eiz

ieiz . 1 + eiz

− 1) − δ(t)]. By direct

2i c, 1 + eiz

c ∈ C,

which is not a Herglotz-Nevanlinna function. If, however, we set ξ = 0, then g(0) = √12 [δ(t − 1) − δ(t)] and the resulting system is an impedance one with ˆ z (A)g(0), g(0))c = −i tanh(iz/2)c, VΔ (z)c = (R

c ∈ C,

that is a Herglotz-Nevanlinna function. The following theorem is similar to Theorem 6.3.7 for the case of impedance systems. Theorem 6.3.8. Let Δ be a system of the form (6.49) where A is a self-adjoint ˙ = H and ﬁnite equal deﬁciency indices, extension of an operator A˙ with Dom(A) Ran(K) ⊂ F, where F is deﬁned by (4.69). Then Δ is an impedance system associated with a scattering L-system Θ (J = I) if and only if −HM + M ∗ H ∗ + iI H − (M ∗ H ∗ + iI)M ∗ + 2iKK ∗ = R−1 PM , M (HM − iI) − H ∗ iI − M H + H ∗ M ∗ where M is an arbitrary contraction such that (I − M ∗ M ) is invertible, H = i(I − M ∗ M )−1 [(I − M ∗ U )(I − U ∗ M )−1 U ∗ − M ∗ ],

170

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

R is a Riesz-Berezansky operator, and U ∈ [Ni , N−i ] is the parameter in the von Neumann formula (1.13) describing A such that U ∗ M − I is a homeomorphism. Most of the proof of Theorem 6.3.8 directly follows from Theorem 6.3.7. The rest of the proof becomes obvious when in the next section we will show that the impedance function of an L-system is Herglotz-Nevanlinna.

6.4 Realizable Herglotz-Nevanlinna operator-functions. Class N (R) As we mentioned at the end of Section 6.1, an operator-valued Herglotz-Nevanlinna function, acting on Hilbert space E (dim E < ∞) admits an integral representation +∞ 1 t V (z) = Q + zX + − dG(t), (6.52) t−z 1 + t2 −∞ where Q = Q∗ , X ≥ 0 in the Hilbert space E, G(t) is a non-decreasing operatorfunction on (−∞, +∞) for which +∞

−∞

(dG(t)f, f )E < ∞, 1 + t2

∀f ∈ E.

Deﬁnition 6.4.1. An operator-valued Herglotz-Nevanlinna function V (z) ∈ [E, E], (dim E < ∞) belongs to the class N (R) if in the representation (6.52) we have i)

X = 0, +∞

ii)

Qf = −∞

for all f ∈ E such that

t dG(t)f 1 + t2

+∞ (dG(t)f, f )E < ∞.

(6.53)

(6.54)

−∞

Hence the integral representation (6.52) for the class N (R) becomes +∞

V (z) = Q + −∞

1 t − t−z 1 + t2

dG(t).

Remark 6.4.2. It can be easily proved that V (z) ∈ N (R) if and only if 1. (Im V (i)f, f )E ≥ 0 for all f ∈ E,

(6.55)

6.4. Realizable Herglotz-Nevanlinna operator-functions. Class N (R)

171

2. the equality lim

η→+∞

Im V (iη) = 0, η

holds, 3. if f ∈ E \ {0} and lim ηIm (V (iη)f, f )E < ∞, then lim V (iη)f = 0. η→+∞

η→+∞

Theorem 6.4.3. Let Θ be an L-system of the form (6.35)–(6.36). Then its impedance function VΘ (z) of the form (6.47), (6.48) belongs to the class N (R). Moreover, VΘ (z) is also the transfer function of the associated to Θ impedance system ΔΘ of the form (6.51). Proof. First we will show that VΘ (z) is a Herglotz-Nevanlinna function. Let Gz be a neighborhood of a point z, (Im z = 0) and z, ζ ∈ Gz . Then, VΘ (z) − VΘ (ζ) = K ∗ (Re A − zI)−1 K − K ∗ (Re A − ζI)−1 K = (z − ζ)K ∗ (Re A − zI)−1 (Re A − ζI)−1 K, and

VΘ (z) − VΘ (ζ) = K ∗ (Re A − zI)−1 (Re A − ζI)−1 K, z−ζ

all z, ζ ∈ Gz . Therefore, letting z → ζ we can say that VΘ (z) is holomorphic in Gz . Now we can conclude that VΘ (z) is holomorphic in any one of the half-planes. It is obvious that VΘ∗ (z) = VΘ (¯ z ). Furthermore, ImVΘ (z) = Im z K ∗ (Re A − z¯I)−1 (Re A − zI)−1 K. Let now Dz = (Re A−zI)−1 K, then it is easy to see that the adjoint operator is given by Dz∗ = K ∗ (Re A− z¯I)−1 . Therefore, we have ImVΘ (z) = (Im z)Dz∗ Dz which implies that Im VΘ (z) ≥ 0 when Im z > 0. Hence, we can conclude that VΘ (z) is an operator Herglotz-Nevanlinna function and admits representation (6.52). Now we observe that by construction the function VΘ (z) of the form (6.47), (6.48) is the transfer function of the associated impedance system ΔΘ of the form ˆ where Aˆ is the quasi-kernel of the (6.51) whose state space operator A = A, operator Re A of our system Θ and operator K of the system ΔΘ is equal to the channel operator K of the system Θ. At this point all we have to show that the conditions i) and ii) in the Deﬁˆ nition 6.4.1 of the class N (R) hold. Let E(δ) be the extended canonical spectral function of the self-adjoint operator A of the system ΔΘ . According to Theorem 3.3.8 there is a t-self-adjoint bi-extension A of A˙ whose quasi-kernel is A. This allows us to apply Lemma 6.2.9 for all f, g ∈ E to get Dz∗

+∞

(VΘ (z)f, g)E = −∞

1 t − t−z 1 + t2

ˆ ˆ g)E , (dG(t)f, g)E + (Qf,

(6.56)

172

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

where

ˆ ˆ G(δ) = K ∗ E(δ)K,

δ ∈ ℵ (ℵ is the set of all ﬁnite intervals on the real axis), and ˆ = 1 K ∗ (R ˆ i (A) + R ˆ −i (A))K = 1 [VΘ (−i) + V ∗ (−i)]. Q Θ 2 2 From Theorem 6.2.10, we have that for all f ∈ E with Kf ∈ L, ∞ ˆ (dG(t)f, f )E < ∞, −∞

and αKf 2−

+∞

≤ −∞

ˆ (dG(t)f, f )E ≤ βKf 2−, 2 1+t

(6.57)

where α > 0 and β > 0 are constants. Moreover, (6.28) implies that +∞ ˆ +∞ ˆ , , (d G(t)f, f ) (dG(t)g, g)E , ˆ , E · . , Qf, g , ≤ C 2 1+t 1 + t2 E −∞ −∞

(6.58)

By (6.56) we have for any f, g ∈ E, +∞

ˆ g)E + (VΘ (z)f, g)E = (Qf, −∞

1 t − t−z 1 + t2

ˆ (dG(t)f, g)E .

(6.59)

On the other hand (6.52) implies +∞

(VΘ (z)f, g)E = (Qf, g)E +z(Xf, g)E + −∞

1 t − t−z 1 + t2

(dG(t)f, g)E . (6.60)

ˆ g)E , (Xf, g)E = 0, and Comparing (6.59) and (6.60) we get (Qf, g)E = (Qf, ˆ (G(δ)f, g) = (G(δ)f, g) (δ ∈ ℵ), for all f, g ∈ E. Taking into account the uniqueˆ ness of representation (6.56), we ﬁnd that X = 0 and G(δ) = G(δ) (δ ∈ ℵ). Thus, +∞ 1 t VΘ (z) = Q + − dG(t). (6.61) t−z 1 + t2 −∞

ˆ Since E(δ) coincides with E(δ) on L, then for any h ∈ G, we have +∞ (dG(t)h, h)E < ∞, Kh ∈ L. −∞

6.4. Realizable Herglotz-Nevanlinna operator-functions. Class N (R)

173

From Theorem 6.2.10 we get +∞ (dG(t)h, h)E = ∞, Kh ∈ / L. −∞

Further, since Q=

1 1 ∗ ˆ ˆ −i (A))K , [VΘ (i) + VΘ (−i)] = K (Ri (A) + R 2 2

we have Ran(Q) ⊆ Ran(K ∗ ) ⊆ E. Now formula (6.58) yields |(Qf, g)E | ≤ Cf E · gE ,

f, g ∈ E.

On the other hand, if Kh ∈ L, then 1 ∗ K ((A + iI)−1 + (A − iI)−1 )Kh 2 +∞ +∞ t t ∗ =K dE(t)Kh = dG(t)h. 2 1+t 1 + t2

Qh =

−∞

−∞

Thus we conclude that VΘ (z) belongs to the class N (R). This completes the proof. Corollary 6.4.4. Let Θ be an L-system and VΘ (z) be its impedance function. Then the channel operator K of Θ is invertible if and only if +∞

(Im VΘ (i)f, f )E = −∞

(dG(t)f, f )E > 0, 1 + t2

∀f ∈ E \ {0},

(6.62)

where G(t) is the measure from representation (6.61). Proof. By Theorem 6.4.3 VΘ (z) ∈ N (R) and has the representation (6.61). The proof follows directly from (6.57). Remark 6.4.5. Let Θ be an L-system from the statement of Theorem 6.4.3 and VΘ (z) be its impedance function. Let G = ker(K) = {f ∈ E | Kf = 0},

G ⊥ = E G.

(6.63)

Suppose h = h1 +h2 ∈ E, h1 ∈ G, h2 ∈ G ⊥ , and K2 := K G ⊥ . Then Kh = Kh2 = K2 h2 and for u ∈ H+ we have (K ∗ u, h)E = (u, Kh) = (u, K2 h2 ) = (K2∗ u, h2 )E = (K2∗ u, h1 + h2 )E = (K2∗ u, h).

174

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

Hence, K ∗ = K2∗ and thus VΘ (z)h = K ∗ (Re A − zI)−1 K(h1 + h2 ) = K ∗ (Re A − zI)−1 K2 h2 = K2∗ (Re A − zI)−1 K2 h2 ,

VΘ (z)h1 = 0.

Consequently, since G is invariant under VΘ (z) and VΘ (¯ z ), VΘ (z) can be written in a diagonal block-matrix form with respect to decomposition (6.63) 0 0 VΘ (z) = , VΘ,2 (z) = K2∗ (Re A − zI)−1 K2 , (6.64) 0 VΘ,2 (z) where K2 is invertible. Theorem 6.4.6. Let A˙ be a closed symmetric operator with equal and ﬁnite deﬁciency indices, acting on a Hilbert space H. Let Aˆ be a self-adjoint extension of ˆ Suppose E is a A˙ and B be a t-self-adjoint bi-extension of A˙ with quasi-kernel A. ˙ B . Deﬁne Hilbert space, K ∈ [E, H− ], ker(K) = {0}, and Ran(K) = L+L V (z) = K ∗ (B − zI)−1 K. Assume J ∈ [E, E], J = J ∗ = J −1 , and the operator I + iV (−i)J is invertible. Deﬁne the operator A = B + iKJK ∗. ˙ −i ∈ ρ(T ). Moreover, if we form Then A is a (∗)-extension of some T ∈ Λ(A), the L-system, A K J Θ= , H+ ⊂ H ⊂ H− E then

WΘ (z) = (I + iV (z)J)−1 (I − iV (z)J),

for all z ∈ ρ(T ), Im z = 0. ˙ B and (4.43),(4.44), (4.45) we get that Proof. From ker(K) = {0}, Ran(K) = L+L ˙ The equality the dimension of E is equal to the deﬁciency number of A. I + iV (−i)J = J(I + iJV (−i))J, implies that I + iJV (−i) is invertible. Let Γ = (I + iJV (−i))−1 . The equality (B + iI)−1 KE = N−i yields that ker(A + iI) ⊆ N−i . Moreover, since (A + iI)(B + iI)−1 KΓ = KΓ + iKJK ∗ (B + iI)−1 KΓ = K(I + iJV (−i))Γ = K we obtain ker(A + iI) = {0} and (A + iI)−1 K = (B + iI)−1 KΓ. Let Υ = (I − iJV ∗ (−i))−1 = (I − iJV (i))−1 .

6.4. Realizable Herglotz-Nevanlinna operator-functions. Class N (R)

175

Similarly, it can be shown that ker(A∗ − iI) = {0} and (A∗ − iI)−1 K = (B − iI)−1 KΥ. Suppose that B is of the form (3.13), that is

i −1 + ∗ + ˙ + B = AP + A P + R S − J PM , ˙ M Dom(A) 2

where S is a (1)-self-adjoint operator in M. For ϕ ∈ Ni we have A˙ ∗ ϕ = iP ϕ and hence + −1 i ∗ + ˙ + (A + iI)ϕ = AP + A P + R S − J PM ϕ + iϕ ˙ M 2 Dom(A) + −1 ∗ + ˙ + ˙ + = (P AP S − 2i J PM ϕ ˙ + A PM )ϕ + (I − P )APDom(A) ˙ ϕ + iϕ + R Dom(A) + −1 + i ˙ = 2iϕ − i(I − P )ϕ + (I − P )AP S − 2 J PM ϕ. ˙ ϕ+R Dom(A) The vector

i −1 + ˙ + −i(I − P )ϕ + (I − P )AP ϕ + R S − J PM ϕ, ˙ Dom(A) 2

˙ B = Ran(K). Hence, (A + iI)ϕ − 2iϕ ∈ Ran(K). Therefore, there belongs to L+L exists h ∈ N−i such that (A + iI)ϕ − 2iϕ = (A + iI)h, i.e., (A + iI)(ϕ − h) = 2iϕ. Hence, (A + iI)H+ ⊃ Ni . Since ˙ = (A˙ + iI)Dom(A), ˙ (A + iI)Dom(A) ˙ ⊕ Ni = H, we have (A + iI)H+ ⊃ H. and (A˙ + iI)Dom(A) ˙ ∗ −iI)H+ ⊃ H. Therefore using Theorem 2.1.5 Similarly, one can show that (A we conclude that the operators (A + iI)−1 and (A∗ − iI)−1 are (−, ·)-continuous. Let us deﬁne Dom(T ) = (A + iI)−1 H, ∗

−1

Dom(T1 ) = (A − iI)

H,

T = A Dom(T ), T1 = A∗ Dom(T1 ).

(6.65)

It is easy to see that Dom(T ) and Dom(T1 ) are dense in H and that the operators (A + iI)−1 H and (A∗ − iI)−1 H are (·, ·)-continuous. The points (−i) and (i) are regular points for the operators T and T1 respectively. This implies that T1 = T ∗ . The operators T and T ∗ are quasi-kernels of operators A and A∗ , respectively, and Re A = B is a self-adjoint bi-extension of ˙ Let dim Ni = dim N−i = r, dim L = d. Then the operator A. dim Ni = dim N−i = r − d,

dim N = d,

dim M = 2r − d.

176

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

˙ B ) = r, we ﬁnd that dim(ker(K ∗ ) ∩ M) = r − d. Since dim Ran(K) = dim(L+L ∗ Because ker(K ) ⊇ JRLB and dim(JRLB ) = r − d, we get that A˙ is a maximal ˙ common symmetric part of T and T ∗ . It follows then that T ∈ Λ(A). Finally for the L-system A K J Θ= , H+ ⊂ H ⊂ H− E we have V (z) = VΘ (z) = K ∗ (Re A − zI)−1 K. Now equalities (6.48) give WΘ (z) = (I + iV (z)J)−1 (I − iVΘ (z)J) for all non-real regular points z of T . This completes the proof.

6.5 Realization of the class N (R) In this section we establish the main realization results. Theorem 6.5.1. Let an operator-valued function V (z) in a ﬁnite-dimensional Hilbert space E belong to the class N (R) such that (6.62) holds. Then V (z) is realizable as a transfer function of an impedance system Δ of the form (6.49). Proof. We will use several steps to prove this theorem. Step 1. First we construct a model Hilbert space and a model self-adjoint operator. Let C00 (E, (−∞, +∞)) be the set of continuous compactly supported vectorvalued functions f (t) (−∞ < t < +∞) with values in a ﬁnite dimensional Hilbert space E. We introduce an inner product (·, ·) deﬁned by +∞ (f, g) = (dG(t)f (t), g(t))E , −∞

for all f, g ∈ C00 (E, (−∞, +∞)). In order to construct a Hilbert space, we identify with zero all functions f (t) such that (f, f ) = 0. Then we make the completion and obtain the new Hilbert space L2G (E). Let us note that the set C00 (E, (−∞, +∞)) is dense in L2G (E). Moreover, if f (t) is continuous and +∞ (dG(t)f (t), f (t))E < ∞,

(6.66)

−∞

L2G (E).

then f (t) belongs to The characterization of the Hilbert space L2G (E) has been obtained by I.S. Kac [158]. If we set σ(t) = trace(G(t)), then σ(t) is also a non-decreasing function and, moreover, the operator measure dG is absolutely continuous with respect to the scalar measure dσ. By the RadonNikodym theorem there exists a dσ-measurable density Ψ(t) ∈ [E, E], Ψ(t) ≥ 0

6.5. Realization of the class N (R)

177

almost everywhere with respect to dσ, such that G(Δ) = Ψ(t)dσ(t), Δ ∈ ℵ. Δ

˜ 2 (E) be the set of all dσ-measurable E-valued functions f on R such that Let L G f 20 :=

R

(Ψ(t)f (t), f (t))E dσ(t) < ∞.

(6.67)

If we set N0 = {f : ||f ||0 = 0} , then the equality 2G (E)/N0 L2G (E) = L holds. Notice that

0< R

(6.68)

(Ψ(t)h, h)E dσ(t) < ∞, 1 + t2

for all h ∈ E \ {0}. Let D0 be the set of the continuous vector-valued (with values in E) functions f (t) such that in addition to (6.66), we have +∞ t2 (dG(t)f (t), f (t))E < ∞. −∞

Since C00 ⊂ D0 , it follows that D0 is dense in L2G (E). We introduce an operator A on D0 in the following way Af (t) = tf (t).

(6.69)

Below we denote again by A the closure of the symmetric operator A in (6.69). Now A is a self-adjoint operator in L2G (E). Let H+ = Dom(A) and deﬁne the inner product (f, g)H+ = (f, g) + (Af, Ag) (6.70) for all f, g ∈ H+. It is clear that H+ is a Hilbert space with norm · H+ generated by the inner product (6.70). We equip the space L2G (E) with spaces H+ and H− and get H+ ⊂ L2G (E) ⊂ H− . By R we denote the corresponding Riesz-Berezansky operator, R ∈ [H− , H+ ].

178

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems Consider the following subspaces of the space E: ⎧ ⎫ +∞ ⎨ ⎬ E1 = h ∈ E : (dG(t)h, h)E < ∞ , ⎩ ⎭ −∞ ⎧ ⎫ +∞ ⎨ ⎬ (dG(t)f, h)E E2 = f ∈ E : = 0 for all h ∈ E . 1 ⎩ ⎭ 1 + t2

(6.71)

−∞

Clearly, the direct decomposition ˙ 2 E = E1 +E holds. If h ∈ E1 , then (6.66) implies that the function h(t) = h is an element of the space L2G (E). On the other hand, if h ∈ E and h ∈ / E1 , then h(t) does not belong to L2G (E). It can be shown that any function h(t) = h ∈ E can be identiﬁed with an element of H− . Indeed, since for all h ∈ E, +∞

−∞

the function

(dG(t)h, h)E < ∞, 1 + t2

(6.72)

h h(t) = √ , 1 + t2

belongs to the space L2G (E). Letting f (t) ∈ D0 , we have +∞ (1 + t2 )(dG(t)f (t), f (t))E < ∞. −∞

Therefore, the function f (t) =

√ 1 + t2 f (t) belongs to the space L2G (E) and hence

+∞ (f , h) = (dG(t)f (t), h(t))E . −∞

Furthermore, 1 2 2 +∞ 2 |(f , h)| ≤ f · h = 3 (1 + t2 )(dG(t)f (t), f (t))E −∞

1 2 2 +∞ (dG(t)h, h) 2 =3 f H+ . 1 + t2 −∞

1 2 2 +∞ (dG(t)h), h) 2 ·3 1 + t2 −∞

6.5. Realization of the class N (R) Also,

179

+∞ +∞ + h 2 (dG(t)f (t), h(t))E = 1 + t dG(t)f (t), √ 1 + t2 E

−∞

−∞

+∞ = (dG(t)f (t), h(t))E = (f , h). −∞

Therefore,

+∞ Fh (f ) = (dG(t)f (t), h(t))E , −∞

is a continuous linear functional on H+ , for f ∈ D0 . Since D0 is dense in H+ , h(t) = h belongs to H− . We calculate the Riesz-Berezansky operator R on the vectors h(t) = h, h ∈ E. By the deﬁnition of R, for all f ∈ H+ we have (f, h) = (f, Rh)H+ . Hence, for all f ∈ D0 , +∞ +∞ h(t) 2 (f, h) = (dG(t)f (t), h(t))E = (1 + t ) dG(t)f (t), 1 + t2 E −∞ −∞ h(t) = f, = (f, Rh)H+ . 1 + t2 H +

Thus

h , h ∈ E. (6.73) 1 + t2 Let us mention some properties of the operator A. It is easy to see that for all g ∈ H+ , we have that Ag ≤ gH+ . Taking this into account we obtain Rh =

Af H− = sup

g∈H+

|(Af, g)| |(f, Ag)| f · Ag = sup ≤ sup ≤ f . gH+ gH+ g∈H+ gH+ g∈H+

Hence, the operator A is (·, −)-continuous. Denote by A the extension of the operator A to H with respect to (·, −)-continuity. Now, (A − zI)−1 g − (A − ζI)−1 g = (z − ζ)(A − zI)−1 (A − ζI)−1 g, holds for all g ∈ H− and in particular (A − iI)−1 g − (A + iI)−1 g = 2i(A − iI)−1 (A + iI)−1 g, and

(A − iI)−1 g2 = (A + iI)−1 g2 ,

180

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

for all g in H− . It follows from (6.72) that the element f (t) =

f , t−z

f ∈ E,

belongs to the space L2G (E). It is easy to show that, for all h ∈ E, (A − zI)−1 h =

h , t−z

(Imz = 0).

(6.74)

Step 2. Now we are going to construct a model symmetric operator and its adjoint. Let H+ be the Hilbert space constructed in Step 1 and let ˙ = {f ∈ H+ : (f, h) = 0 Dom(A)

for all h ∈ E} = H+ RE,

(6.75)

˙ where by we mean orthogonality in H+ . We deﬁne an operator A˙ on Dom(A) by the following expression: ˙ A˙ = A Dom(A).

(6.76)

˙ Obviously, A˙ is a closed symmetric operator. We note that if E1 = 0, then Dom(A) 2 ˙ is dense in LG (E). Deﬁne H0 = Dom(A) and let P be the orthogonal projection ˙ = dim E1 < ∞, Theorem 2.4.1 and in H = L2G (E) onto H0 . Since codim Dom(A) ˙ Theorem 4.1.3 yield that operators P A and P A are closed, i.e., A˙ is a regular symmetric operator and A is its regular self-adjoint extension. Deﬁne A1 = A Dom(A1 ),

Dom(A1 ) = H+ RE1 .

The following obvious inclusions hold: A˙ ⊂ A1 ⊂ A. It is easy to see that ˙ ⊕ RE2 , Dom(A1 ) = H0 and A1 is a closed symmetric operaDom(A1 ) = Dom(A) tor. Indeed, if we identify the space E with the space of functions h(t) = h, h ∈ E, then we would obtain L2G (E) H0 = E1 . Since +∞

−∞

(dG(t)h, g)E = 0, 1 + t2

and

g , 1 + t2 for all h ∈ E1 , g ∈ E2 , we ﬁnd that E1 is (·)-orthogonal to RE2 and hence Dom(A1 ) = H0 . We denote by A∗1 the adjoint of the operator A1 . Now we are going to ﬁnd the defect subspaces Ni and N−i of the operator ˙ Since the subspace E ∈ H− is (·)-orthogonal to Dom(A), ˙ we have that (A ± A. iI)−1 E = N±i . Moreover, by (6.74) we have Rg =

(A ± iI)−1 h =

h , t±i

h ∈ E.

6.5. Realization of the class N (R)

Therefore N±i =

f (t) ∈

181

h f (t) = , t±i

L2G (E),

4 h∈E .

Similarly, the defect subspaces of the operator A1 are

4 h 0 2 N±i = f (t) ∈ LG (E), f (t) = , h ∈ E1 . t±i Obviously, N0z ⊂ D0 because +∞

−∞

t2 (dG(t)h, h)E ≤ K(z) |t − z|2

+∞ (dG(t)h, h)E < ∞,

h ∈ E1 .

−∞

Taking into account that ˙ N0i N0−i , Dom(A∗1 ) = Dom(A) we can conclude that Dom(A∗1 ) ⊆ Dom(A). At the same time, the inclusion A1 ⊂ A implies that Dom(A∗1 ) ⊃ Dom(A). Therefore we obtain Dom(A∗1 ) = Dom(A) and P A = A∗1 . Since A is the self-adjoint extension of operator A˙ we ﬁnd by (1.33) that ˙ (I − U )Ni , Dom(A) = Dom(A) for some admissible isometric operator U acting from Ni into N−i . It is easy to check that U (A − iI)−1 h = (A + iI)−1 h, for all h in E. Consequently, the operator U has the form h h U = , h ∈ E. t−i t+i Straightforward calculations show that h h h 2it A(I − U ) =t −t = 2 h. t−i t−i t+i t +1 ˙ The linear manifold H+ = Dom(A˙ ∗ ) becomes a Hilbert Let A˙ ∗ be the adjoint to A. space with the inner product (x, y)+ = (x, y) + (A˙ ∗ x, A˙ ∗ y),

x, y ∈ H+ .

Let H+ ⊂ L2G (E) ⊂ H− be the rigged triplet and let R be the corresponding Riesz-Berezansky operator. Since P A is a closed operator, then H+ is a subspace of H+ . By Theorem 2.5.7, H+ = Dom(A) (U + I)Nˆi , where ˆi = {ϕ ∈ Ni , (U − I)ϕ ∈ H0 }, N

(6.77)

182

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

˙ and U is an admissible isometric operator related to A. Taking H0 = Dom(A), into account that h −2ih (U − I) = 2 , h ∈ E, t−i t +1

we conclude that ˆi = N

4 g , g ∈ E2 = E E1 . t−i

Therefore, Dom(A˙ ∗ ) = Dom(A)

tg 2 t +1

4 ,

g ∈ E2 .

Step 3. In this step we will construct a special self-adjoint bi-extension whose quasi-kernel coincides with the operator A. Applying (2.15), we have ˙ ⊕ N ⊕ N ⊕ N, H+ = Dom(A) i −i ˙ N = RE1 , and where N±i are semi-deﬁciency spaces of the operator A, ˙ ⊕ E1 = H = L2G (E). Dom(A) We begin by setting + + (f, g)1 = (f, g)+ + (PN f, PN g)+ ,

for all f, g ∈ H+ .

+ Here PN is an orthoprojection of H+ onto N. Obviously, the norm ·1 is equivalent to · + . We denote by H+1 the space H+ equipped with the norm · 1 , so that H+1 ⊂ H ⊂ H−1 is the corresponding rigged space with Riesz-Berezansky operator R. By Theorem 2.5.5 there exists a (1)-isometric operator U such that

˙ ⊕ (U + I)(N ⊕ N), Dom(A) = Dom(A) i where Dom(U) = Ni ⊕ N, Ran(U) = N−i ⊕ N and (−1) is a regular point for the operator U. (For further convenience we have substituted U in the statement of Theorem 2.5.5 by −U). Moreover, ⎧ + −1 ˙∗ ⎪ ⎪ ⎨ϕ = i(I + PNi )(A + iI) χ, + −1 ˙∗ Uϕ = i(I + PN U χ, )(A − iI) −i ⎪ ⎪ ⎩ϕ ∈ Dom(U), χ ∈ N . i

Here U is the isometric operator described in (6.77) of Step 2. Consequently we obtain ⎧ + i ˙∗ ⎪ ⎨χ = − 2 (A + iI)(I + PN )ϕ, + U χ = − 2i (A˙ ∗ − iI)(I + PN )Uϕ, ⎪ ⎩ ϕ ∈ Dom(U), χ ∈ Ni .

6.5. Realization of the class N (R) It follows that

183

+ χ − U χ = ϕ + Uϕ + iA∗ PN (U − I)ϕ, + χ + U χ = ϕ − U ϕ − iA∗ PN (I + U)ϕ.

+ Since χ − U χ = ϕ + Uϕ + iA∗ PN (U − I)ϕ, we ﬁnd that χ − U χ ∈ H0 if and only if + + ˙ ⊂ H0 PN (U + I)ϕ = 0. This follows from the fact that A˙ ∗ PN (U − I)ϕ ∈ Dom(A) and from the direct decomposition H = H0 N (see Proposition 2.4.2). Let us + note that if PN (U + I)ϕ = 0, then χ + U χ = ϕ − U ϕ. Thus, + + Nˆi = {f = (A˙ ∗ + iI)(I + PN )ϕ, PN (U + I)ϕ = 0}.

Let + N = ker PN (I + U).

Then we have H+ = Dom(A) (I − U )N.

(6.78)

We denote by P the projection operator of H+ onto Dom(A) with respect to the decomposition (6.78). Let P = I−P. Since Dom(A) = H+ , we have P ∈ [H+ , H+ ]. We will denote by P ∗ ∈ [H− , H− ] the adjoint operator to P, i.e., (Pf, g) = (f, P ∗ g), ˆi , then φ + U φ = (I − U)ϕ, for ϕ ∈ N , and for all f ∈ H+ , g ∈ H− . If φ ∈ N + + + A˙ ∗ (I − U )ϕ = iPN ϕ + iPN Uϕ + APN (I − U)ϕ i

−i

+ = i(U + I)ϕ + A˙ ∗ PN (I − U)ϕ.

This implies A˙ ∗ (I + U )φ = i(φ − U φ). Hence tg g ∗ ˙ A =− 2 , 2 t +1 t +1

g ∈ E2 .

(6.79)

Let Q be the operator in the deﬁnition of the class N (R). We introduce a new operator Φ acting in the following way: Φf = −Q R−1 A˙ ∗ Pf,

f ∈ H+ .

(6.80)

In order to show that Φ ∈ [H+ , E], we consider the following calculation for f ∈ H+ : |(Φf, g)E | |(QR−1 A˙ ∗ Pf, g)E | = sup gE gE g∈E g∈E −1 ˙ ∗ |(R A Pf, Qg)E | R−1 A˙ ∗ Pf E · QgE = sup ≤ sup gE gE g∈E g∈E ∗ ∗ ˙ ˙ ≤ βA Pf H+ ≤ αA Pf H+ , α, β - positive constants.

Φf E = sup

184

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

Here we used that Pf ⊂ Dom(A), for all f ∈ H+ , formulas (6.73) and (6.79), and the equivalence of the norms · H+ and · + = · H+ . For f ∈ H+ , we have Pf = (I − U )ϕ, ϕ ∈ N and + A˙ ∗ Pf = i(U + I)ϕ + iA∗ PN (U − I)ϕ.

Hence + + + A˙ ∗ PN (U − I)ϕ2+ = A˙ ∗ PN (U − I)ϕ2 + A˙ ∗ A˙ ∗ PN (U − I)ϕ2 + + = A˙ ∗ PN (U − I)ϕ2 + P PN (U − I)ϕ2 ≤ A˙ ∗ P + (U − I)ϕ2 + P + (U − I)ϕ2

=

N + PN (U

−

I)ϕ2+ ,

N

and + + i(U + I)ϕ + iA˙ ∗ PN (U − I)ϕ2+ = A˙ ∗ PN (U − I)ϕ2+ + ϕ + Uϕ2+ + ≤ PN (U − I)ϕ2+ + ϕ + Uϕ2+ = ϕ − U ϕ2+ .

This implies that there exists a constant a > 0 such that A˙ ∗ Pf ≤ Pf + ≤ af +,

∀f ∈ H+ .

Therefore, for some constant b > 0 we have Φf ≤ bf +, ∀f ∈ H+ . Thus, Φ ∈ [H+ , E]. Let Φ∗ be the adjoint operator to Φ, i.e., Φ∗ ∈ [E, H− ] and for all f ∈ H+ , g ∈ E, (Φf, g)E = (f, Φ∗ g). Since Φ(Dom(A)) = 0, Ran(Φ∗ ) is (·)-orthogonal to Dom(A). Letting M = N−i ⊕ Ni ⊕ N and using (6.78) we obtain M = (U + I)(Ni ⊕ N) (I − U )N. In the space M we deﬁne an operator S in the following way: S(ϕ + Uϕ) =

i (I − U)ϕ, 52

ϕ ∈ Ni ⊕ N,

6 i + + S(ψ − U ψ) = −R(Φ∗ + P ∗ )R−1 A˙ ∗ + (PN − P ) (ψ − Uψ), N−i i 2

(6.81)

where ψ ∈ N . In order to show that S is a (1)-self-adjoint operator on M we must verify the equalities (a) (S(ϕ + Uϕ), ϕ + Uϕ)1 = (ϕ + Uϕ, S(ϕ + Uϕ))1 , ϕ ∈ Ni ⊕ N, (b) (S(ψ − U ψ, ψ − Uψ)1 = (ψ − Uψ, S(ψ − Uψ))1 , ψ ∈ N, (c) (S(ϕ + Uϕ), ψ − U ψ)1 = (ϕ + Uϕ, S(ψ − Uψ))1 ϕ ∈ Ni ⊕ N, ψ ∈ N. The equality (a) easily follows from the (1)-unitary property of the operator U. + Let us check (b). Since PN (ψ + Uψ) = 0, for ψ ∈ N we get + + PN (ψ − U ψ) = ψ + Uψ, ψ ∈ N − PN i

−i

6.5. Realization of the class N (R)

185

and (ψ + Uψ), ψ − U ψ)1 = 0. Since P(I − U)N = 0, we have (RP ∗ R−1 A˙ ∗ (ψ − U ψ), ψ − U ψ) = (R−1 A˙ ∗ (ψ − U ψ), P(ψ − Uψ)) = 0. This allows us to consider only the term of (6.81) that contains Φ∗ . Because Q is a self-adjoint operator in E, we have (S(ψ − U ψ), ψ − U ψ)1 = (−RΦ∗ R−1 A˙ ∗ (ψ − Uψ), (ψ − U ψ)1 = (R−1 A˙ ∗ (ψ − U ψ), −Φ(ψ − U ψ))E = (R−1 A˙ ∗ (ψ − U ψ), QR−1 A˙ ∗ P(ψ − U ψ))E = (QR−1 A˙ ∗ (ψ − U ψ), R−1 A˙ ∗ (ψ − U ψ))E = ((ψ − U ψ), −Φ∗ R−1 A˙ ∗ (ψ − U ψ))E = ((ψ − U ψ), −RΦ∗ R−1 A˙ ∗ (ψ − U ψ))1 = ((ψ − U ψ), S(ψ − U ψ))1 . + + Now we will show that (c) holds. The relation PN (ψ + Uψ) = 0 implies PN ψ= + −PN Uψ. Also, (ϕ, ψ)1 = (Uϕ, Uψ)1 , since U is a (1)-isometric mapping. Hence,

i ((I − U )ϕ, ψ − U ψ)1 2 i i = i(ϕ, ψ)1 − (ϕ, Uψ)1 − (Uϕ, ψ)1 2 2 i i + + = i(ϕ, ψ)1 − (ϕ, PN Uψ)1 − (Uϕ, PN ψ)1 2 2 i + = i(ϕ, ψ)1 + (PN (I − U )ϕ, ψ)1 . 2

(S(ϕ + Uϕ), ψ − U ψ)1 =

Also, note that i + i + ϕ + Uϕ, (PN = − (ϕ + Uϕ, ψ + Uψ)1 , − P )(ψ − U ψ) N−i i 2 2 1 and (ϕ + Uϕ, S(ψ − U ψ)1 = (ϕ + Uϕ, −R(Φ∗ + P ∗ )R−1 A˙ ∗ (ψ − Uψ))1 i − (ϕ + Uϕ, ψ + Uψ)1 . 2 Next, recall that Ran(Φ∗ ) is (·)-orthogonal to Dom(A) and ˙ ⊕ (U + I)(N ⊕ N). ϕ + Uϕ ∈ Dom(A) = Dom(A) i Moreover, + + ˙∗ + A˙ ∗ (ψ − U ψ) = iPN ψ + iPN Uψ + A PN (ψ − U ψ) i −i = i(ψ + Uψ) + A˙ ∗ P + (ψ − U ψ) ∈ Dom(A). N

186

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

Thus, (ϕ + Uϕ, RΦ∗ R−1 A˙ ∗ (ψ − Uψ))1 = (ϕ + Uϕ, Φ∗ R−1 A˙ ∗ (ψ − Uψ)) = 0, (ϕ + Uϕ, −RP ∗ R−1 A˙ ∗ (ψ − Uψ))1 = −(ϕ + Uϕ, A˙ ∗ (ψ − Uψ))H+ = −(ϕ + Uϕ, A˙ ∗ (ψ − Uψ)) − (A(ϕ + Uϕ), AA˙ ∗ (ψ − U ψ)). Applying Theorem 2.5.5 we obtain + A(ϕ + Uϕ) = A˙ ∗ (ϕ + Uϕ) + i(A˙ A˙ ∗ + I)PN (I − U)ϕ, + A˙ ∗ (ψ − U ψ) = i(I + U)ψ + A˙ ∗ PN (I − U)ψ, + + AA˙ ∗ (ψ − Uψ) = A˙ A˙ ∗ PN (I − U)ψ + iA˙ ∗ (U + I)ψ − (A˙ A˙ ∗ + I)PN (I − U)ψ + ∗ ˙ = iA (U + I)ψ − P (I − U )ψ. N

The above identities yield + (ϕ + Uϕ, A˙ ∗ (ψ − U ψ))H+ = (ϕ + Uϕ, i(ψ + Uψ))1 − i(PN (I − U )ϕ, ψ)1 .

Hence, (ϕ + Uϕ, −RP ∗ R−1 A˙ ∗ (ψ − U ψ)) = i(ϕ + Uϕ, ψ + Uψ)1 + i(PN+ (I − U )ϕ, ψ)1 i i ϕ + Uϕ, (ψ + Uψ) = − (ϕ + Uϕ, ψ + Uψ)1 , 2 2 1 and (ϕ + Uϕ, S(ψ − U ψ))1 i + = i(ϕ + Uϕ, ψ + Uψ)1 + i(PN (I − U )ϕ, ψ)1 − (ϕ + ϕ, ψ + Uψ)1 2 i i + = i(ϕ, ψ)1 + (Uϕ, ψ)1 + (ϕ, Uψ)1 + i(PN (I − U)ϕ, ψ)1 2 2 i + = i(ϕ, ψ)1 + (PN (I − U )ϕ, ψ)1 2 = (S(ϕ + Uϕ), ψ − Uψ)1 . This shows that S is a (1)-self-adjoint operator in M. Applying (3.13) we obtain that a self-adjoint bi-extension of the operator A˙ is deﬁned by the formula i −1 + + ∗ ˙ ˙ B = APDom(A) (S − J) PM , (6.82) ˙ + A +R 2 where S is given by (6.81). Obviously, if f = g + (U + I)ϕ, ϕ ∈ Ni ⊕ N, and ˙ then Bf = Af . This means that the quasi-kernel of the operator B g ∈ Dom(A), coincides with A and hence B is t-self-adjoint.

6.5. Realization of the class N (R)

187

Step 4. Deﬁne a linear operator K ∈ [E, H− ] by the formula ˆ E1 h, Kh = (P ∗ + Φ∗ )PE2 h + IP

h ∈ E,

(6.83)

where PE2 and PE1 are projections of the space E onto E2 and E1 in (6.71), respectively, and Iˆ is an embedding of E into H− . Let z be a regular point of the ˆ z = (B − zI)−1 . Also, note that operator A. By Theorem 4.5.12 we have R ˆ z f, g) = (f, (A − z¯I)−1 g), (R

∀f ∈ H− , g ∈ H.

As it was shown in Step 1 (see (6.74)) (A − zI)−1 h =

h , t−z

∀h ∈ E,

where E is considered as a subspace of H− . Clearly, ˆ z P ∗ h, g) = (P ∗ h, (A − z¯I)−1 g) (R = (h, (A − z¯I)−1 g) = ((A − zI)−1 h, g), ∀h ∈ E, g ∈ H = L2G (E). It follows that ˆz P ∗h = R

h , t−z

∀h ∈ E.

Since Φ(Dom(A)) = 0, then Φ(A − z¯I)−1 g = 0 for all g ∈ H, and we have ˆ z Φ∗ h, g) = Φ∗ h, (A − z¯I)−1 g = (h, Φ(A − z¯I)−1 g) = 0. (R Thus, h , h ∈ E. t−z If h = h1 + h2 , where h1 ∈ E1 and h2 ∈ E2 , then ˆ z (Φ∗ + P ∗ )h = R ˆ z P ∗h = R

ˆ z Kh1 = R ˆ z P ∗ h1 = h1 , h1 ∈ E1 , R t−z h 2 ˆ z Kh2 = R ˆ z h2 = R , h2 ∈ E2 , t−z This implies that the operator K is invertible. Indeed, if Kh = 0, then ˆ 2, (P ∗ + Φ∗ )h1 = −Ih ˆ z Kh = 0. Hence, R ˆ z (P ∗ + Φ∗ )h1 = −R ˆ z h2 . That is, and R h1 h2 = , t−z t−z

h = h1 + h2 ,

ˆ z K ∈ [E, H+ ], since R ˆ z maps which implies that h = 0. We should also note that R Ran(K) into H+ continuously.

188

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

Step 5. Let K ∗ ∈ [H+ , E] be the adjoint to the operator K in (6.83), i.e., (Kf, g) = (f, K ∗ g), f ∈ E, g ∈ H+ . We consider an operator-valued function Vˆ deﬁned by ˆ z K, Imz = 0. Vˆ (z) = K ∗ R (6.84) ˆ z Kg, Kh) We will prove that V (z) = Vˆ (z), Im z = 0. Obviously, (Vˆ (z)g, h)E = (R for g ∈ E, h ∈ E, g = g1 + g2 , h = h1 + h2 , where g1 , h1 ∈ E1 , g2 , h2 ∈ E2 . Therefore, ˆ z Kg, Kh) = (R ˆz (P ∗ + Φ∗ )g2 + R ˆ z g1 , (P ∗ + Φ∗ )h2 + Ih ˆ 1) (R ˆ z P ∗ g2 + R ˆ z g1 , (P ∗ + Φ∗ )h2 + Ih ˆ 1) = (R ˆz P ∗ g2 , P ∗ h2 ) + (R ˆ z P ∗ g 2 , Φ∗ h 2 ) + ( R ˆ z P ∗ g 2 , h1 ) = (R ˆ z g1 , P ∗ h2 ) + (R ˆ z g1 , Φ∗ h2 ) + (R ˆ z g 1 , h1 ) + (R ˆ z P ∗ g2 , h2 ) + (ΦR ˆ z P ∗ g2 , h2 )E + (R ˆ z P ∗ g 2 , h1 ) = (P R ˆ z g1 , h2 ) + (ΦR ˆz g1 , h2 )E + (R ˆ z g1 , h1 ). + (R We also have

ˆ z P ∗ g2 = g2 ∈ R / H+ . t−z

Consider an element g2 tg2 g2 ztg2 − 2 = + , 2 t−z t +1 (t − z)(t + 1) (t − z)(t2 + 1) Clearly,

+∞

|t −

−∞

and

+∞

−∞

t2 z|2 (t2

+ 1)

·

g2 ∈ E2 .

(dG(t)g2 , g2 )E < ∞, 1 + t2

|z|2 t4 (dG(t)g2 , g2 )E · < ∞. |t − z|2 (t2 + 1) 1 + t2

Hence

g2 tg2 − 2 ∈ Dom(A). t−z t +1

Moreover, tg2 ∈ (I − U )N, +1

t2

g2 ∈ E2 .

This implies

P

g2 t−z

4 =

g2 tg2 − 2 , t−z t +1

P

g2 t−z

4 =

tg2 . t2 + 1

6.5. Realization of the class N (R)

189

Consequently, +∞

∗

ˆ z P g 2 , h2 ) = (P R −∞

1 t − 2 t−z t +1

(dG(t)g2 , h2 )E .

We also have that ˆ z P ∗ g2 , h2 )E = −(QR−1 A˙ ∗ PR ˆ z P ∗ g2 , h2 )E = −(R−1 A˙ ∗ PR ˆ z Pg2 , Qh2 )E . (ΦR From (6.73) and (6.79) ˆ z P ∗ g2 = R−1 − g2 R−1 A˙ ∗ PR = −g2 , t2 + 1 ˆ z P ∗ g2 , h2 )E = (g2 , Qh2 )E = (Qg2 , h2 )E . Furthermore, implying (ΦR +∞

∗

1 t−z

ˆ z P g 2 , h1 ) = (R −∞ +∞

= −∞ +∞

= −∞ +∞

= −∞

1 t−z 1 t−z

(dG(t)g2 , h1 )E

(dG(t)g2 , h1 )E − (g2 , Qh1 )E + (Qg2 , h1 )E

⎛

+∞

(dG(t)g2 , h1 ) − ⎝g2 ,

−∞

1 t − 2 t−z t +1

⎞ t dG(t)h1 ⎠ + (Qg2 , h1 )E t2 + 1

E

(dG(t)g2 , h1 )E + (Qg2 , h1 )E .

ˆ z g1 = 0, we have Since ΦR +∞

ˆ z g 1 , h2 ) = (R −∞ +∞

= −∞

1 t−z

(dG(t)g1 , h2 )E − (Qg1 , h2 )E + (Qg1 , h2 )E

1 t − 2 t−z t +1

(dG(t)g1 , h2 )E + (Qg1 , h2 )E .

Similarly, +∞

ˆ z g 1 , h1 ) = (R −∞

1 t − 2 t−z t +1

(dG(t)g1 , h1 )E + (Qg1 , h1 )E .

190

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

Finally, +∞

ˆ z g, h) = (R −∞

1 t − 2 t−z t +1

(dG(t)g, h)E + (Qg, h)E ,

and hence, +∞

(Vˆ (z)g, h) = −∞

1 t − 2 t−z t +1

(dG(t)g, h)E + (Qg, h)E .

(6.85)

Since the integral representation (6.85) of the function Vˆ (z) completely matches the one of V (z) ∈ N (R) in (6.55), we conclude that Vˆ (z) ≡ V (z). But by (6.84), Vˆ (z) is the transfer function of the impedance system Δ of the form (6.49) with the self-adjoint state space operator A that is the quasi-kernel of operator B deﬁned by (6.82) and operator K is deﬁned by (6.83). Therefore, we have constructed an impedance system whose transfer function coincides with V (z). This completes the proof. Theorem 6.5.2. Let an operator-valued function V (z) in a Hilbert space E belong to the class N (R) such that (6.62) holds. Then V (z) can be realized as the impedance function of an L-system Θ of the form (6.31)–(6.36) with invertible channel operator K and preassigned directing operator J for which I + iV (−i)J is invertible. Proof. In the proof of this theorem we will heavily rely on the construction developed in the proof of Theorem 6.5.1. Let D = KJK ∗ ,

(6.86)

where J ∈ [E, E] satisﬁes J = J ∗ = J −1 and K is deﬁned by (6.83). Since ˙ then Dφ = 0 for all φ ∈ Dom(A). ˙ Moreover, Ran(K) is orthogonal to Dom(A), (Df, g) = (f, Dg) for all f ∈ H+ , g ∈ H+ . We deﬁne an operator A by A = B + iD,

(6.87)

where B is deﬁned by (6.82) and D by (6.86). We will prove that A is a (∗)-extension ˆ ±i Kg = Kg, for of some operator T of the class Λ. Let us show that (B + iI)R all g ∈ E, where B is a t-self-adjoint bi-extension deﬁned by (6.82). By Theorem 4.5.12, the equation (B − zI)x = f has a unique solution x for any 5 6 i + i + −1 f ∈ Ran R S − PN + PN + E1 . 2 i 2 −i We are going to show that in fact 5 6 i + i + Ran(K) = Ran R−1 S − PN + P + E1 . 2 i 2 N−i

6.5. Realization of the class N (R)

191

If ψ ∈ N , then i + i + S − PN + PN (ψ − U ψ) = R(Φ∗ + P ∗ )R−1 A˙ ∗ (ψ − U ψ). 2 i 2 −i Using (6.79) we can conclude that R−1 (I − U)N = E2 , and hence 5 6 i + i + −1 Ran R S − PN + PN (I − U)N = (Φ∗ + P ∗ )E2 . 2 i 2 −i + + Letting P + = PN + PN , we have i

−i

i + i + P + S − PN PN (I + U)ϕ = 0, ϕ ∈ M. + 2 i 2 −i + i + ˆz = Therefore, E1 + Ran R−1 S − 2i PN P = Ran(K). Since R + 2 N i

−i

(B − zI)−1 , the above calculations imply ˆ z Kg, (B − zI)−1 Kg = R ˆ z KE = Nz is the defect space of the for all g ∈ E. For Im z = 0 we have that R ˙ Therefore (B + iI)R ˆ ±i Kg = Kg and R ˆ ±i KE = N±i . It follows that operator A. V (z) = K ∗ (B − zI)−1 K, Im z = 0. Now we apply Theorem 6.4.6.

Remark 6.5.3. In the proof of Theorem 6.5.2 above we have shown that (−i) is a regular point for the constructed operator T . Hence VΘ (z) is a linear-fractional transformation of the form (6.48) of the transfer function WΘ (z) in some neighborhood of the point (−i). Now we will show that condition (6.62) that was used in the statements of Theorems 6.5.1 and 6.5.2 can be released. Theorem 6.5.4. Let an operator-valued function V (z) in a Hilbert space E belong to the class N (R). Then V (z) can be realized as: 1. transfer function of an impedance system Δ of the form (6.49), 2. impedance function of an L-system Θ of the form (6.35)–(6.36) with a preassigned directing operator J for which I + iV (−i)J is invertible. Proof. Let V (z) ∈ N (R) and have a representation (6.55). Deﬁne the subspaces G and G ⊥ of E by the formula

4 +∞ (dG(t)f, f )E G= f ∈E| = 0 , G ⊥ = E G. (6.88) 2 1 + t −∞

192

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

The deﬁnition of G in (6.88) implies that, for any interval δ ∈ ℵ, δ

(dG(t)f, f )E = 0. 1 + t2

Taking into account that 1/(1+t2) increases on the left real semi-axis and decreases on the right semi-axis, we have (G(t)f, f )E = (G(δ)f, f )E = 0, δ

for all f ∈ G. Thus G(δ)f = 0, ∀f ∈ G, δ ∈ ℵ. Considering this and integral representation (6.53) of Q we have that both operator Q and the measure G(δ) in the representation (6.55) have block-matrix form with respect to decomposition (6.88), 0 0 0 0 Q= , G(δ) = , δ ∈ ℵ. (6.89) 0 Q2 0 G2 (δ) Consequently, our function V (z) can also be written as V (z) =

0 0 0 V2 (z)

+∞

,

V2 (z) = Q2 + −∞

1 t − t−z 1 + t2

dG2 (t),

(6.90)

where Q2 and G2 (t) are described by (6.89). But by construction V2 (z) is such that the inequality (6.62) holds. Therefore, by Theorem 6.5.1, V2 (z) can be realized as a transfer function of an impedance system Δ of the form (6.49), and by Theorem 6.5.2 as an impedance function of an L-system Θ of the form (6.36)-(6.36) with a preassigned directing operator J for which I + iV (−i)J is invertible. In order to realize the function V (z) by the same type of systems that realize V2 (z), we only need to replace the input-output space G of the realizing systems by E and 0 channel operator K2 ∈ [G, H− ] by K ∈ [E, H− ], where K = . Clearly, K2 Ran(K) = Ran(K2 ) = Ran(Im A). Theorem 6.5.5. Let an operator-valued function V (z) in a Hilbert space E belong to the class N (R). Then V (z) can be realized as an impedance function of a scattering (J = I) L-system Θ of the form (6.35)–(6.36). Proof. It can be seen that when J = I the invertibility condition for the operator I +iV (−i)J is satisﬁed automatically. Indeed, it follows from representation (6.55) that +∞ dG(t) V (−i) = −i + Q = −iB + Q, 1 + t2 −∞

6.5. Realization of the class N (R)

193

where B ≥ 0 is the integral in the above formula. Suppose (I + iV (−i))f = 0 for some f ∈ E. Then (I + iV (−i))f = f + Bf + iQf = 0,

f ∈ E,

and ((I + iV (−i))f, f ) = (f, f ) + (Bf, f ) + i(Qf, f ) = 0 for f ∈ E. Consequently, (f, f ) + (Bf, f ) = 0 and, since B is a non-negative operator, we have f = 0. Thus, I + iV (−i) is invertible. The rest follows from Theorem 6.5.2. The following theorem deals with the realization of two realizable operatorvalued Herglotz-Nevanlinna functions that diﬀer from each other only by constant terms in representation (6.55). Theorem 6.5.6. Let the operator-valued functions +∞

V1 (z) = Q1 + −∞

and

+∞

V2 (z) = Q2 + −∞

1 t − t − z 1 + t2

1 t − t − z 1 + t2

dG(t),

(6.91)

dG(t),

belong to the class N (R). Then they can be realized as impedance functions of L-systems A1 K1 I Θ1 = , (A1 ⊃ T1 ), (6.92) H+ ⊂ H ⊂ H− E

and Θ2 =

A2 H+ ⊂ H ⊂ H−

K2

I , E

(A2 ⊃ T2 ),

respectively, so that the operators T1 and T2 acting on the Hilbert space H are both extensions of the symmetric operator A˙ deﬁned in this Hilbert space. Proof. Applying Theorem 6.5.2 to the function V1 (z), we obtain a system Θ1 of the type (6.92). The corresponding symmetric operator A1 constructed in Steps 1 and 2 of the proof of Theorem 6.5.2 satisﬁes the formulas (6.75) and (6.76). The construction of A1 doesn’t involve the operator Q1 from (6.91). It is easy to see (1) (1) that the corresponding rigged Hilbert space H+ ⊂ H(1) ⊂ H− was built without the use of the operator Q1 too. Similarly, if we apply Theorem 6.5.2 to the function V2 (z) we get the corresponding symmetric operator A2 = A1 and the same rigged Hilbert space. This happens because the operator-functions V1 (z) and V2 (z) diﬀer from each other only by the constant terms Q1 and Q2 . Setting A˙ = A1 = A2 , we can conclude ˙ that T1 and T2 are both extensions of the symmetric operator A.

194

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

6.6 Minimal realization and the theorem on bi-unitary equivalence Following Chapter 5, we call a closed linear operator in a Hilbert space H a prime operator if there exists no non-trivial reducing invariant subspace of H on which it induces a self-adjoint operator. A rigged canonical system Θ of the form (6.31)(6.36) is said to be a minimal L-system if its symmetric operator A˙ is a prime operator. Theorem 6.6.1. Let the operator-valued function V (z) belong to the class N (R). Then it can be realized as an impedance function of a minimal L-system Θ of the form (6.31)–(6.36) with a preassigned directing operator J for which I + iV (−i)J is invertible. Proof. Theorem 6.5.2 provides us with a possibility of realization for a given operator-valued function V (z) from the class N (R). Let us assume that its symmetric operator A˙ has a non-trivial reducing invariant subspace H1 ⊂ H on which it generates the self-adjoint operator A1 . Then we can write the (·)-orthogonal decomposition H = H 0 ⊕ H1 , A˙ = A˙ 0 ⊕ A1 , where A˙ 0 is an operator induced by A˙ on H0 . Now let us consider an operator T ⊃ A˙ as in the deﬁnition of the system Θ. We have T = T0 ⊕ A1 , where T0 ⊃ A˙ 0 . Indeed, since A1 is a self-adjoint operator it can not be extended any further. Clearly, Dom(A1 ) = H1 . Similarly, T ∗ = T0∗ ⊕ A1 , where T0∗ ⊃ A˙ 0 . Furthermore, 0 1 H+ = H+ ⊕ H+ = Dom(A˙ ∗0 ) ⊕ Dom(A1 ). We now show that the same holds in the (+)-orthogonality sense. Indeed, if f0 ∈ 0 1 H+ , f1 ∈ H+ = Dom(A1 ), then (f0 , f1 )+ = (f0 , f1 ) + (A˙ ∗ f0 , A˙ ∗ f1 ) = (f0 , f1 ) + (A˙ ∗0 f0 , A1 f1 ) = 0 + 0 = 0. Consequently, we have 0 1 0 1 H+ ⊂ H ⊂ H− = H+ ⊕ H+ ⊂ H 0 ⊕ H1 ⊂ H− ⊕ H− .

Similarly, we obtain A = A0 ⊕ A1 and A∗ = A˙ 0 ⊕ A1 . Therefore, A − A∗ (A0 ⊕ A1 ) − (A∗0 ⊕ A1 ) A0 − A∗0 A1 − A1 A0 − A∗0 = = ⊕ = ⊕ O, 2i 2i 2i 2i 2i 1 1 where O is the zero operator in [H+ , H− ]. This implies that

KJK ∗ = K0 JK0∗ ⊕ O.

6.6. Minimal realization and the theorem on bi-unitary equivalence

195

0 Let P+0 be an orthoprojection operator of H+ onto H+ and set K = K0 . Now ∗ ∗ 0 K = K0 P+ , since for all f ∈ E, g ∈ H+ , then we have

(Kf, g) = (K0 f, g) = (K0 f, g0 + g1 ) = (K0 f, g0 ) + (K0 f, g1 ) = (K0 f, g0 ) = (f, K0∗ g0 ) = (f, K0∗ P+0 g). Next, consider h ∈ E and φ = φ0 + φ1 in H+ such that (A − zI)P+0 φ = Kh. Then (A0 ⊕ A1 − zI)P+0 φ = K0 h, A0 φ0 − zφ0 = K0 h, (A − zI)φ0 = K0 h, φ0 = (A0 − zI)−1 K0 h. On the other hand, φ0 = (A − zI)−1 Kh. Therefore (A − zI)−1 Kh = (A0 − zI)−1 K0 h, and

K ∗ (A − zI)−1 Kh = K0∗ (A0 − zI)−1 K0 h.

This means that the transfer operator-functions of our system Θ and of the system A0 K0 J Θ0 = (6.93) H+ ⊂ H ⊂ H− E coincide. This proves the statement of the theorem.

Corollary 6.6.2. If an operator-valued function V (z) belongs to the class N (R), then V (z) can be realized as an impedance function of a minimal scattering (J = I) L-system Θ of the form (6.36). The proof of the corollary immediately follows from Theorems 6.5.5 and 6.6.1. Remark 6.6.3. Let Θ be an L-system of the form (6.31)-(6.36). Then the minimal L-system Θ0 of the form (6.93) constructed in the proof of Theorem 6.6.1 is called the principal part of the L-system Θ. It follows directly from the proof of Theorem 6.6.1 that both the transfer and impedance functions of any L-system Θ coincide with the transfer and impedance functions of its principal part Θ0 , respectively. The following lemma gives a criterion of primeness for a closed symmetric operator. Lemma 6.6.4. A closed symmetric operator A˙ is prime if and only if N = c.l.s. Nz = H, z=z

˙ where Nz is the deﬁciency subspace of A.

(6.94)

196

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

Proof. Let M =

7 z=z

˙ It is easy to see that Mz , where Mz = (A˙ − zI)Dom(A).

H = N ⊕ M. We show that ˙ for all z, Im z = 0. (A˙ − zI)−1 M = M ∩ Dom(A)

(6.95)

˙ If Im z = 0 and Let Im z0 = 0 and φ ∈ M. Then hz0 = (A˙ − z0 I)−1 φ ∈ Dom(A). z = z0 , then (A˙ − zI)hz0 = (A˙ − z0 I)hz0 + (z0 − z)hz0 = φ + (z0 − z)hz0 . ˙ ∩ Mz Since (A˙ − zI)hz0 ∈ Mz and φ ∈ Mz , we get hz0 ∈ Mz . Thus, hz0 ∈ Dom(A) for all z = z0 , Im z = 0. Let T0 be a quasi-self-adjoint extension of A˙ having a regular point z¯0 . For example, ˙ +N ˙ + z¯0 ψz0 , h ∈ Dom(A), ˙ ψz0 ∈ Nz0 . ˙ z0 , T0 (h + ψz0 ) = Ah Dom(T0 ) = Dom(A) One can easily verify that the operator Uzz0 = (T0 − z¯0 I)(T0 − z¯I)−1 = I + (¯ z0 − z¯)(T0 − z¯I)−1 maps Nz¯0 into Nz¯ for all z¯ ∈ ρ(T0 ). Clearly, lim Uzz0 f = f

z→z0

for all f ∈ H.

Further, for all fz¯0 ∈ Nz¯0 , using that hz0 = (A˙ − z0 I)−1 φ ∈ Mz , z = z0 , we have (hz0 , Uzz0 fz¯0 ) = 0. It follows that (hz0 , fz¯0 ) = lim ((A˙ − z0 I)−1 φ, Uzz0 fz¯0 ) = 0. z→¯ z0

˙ we get Therefore, hz0 ∈ Mz for all z, Im z = 0. Since hz0 ∈ Dom(A), ˙ (A˙ − z0 I)−1 M ⊆ M ∩ Dom(A). ˙ then since Conversely, if f ∈ M ∩ Dom(A), φ = (A˙ − z0 I)f = (A˙ − zI)f + (z − z0 )f, ˙ ⊆ M and thus, (6.95) holds. This means that we have (A˙ − z0 I)(M ∩ Dom(A)) ˙ ˙ the symmetric operator A (M ∩ Dom(A)), as acting on M, has regular points ˙ (M ∩ Dom(A)) ˙ is a in the open upper and lower half plains. It follows that A self-adjoint operator in M. Remark 6.6.5. One can see from the proof of Lemma 6.6.4 that if G is an open subset of C such that C+ ∩ G = ∅, and C− ∩ G = ∅, then c.l.s. Nz = H implies z∈G

that A˙ is prime.

6.6. Minimal realization and the theorem on bi-unitary equivalence

197

Lemma 6.6.6. Let A˙ be a closed, symmetric operator with equal deﬁciency indices in the Hilbert space H. Let A be an arbitrary self-adjoint extension of A˙ and let E(t) and Rλ = Rλ (A) be the resolution of identity and the resolvent of A, respectively. Then for z ∈ C, Im z = 0, the following statements are equivalent: (i) A˙ is prime; (ii) c.l.s.{Rλ Nz0 , Im λ = 0} = H; ˙ (iii) c.l.s{E(Δ)Nz0 , Δ ∈ ℵ} = H, where Nz0 is the defect subspace of A. Proof. Let L be a subspace of H. Deﬁne H0 = c.l.s{E(Δ)L, Δ ∈ ℵ},

Since Rλ =

R

H0 ,

H0 = c.l.s.{Rλ L, Im λ = 0}. dE(t) , t−λ

H0

we get Rλ L ⊂ and hence, ⊆ H0 . On the other hand, clearly, the subspace H0 reduces Rλ for all non-real λ, and hence H0 and H H0 are invariant with respect to A. In addition, in view of the equality lim iηRiη f = −f,

η→∞

f ∈ H,

we get L ⊂ H0 . It follows that E(Δ)L ⊂ H0 for all Δ ∈ ℵ. Hence, H0 ⊆ H0 . Thus, H0 = H0 . Taking L = Nz0 , we get (ii) ⇐⇒ (iii). Let us prove. (i) ⇐⇒ (ii). The latter follows from (I − (λ − z0 )Rλ ) Nz0 = Nλ , which leads to the equality Rλ1 Nz0 + Rλ2 Nz0 + · · · + Rλn Nz0 = Nz0 + Nλ1 + · · · + Nλn .

With the help of Lemma 6.6.4 we can oﬀer an alternative proof of Theorem 6.6.1 and provide a valuable property of the model symmetric operator A˙ constructed in the proof of Theorem 6.5.1. Theorem 6.6.7. The symmetric operator A˙ of the form (6.75)–(6.76) is prime. Consequently, any operator-valued function V (z) ∈ N (R) can be realized as impedance function of a minimal L-system Θ of the form (6.31)–(6.36) with the prime symmetric operator A˙ of the form (6.75)–(6.76) and a preassigned directing operator J for which I + iV (−i)J is invertible. Proof. Let G(t) be the function from the integral representation of V (z), σ(t) = trace(G(t)). Recall that in the proof of Theorem 6.5.1 by Ψ(t) we used the RadonNikodym derivative dG(t) Ψ(t) = . dσ(t)

198

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

Let L2G (E) be the Hilbert space constructed in the proof of Theorem 6.5.1. Let A be a self-adjoint operator in L2G (E) deﬁned by (6.69), i.e., 8 9 Dom(A) = f ∈ L2G (E) : tf (t) ∈ L2G (E) , Af (t) = tf (t), and A˙ be a symmetric operator of the form (6.75)–(6.76). The defect subspaces of A˙ are described in the second step of the proof of Theorem 6.5.1 and are given by e Nz = , e ∈ E , Im z = 0. t−z Suppose that there is a function h ∈ L2G (E) such that (Nz , h)L2G (E) = 0, Im z = 0. Then using (6.67) and (6.68), we get (Ψ(t)e, h(t))E dσ(t) = 0, e ∈ E, Im z = 0. t−z R Let γe (t) = (Ψ(t)e, h(t))E = (e, Ψ(t)h(t))E , t ∈ R. Then we have

R

It follows that R

γe (t)dσ(t) =0 t−z

for all z such that Im z = 0.

1 1 − γe (t)dσ(t) = 0, Im z = 0, Im ξ = 0. t−z t−ξ

Then

R

1 γe (t)dσ(t) = 0. (t − z)(t − ξ)

Fix ξ = i and set fe (t) := Hence

R

Since

γe (t) . t−i

fe (t) dσ(t) = 0, Im z = 0. t−z

Ψ(t)e (Ψ(t)e, e)E |fe (t)| = , h(t) ≤ (Ψ(t)h(t), h(t))E t−i |t − i|2 E 1 (Ψ(t)e, e)E ≤ + (Ψ(t)h(t), h(t))E , 2 1 + t2

(6.96)

6.6. Minimal realization and the theorem on bi-unitary equivalence and

R

(Ψ(t)e, e)E dσ(t) < ∞, 1 + t2

199

(Ψ(t)h(t), h(t))E dσ(t) < ∞, R

the function fe (t) belongs to the Banach space L1 (R, dσ). Relations (6.96) yield 1 1 − fe (t)dσ(t) = 0, Im z = 0. t − z t − z¯ R Set

t μe (t) :=

fe (τ )dσ(τ ). −∞

Then we have

R

y dμe (t) = 0, y = 0. (x − t)2 + y 2

Because the complex measure dμe has ﬁnite variation, the latter equality yields (see [166]) g(t) dμe (t) = 0 R

for all continuous and bounded on R functions g. In particular, g(t) dμe (t) = 0 t−z R

for all non-real z and all bounded and continuous on R functions g. Take a continuous function g with a compact support and deﬁne a new measure t νe (t) :=

g(t)dμe (t). −∞

Then dνe has a compact support and bounded variation. Since dνe (t) = 0 for all nonreal z, t−z R

we obtain dνe = 0 [132]. It follows that γe (t) = 0 almost everywhere with respect to dσ. Since e ∈ E is arbitrary, we have h(t) ∈ ker(Ψ(t)) almost everywhere with respect to dσ. Due to (6.67) and (6.68) we get h = 0 in L2G (E). Thus, c.l.s.{Nz , Im z = 0} = L2G (E), and A˙ is prime.

200

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

The lemma below establishes the link between the primeness of operators T ˙ and A. Lemma 6.6.8. If A˙ is a maximal common symmetric part of T and T ∗ , then operator T is prime if and only if A˙ is prime. Proof. Let H1 be a non-trivial reducing invariant subspace for T . Then H = H1 ⊕ H2 and T1 = T H1 is a self-adjoint in H1 operator. Let Pk (k = 1, 2) be the orthogonal projection operator on subspaces Hk , (k = 1, 2), respectively. Furthermore, for all g ∈ Dom(T ), Pk g ∈ Dom(T ) and T Pk g ∈ Hk , (k = 1, 2). If f ∈ H1 ∩ Dom(T ), then for all g ∈ Dom(T ) we have (T f, g) = (T f, P1 g) = (f, T P1 g) = (f, T P1 g) + (f, T P2 g) = (f, T g). This implies that, f ∈ Dom(T ∗ ) and T ∗ f = T f . Since ˙ = Dom(T ) ∩ Dom(T ∗ ), Dom(A) ˙ and consequently A˙ induces a self-adjoint then we have Dom(T ) ∩ H1 ⊂ Dom(A) 1 ˙ operator on H . Clearly, if A is not prime, then T is not prime either. The next statement immediately follows from Remark 6.6.5 and Lemma 6.6.8. Lemma 6.6.9. Let T be a quasi-self-adjoint extension of A˙ such that ρ(T ) ∩ C+ and ρ(T ) ∩ C− are non-empty. Let also G be an open subset of ρ(T ) such that G ∩ C− = ∅ and G ∩ C+ = ∅. Then the operator T is prime if and only if c.l.s. Nz = H, z∈G

where Nz is the deﬁciency subspace of A˙ that is the maximal common symmetric part of T and T ∗ . Theorem 6.6.10. Let A1 Θ1 = H+1 ⊂ H1 ⊂ H−1

K1

J E

and Θ2 =

H+2

A2 ⊂ H2 ⊂ H−2

K2

J , E (6.97)

be two minimal L-systems with A1 ⊃ T1 ⊃ A˙ 1 , A2 ⊃ T2 ⊃ A˙ 2 ,

A∗1 ⊃ T1∗ ⊃ A˙ 1 , A∗ ⊃ T ∗ ⊃ A˙ 2 . 2

2

Let also operators T1 and T2 be such that (ρ(T1 ) ∩ ρ(T2 )) ∩ C± = ∅. If the transfer functions WΘ1 (z) and WΘ2 (z) satisfy the condition WΘ1 (z) = WΘ2 (z),

z ∈ ρ(T1 ) ∩ ρ(T2 ),

(6.98)

then there exists an isometric operator U from H1 onto H2 such that U+ = U |H+1 ∗ −1 is an isometry from H+1 onto H+2 , U− = (U+ ) is an isometry from H−1 onto H−2 , and U T1 = T2 U, U− A1 = A2 U+ , U− K1 = K2 . (6.99)

6.6. Minimal realization and the theorem on bi-unitary equivalence

201

Proof. It follows from (6.44) and (6.98) that K1∗ (A1 − zI)−1 K1 = K2∗ (A2 − zI)−1 K2 .

(6.100)

Since for j = 1, 2 we have ¯ −1 = (A∗ − ζI) ¯ −1 [(A∗ − ζI) ¯ − (Aj − zI)](Aj − zI)−1 (Aj − zI)−1 − (A∗j − ζI) j j ∗ −1 −1 ∗ ¯ ¯ −1 Kj JK ∗ (Aj − zI)−1 , = (z − ζ)(Aj − ζI) (Aj − zI) − 2i(Aj − ζI) j then ¯ −1 (A1 − zI)−1 K1 = K ∗ (A1 − zI)−1 K1 (z − ζ)K1∗ (A∗1 − ζI) 1 ∗ ∗ −1 ∗ ∗ ¯ ¯ −1 K1 JK ∗ (A1 − zI)−1 K1 − K (A − ζI) K1 + 2iK (A − ζI) 1 1 1 1 ∗ −1 ∗ ∗ ¯ −1 K2 = K2 (A2 − zI) K2 − K2 (A2 − ζI) ¯ −1 K2 JK ∗ (A2 − zI)−1 K2 + 2iK2∗(A∗2 − ζI) 2 ∗ ∗ −1 ¯ = (z − ζ)K2 (A2 − ζI) (A2 − zI)−1 K2 .

1

Therefore, ¯ −1 (A1 − zI)−1 K1 = K ∗ (A∗ − ζI) ¯ −1 (A2 − zI)−1 K2 . K1∗ (A∗1 − ζI) 2 2

(6.101)

Now we can apply (4.45) and get (A1 − zI)−1 K1 E = N(1) z ,

(A2 − zI)−1 K2 E = N(2) z .

(6.102)

Because the systems Θ1 and Θ2 are minimal, then by Lemma 6.6.8 the operators T1 and T2 are prime. Thus, using (ρ(T1 ) ∩ ρ(T2 )) ∩ C± = ∅ yields c.l.s.

z∈ρ(T1 )∩ρ(T2 )

N(1) z = H1 ,

c.l.s.

z∈ρ(T1 )∩ρ(T2 )

N(2) z = H2 .

Deﬁne linear manifolds Dj = l.s.{(Aj − zI)−1 Kj E, z ∈ ρ(T1 ) ∩ ρ(T2 )}, j = 1, 2. Then D1 and D2 are dense in H1 and H2 , respectively. Now deﬁne a linear operator U : D1 → D2 : # # U (A1 − zj I)−1 K1 fj = (A2 − zj I)−1 K2 fj , where {zj } ⊂ ρ(T1 ) ∩ ρ(T2 ), {fj } ⊂ E. Then equality (6.101) yields that the operator U is isometric and maps D1 onto D2 . It follows that U admits a unitary continuation mapping from H1 onto H2 . We preserve the notation U for this continuation. Taking into account Hilbert’s identity for resolvents (Aj − zI)−1 − (Aj − ξI)−1 = (z − ξ)(Aj − ξI)−1 (Aj − zI)−1 ,

(j = 1, 2)

202

Chapter 6. Herglotz-Nevanlinna Functions and Rigged Canonical Systems

we get that ) ) (A1 − zj I)−1 K1 fj = (T2 − zI)−1 (A2 − zj I)−1 K2 fj ) = (T2 − zI)−1 U (A1 − zj I)−1 K1 fj ,

U (T1 − zI)−1

for each z ∈ ρ(T1 ) ∩ ρ(T2 ). Hence, U (T1 − zI)−1 = (T2 − zI)−1 U,

(6.103)

U T1 = T2 U.

(6.104)

and (1)

(2)

Since U ((A1 − zI)−1 K1 = (A2 − zI)−1 K2 , the operator U maps Nz onto Nz . Therefore U maps (A˙ 1 − z¯I)Dom(A˙ 1 ) onto (A˙ 2 − z¯I)Dom(A˙ 2 ). Relations (6.103) and (6.104) imply U Dom(A˙ 1 ) = Dom(A˙ 2 ), and U Dom(A˙ ∗1 ) = Dom(A˙ ∗2 ),

U A˙ 1 = A˙ 2 U,

U A˙ ∗1 = A˙ ∗2 U. Because

(f, g)H+1 = (f, g)H1 + (A˙ ∗1 f, A˙ ∗1 g)H1 ,

(f, g)H+2 = (f, g)H2 + (A˙ ∗2 f, A˙ ∗2 g)H2 ,

the operator U is an isometry of H+1 onto H+2 . Since Re Aj , (j = 1, 2) is a self-adjoint bi-extension of the symmetric operator Aj , the operators :j f = Re Aj f, A

:j ) = {f ∈ H+j : Re Aj f ∈ Hj }, D(A

(6.105)

are self-adjoint in Hj , (j = 1, 2). These operators are self-adjoint extensions of A˙ j (j = 1, 2). Let U+ = U H+1 ∈ [H+1 , H+2 ]. ∗ ∗ Then U+ ∈ [H−2 , H−1 ] and U+ isometrically maps H−2 onto H−1 . Put U− = ∗ −1 ∗ −1 (U+ ) . Clearly (U+ ) is an extension of U onto H−1 . Thus, we have obtained a triplet of operators (U+ , U, U− ) that maps isometrically the triplet (H+1 , H1 , H−1 ) onto the triplet (H+2 , H2 , H−2 ). Equation (6.103) implies that U+ (A1 − zI)−1 = (A2 − zI)−1 U− . Taking into account that U− |H+1 = U+ , we have (A2 − zI)U+ = −1 U− (A1 − zI). Finally, A2 = U− A1 U+ and −1 Re A2 = U− Re A1 U+ ,

−1 Im A2 = U− Im A1 U+ .

From (6.105) and the description of self-adjoint bi-extensions of the symmetric :2 = U A :1 U −1 . This means that the operators A :2 and operators one concludes A : A1 are unitary equivalent. Corollary 6.6.11. Let Θ1 and Θ2 be the two L-systems from the statement of Theorem 6.6.10. Then the mapping U described in the conclusion of the theorem is unique.

6.6. Minimal realization and the theorem on bi-unitary equivalence

203

A K J Proof. First let us make an observation that if Θ = H+ ⊂H⊂H E is a minimal − L-system such that U− A = AU+ and U− K = K, where U is an isometry mapping described in theorem 6.6.10, then U = I. Indeed, we know (see (6.102)) that (Re A − λI)−1 KE = Nλ .

(6.106)

We have U (Re A − λI)−1 Ke = U+ (Re A − λI)−1 Ke = (Re A − λI)−1 U− Ke ¯ = (Re A − λI)−1 Ke, ∀e ∈ E, λ = λ. Combining the above equation with (6.94) and (6.106) we obtain U = I. Now let Θ1 and Θ2 be the two minimal L-systems from the statement of Theorem 6.6.10. Suppose there are two isometric mappings U1 and U2 guaranteed by Theorem 6.6.10. Then the relations −1 A2 = U−,1 A1 U+,1 ,

lead to

U−,1 K1 = K2 ,

−1 A2 = U−,2 A1 U+,2 ,

−1 −1 A1 U+,1 U+,2 = U−,1 U−,2 A1 ,

U−,2 K1 = K2 ,

−1 U−,1 U−,2 K = K.

Since Θ1 is minimal then U1−1 U2 = I and hence U1 = U2 . This proves the uniqueness of U . Two L-systems of the form (6.97) are called bi-unitarily equivalent if there exists a triplet of operators (U+ , U, U− ) that isometrically maps the triplet (H+1 , H1 , H−1 ) onto the triplet (H+2 , H2 , H−2 ) in a way that (6.99) holds and −1 A2 = U− A1 U+ . For the remainder of the text we will refer to Theorem 6.6.10 as the theorem on bi-unitary equivalence. Corollary 6.6.12. If two L-systems Θ1 and Θ2 satisfying the conditions of Theorem 6.6.10 are bi-unitary equivalent, then their transfer functions WΘ1 (z) and WΘ2 (z) coincide on z ∈ ρ(T1 ) ∩ ρ(T2 ), i.e., (6.98) holds. The corollary is proved by reversing the argument in the proof of Theorem 6.6.10.

Chapter 7

Classes of realizable Herglotz-Nevanlinna functions In this chapter we are going to introduce three distinct subclasses N0 (R), N1 (R), and N01 (R) of the class of functions N (R) realizable as impedance functions of L-systems, that was studied in Chapter 6. We give complete proofs of direct and inverse realization theorems in each subclass. We show that each subclass is characterized by a diﬀerent property of the state-space operator in the corresponding realizing L-system. Based on this partition of the class N (R), we introduce the corresponding structure of subclasses Ω0 (R, J), Ω1 (R, J), and Ω01 (R, J) on the class Ω(R, J) consisting of functions realizable as transfer functions of L-systems. Multiplication and coupling theorems are then proved for each subclass of Ω(R, J). We also establish the connection between the impedance functions of Lsystem and boundary triplets. In addition to that we consider the Krein-Langer Q-functions of a densely deﬁned symmetric operator and show that they belong to the class N0 (R) and thus can be realized as impedance functions of L-systems.

7.1 Sub-classes of the class N (R) and their realizations Let E be a ﬁnite-dimensional Hilbert space. We start the section by introducing the following subclasses of the class N (R) from Section 6.4. We recall that an operator-valued Herglotz-Nevanlinna function V (z) in E belongs to the class N (R) if in formula (6.52), that is, +∞

V (z) = Q + zX + −∞

1 t − t−z 1 + t2

dG(t),

Y. Arlinskii et al., Conservative Realizations of Herglotz–Nevanlinna Functions, Operator Theory: Advances and Applications, DOI 10.1007/978-3-7643-9996-2_7, © Springer Basel AG 2011

205

206

Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

we have X = 0 and

+∞

Qh = −∞

for all h ∈ E such that

t dG(t)h, 1 + t2

+∞ −∞

(dG(t)h, h)E < ∞.

Deﬁnition 7.1.1. An operator-valued Herglotz-Nevanlinna function V (z) in E of the class N (R) is said to be a member of the subclass N0 (R) if in the representation (6.52) +∞ (dG(t)h, h)E = ∞, (h ∈ E, h = 0). (7.1) −∞

Obviously, any function V (z) ∈ N0 (R) has the representation +∞

V (z) = Q + −∞

1 t − t−z 1 + t2

dG(t),

(Q = Q∗ ),

(7.2)

where G(t) satisﬁes (7.1) and Q is an arbitrary self-adjoint operator in the Hilbert space E. Deﬁnition 7.1.2. An operator-valued Herglotz-Nevanlinna function V (z) in E of the class N (R) is said to be a member of the subclass N1 (R) if in the representation (6.52) +∞ (dG(t)h, h)E < ∞, (h ∈ E). −∞

It follows from the deﬁnition of the class N (R) that the operator-valued function V (z) of the class N1 (R) has a representation +∞

V (z) = −∞

1 dG(t). t−z

(7.3)

Deﬁnition 7.1.3. An operator-valued Herglotz-Nevanlinna function V (z) in E of the class N (R) is said to be a member of the subclass N01 (R) if the subspace

4 +∞ E1 = h ∈ E : (dG(t)h, h)E < ∞ (7.4) −∞

possesses a property: E1 = {0}, 1 The

E1 = E.

1

deﬁnition of the class N01 (R) implies that it does not exist if dim E = 1.

7.1. Sub-classes of the class N (R) and their realizations

207

One may notice that N (R) is a union of three distinct subclasses N0 (R), N1 (R) and N01 (R). The following theorem is an analogue of Theorem 6.5.2 for the class N0 (R). Theorem 7.1.4. Let Θ be an L-system of the form (6.36) with an invertible chan˙ Then its impedance nel operator K and a densely-deﬁned symmetric operator A. function VΘ (z) of the form (6.47) belongs to the class N0 (R). Proof. Relying on Theorem 6.4.3 we conclude that an operator-valued function VΘ (z) of the system Θ mentioned in the statement belongs to the class N (R). Since N0 (R) is a subclass of N (R), it is suﬃcient to show that +∞ (dG(t)h, h)E = ∞,

(h ∈ E, h = 0).

−∞

According to Theorem 6.2.10, if for some vector h ∈ E, h = 0 we have that Kh ∈ /L ˙ where L = H Dom(A), then +∞ (dG(t)h, h)E = ∞,

ˆ where G(t) = K ∗ E(t)K,

(7.5)

−∞

ˆ ˆ Here Aˆ is the E(t) is an extended canonical spectral function of the operator A. quasi-kernel of the operator Re A. ˙ = The fact that A˙ is a closed symmetric operator with dense domain (Dom(A) H) implies that L = {0}. Thus, for any h ∈ E such that h = 0 we have Kh ∈ / L, and (7.5) holds. Therefore, VΘ (z) ∈ N0 (R). It directly follows from Theorem 7.1.4 above and Remark 6.4.5 that if the operator K in the statement of Theorem 7.1.4 is not invertible and ker(K) = G, then VΘ (z) can be written as a diagonal block-matrix (6.64) with respect to decomposition (6.63), i.e., 0 0 VΘ (z) = , 0 VΘ,2 (z) where VΘ,2 (z) ∈ N0 (R). Theorem 7.1.5 below is a version of Theorem 6.5.2 for the class N0 (R). Theorem 7.1.5. Let an operator-valued function V (z) in a Hilbert space E belong to the class N0 (R). Then it can be realized as an impedance function of an Lsystem Θ with an invertible channel operator K, a preassigned directing operator ˙ J for which I + iV (−i)J is invertible, a densely deﬁned symmetric operator A, ∗ and Dom(T ) = Dom(T ).

208

Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

Proof. Since N0 (R) is a subclass of N (R), then all conditions of Theorem 6.5.2 are satisﬁed and the operator-valued function V (z) ∈ N0 (R) can be realized as an impedance function of the L-system Θ constructed in the proof of Theorem 6.5.2. ˙ = H and Dom(T ) = Dom(T ∗ ). Thus, all we have to show is that Dom(A) Since V (z) is a member of the class N0 (R), then (7.1) holds for all non-zero h ∈ E and hence E1 = {0}, where E1 is deﬁned by (7.4). As it was shown in Step 2 of the proof of Theorem 6.5.1, any element h ∈ E can be considered as an element h = h(t) ∈ H− and the following formula holds: ˙ = E1 . L2G (E) Dom(A)

(7.6)

The right-hand side of (7.6) is trivial in our case and we can conclude that ˙ = L2 (E) = H. Dom(A) G Following the proof of Theorem 6.5.2 we construct the realizing L-system Θ with the operator T deﬁned by (6.65). Let us show that Dom(T ) = Dom(T ∗ ). Indeed, ˙ then the densely-deﬁned operator A˙ is the maximal common symsince T ∈ Λ(A), metric part of T and T ∗ . Therefore, we can apply Lemmas 4.1.4, 4.2, and Theorem 4.1.11 to get the condition Dom(T ) = Dom(T ∗ ). The proof of the theorem is complete. Corollary 7.1.6. If V (z) belongs to the class N0 (R), then it can be realized as an impedance function of a scattering (J = I) L-system with an invertible chan˙ and Dom(T ) = nel operator K, a densely-deﬁned closed symmetric operator A, ∗ Dom(T ). Proof. The proof immediately follows from Theorems 6.5.5 and 7.1.5.

Similar results for the class N1 (R) can be obtained in the next two theorems. Theorem 7.1.7. Let Θ be an L-system of the form (6.36), where A˙ is a symmetric O-operator and Dom(T ) = Dom(T ∗ ). Then its impedance function VΘ (z) of the form (6.47) belongs to the class N1 (R). Proof. As in Theorem 7.1.4 we already know that the operator-valued function VΘ (z) belongs to the class N (R). Therefore it is enough to show that +∞ (dG(t)h, h)E < ∞, −∞

for all h ∈ E and (7.3) holds. Since it is given that A˙ is a closed symmetric O-operator, we can use Theorem 4.1.12 saying that, for the system Θ, Dom(T ) = Dom(T ∗ ) = H+ = Dom(A˙ ∗ ).

7.1. Sub-classes of the class N (R) and their realizations

209

This fact implies that the (∗)-extension A coincides with operator T . Consequently, A∗ = T ∗ and our system Θ has a form T K J Θ= , H+ ⊂ H ⊂ H− E where

T − T∗ = KJK ∗ . 2i Taking into account that dimE < ∞ and K : E → H− we conclude that dim(Ran(Im T )) < ∞. Let Im T =

T = Re T + iIm T,

T ∗ = Re T − iIm T,

where Re T = (1/2)(T + T ∗ ). In our case the operator K acts from the space E into the space H. Therefore Kh = g ∈ H for all h ∈ E. For the operator-valued function VΘ (z) we can derive an integral representation for all f ∈ E, ⎛ ⎞ +∞ ∗ dE(t) VΘ (z)f, f E = K (Re T − zI)−1 Kf, f E = ⎝K ∗ Kf, f ⎠ t−z −∞ E (7.7) +∞ ∗ d K E(t)Kf, f E = , t−z −∞

where E(t) is the resolution of identity of the operator Re T . Let G(t) = K ∗ E(t)K. Then

+∞

+∞

d(G(t)h, h) = −∞

d(K ∗ E(t)Kh, h) =

−∞ +∞

=

+∞

d(E(t)Kh, Kh) −∞ +∞

d(E(t)g, g) = (g, g) −∞

dE(t) = (g, g) −∞

= (Kh, Kh) = (K ∗ Kh, h) = (Im T h, h) < ∞, for all h ∈ E. Using the last relation and (7.7) we obtain the representation (7.3). Theorem 7.1.8. Let an operator-valued function V (z) in a Hilbert space E belong to the class N1 (R). Then it can be realized as an impedance function of an L-system Θ with a preassigned directing operator J for which I + iV (−i)J is invertible, a symmetric O-operator A˙ with non-dense domain, and Dom(T ) = Dom(T ∗ ).

210

Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

Proof. Similarly to Theorem 7.1.5 we can say that since N1 (R) is a subclass of N (R), then it is suﬃcient to show that operator A˙ is a closed symmetric O-operator with a non-dense domain and Dom(T ) = Dom(T ∗ ). Once again we rely on the proof of Theorem 6.5.1 and use formula (7.6), together with the fact that E1 = E, (where E1 is deﬁned by (6.71)), to obtain ˙ = H = L2 (E). Let N be the semi-defect subspaces of operator A˙ and Dom(A) ±i G 0 N±i be the defect subspaces of operator A1 , described in Step 2 of the proof of Theorem 6.5.1. It was also shown there that

4 h N±i = f (t) ∈ L2G (E), f (t) = , h∈E , t±i N0±i

=

f (t) ∈

L2G (E),

4 h f (t) = , h ∈ E1 , t±i

˙ In our and N±i = N±i N0±i , where N±i are the defect spaces of the operator A. case, however, E = E1 and therefore the above formulas yield that N±i = N0±i . Consequently, N±i = {0} and hence A˙ is an O-operator. Note that A˙ is also a regular symmetric operator. Thus, Theorem 4.1.12 is applicable and gives Dom(T ) = Dom(T ∗ ). This completes the proof of the theorem.

Corollary 7.1.9. If a function V (z) belongs to the class N1 (R), then it can be realized as an impedance function of a scattering (J = I) L-system with a symmetric non-densely deﬁned O-operator A˙ and Dom(T ) = Dom(T ∗ ). Proof. The proof of the corollary follows from Theorems 6.5.5 and 7.1.8.

The following two theorems will complete our framework by establishing direct and inverse realization results for the remaining subclass of realizable operator-valued Herglotz-Nevanlinna functions N01 (R). Theorem 7.1.10. Let Θ be an L-system of the form (6.36) with a symmetric nondensely deﬁned operator A˙ and Dom(T ) = Dom(T ∗ ). Then its impedance function VΘ (z) of the form (6.47) belongs to the class N01 (R). Proof. We know that VΘ (z) belongs to the class N (R). To prove the statement of the theorem we only have to show that in the direct decomposition E = E1 E2 (see Step 1 of the proof of Theorem 6.5.1) both components, deﬁned in (6.71), are non-zero. In other words we have to show the existence of such vectors h ∈ E that +∞ d(G(t)h, h) = ∞, −∞

(7.8)

7.2. Class Ω(R, J). The Potapov-Ginzburg Transformation

211

and vectors f ∈ E, f = 0 that +∞ d(G(t)f, f ) < ∞.

(7.9)

−∞

˙ and L = H H0 . Since Dom(A) ˙ = H0 = H, L is nonLet H0 = Dom(A) −1 empty. K L is obviously a subset of E. Moreover, according to Theorem 6.2.10 for all f ∈ K −1 L, (7.9) holds. Thus, K −1 L is a non-zero subset of E1 . Now we have to show that the vectors satisfying (7.8) make a non-zero subset of E as well. Indeed, the deﬁnition of L-system and condition Dom(T ) = Dom(T ∗ ) implies that Ran(K) = Ran(Im A) ⊆ LA L, where LA was deﬁned by (4.44). Therefore, there exist g ∈ H− , g ∈ / L, f ∈ E such that Kf = g ∈ / L. Then, according to Theorem 6.2.10, for this f ∈ E, (7.8) holds. The proof of the theorem is complete. Theorem 7.1.11. Let an operator-valued function V (z) in a Hilbert space E belong to the class N01 (R). Then it can be realized as an impedance function of an L-system Θ with a preassigned directing operator J for which I + iV (−i)J is ˙ and Dom(T ) = Dom(T ∗ ). invertible, a symmetric non-densely deﬁned operator A, ˙ = H. We have already Proof. Once again all we have to show is that Dom(A) 2 ˙ = E1 . This implies that Dom(A) ˙ is dense mentioned (7.6) that LG (E) Dom(A) in H if and only if E1 = 0. Since the class N01 (R) assumes the existence of non-zero vectors f ∈ E such that (7.9) is true, we can conclude that E1 = 0 and therefore ˙ = H. Dom(A) In the proofs of Theorems 7.1.5 and 7.1.8 we have shown that Dom(T ) = Dom(T ∗ ) in the case when E2 = 0. If E2 = 0, then Dom(T ) = Dom(T ∗ ). The deﬁnition of the class N01 (R) implies that E2 = 0. Thus we have Dom(T ) = Dom(T ∗ ). The proof is complete. Corollary 7.1.12. If a function V (z) belongs to the class N01 (R), then it can be realized as an impedance function of a scattering (J = I) L-system Θ with a symmetric non-densely deﬁned operator A˙ and Dom(T ) = Dom(T ∗ ). Proof. The proof of the corollary follows from Theorems 6.5.5 and 7.1.11.

7.2 Class Ω(R, J). The Potapov-Ginzburg Transformation In this section we introduce the class Ω(R, J) of J-contractive in a half-plane operator-valued functions that are the transfer functions of L-systems.

212

Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

Deﬁnition 7.2.1. An operator-valued function W (z) in a ﬁnite-dimensional Hilbert space E holomorphic in the domain GW in the lower half-plane is said to be a member of the class Ω(R, J), where J ∈ [E, E], J = J ∗ = J −1 , if 1. the operator I − W (z0 ) is invertible for some z0 ∈ GW , 2. the operator-valued function V (z) = i(W (z) + I)−1 (W (z) − I)J,

z ∈ GW ,

(7.10)

admits a holomorphic continuation to the Herglorz-Nevanlinna function from the class N (R) (in the lower half-plane). It follows from condition 2 that, for z ∈ GW , Im V (z) = (W (z) + I)−1 (W (z)JW ∗ (z) − J)(W ∗ (z) + I)−1 = J(W ∗ (z) + I)−1 (W ∗ (z)JW (z) − J)(W (z) + I)−1 J. Therefore, W (z)JW ∗ (z) − J ≤ 0,

W ∗ (z)JW (z) − J ≤ 0,

z ∈ GW ,

i.e., the function W (z) is J-contractive. Theorem 7.2.2. Let W (z) ∈ Ω(R, J). Then there exists an L-system A K J Θ= , H+ ⊂ H ⊂ H− E such that ker(K) = {0}, W (z) = WΘ (z) for z ∈ GW . Proof. It follows from Deﬁnition 7.2.1 that the operator V (z0 ) is invertible. Since V (z) admits a holomorphic continuation to a function from the class N (R), the subspace ker(V (z)) does not depend on the choice z from the lower half-plane (see (6.64), (6.90)). Applying Theorem 6.5.2 we obtain the desired statement. Deﬁnition 7.2.3. An operator-valued function W (z) of the class Ω(R, J) belongs to the class Ω0 (R, J) (resp. Ω1 (R, J), Ω01 (R, J)) if the operator-valued function V (z) deﬁned by (7.10) belongs to the class N0 (R) (resp. N1 (R, J), N01 (R, J)). 2 The theorem below is a version of Potapov-Ginzburg transformation. Together with Corollary 7.2.5 it establishes the relation between contractive and Jcontractive in the half-plane operator-valued functions from the classes Ω(R, J), Ω0 (R, J), Ω1 (R, J), and Ω01 (R, J).

2 The

deﬁnition of the class Ω01 (R) implies that it does not exist if dim E = 1.

7.2. Class Ω(R, J). The Potapov-Ginzburg Transformation

213

Theorem 7.2.4. Let operator-valued function W (z) belong to the class Ω(R, J). Let also P + and P − be a pair of orthogonal projections of the form P+ =

1 (I + J) 2

and

P− =

1 (I − J). 2

Then there exists an operator-function Σ(z) of the class Ω(R, I) such that W (z) = (P + Σ(z) − P − )(P + − P − Σ(z))−1 . Proof. Let

V (z) = i[W (z) + I]−1 [W (z) − I]J.

Since W (z) belongs to Ω(R, J) we have that V (z) belongs to the class N (R) and thus, by Theorem 6.5.2, can be realized as an impedance function of a scattering L-system A K I Θ = . H+ ⊂ H ⊂ H− E The latter implies that ∗

V (z) = VΘ (z) = K (Re A − zI)−1 K = i[WΘ (z) + I]−1 [WΘ (z) − I], for z ∈ ρ(T ) where ∗

WΘ (z) = I − 2iK (A − zI)−1 K . It is clear that WΘ (z) belongs to the Ω(R, I) class. Therefore i[W (z) + I]−1 [W (z) − I]J = i[WΘ (z) + I]−1 [WΘ (z) − I],

z ∈ ρ(T ),

where W (z) ∈ Ω(R, J) and WΘ (z) ∈ Ω(R, I). Now let Σ(z) ≡ WΘ (z). Then (W (z) + I)−1 (W (z) − I)J = [Σ(z) + I]−1 [Σ(z) − I] = [Σ(z) − I][Σ(z) + I]−1 . It follows that [W (z) − I]J[Σ(z) + I] = [W (z) + I][Σ(z) − I]. Taking into account that P + − P − = J and P + + P − = I we obtain [W (z) − I](P + − P − )[Σ(z) + I] = [W (z) + I]J[Σ(z) − I], or W (z)P + Σ(z) − W (z)P − Σ(z) − P + Σ(z) + P − Σ(z) + W (z)P + − W (z)P − − P + + P − = W (z)Σ(z) − W (z) + Σ(z) − I,

214

Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

W (z)[P + Σ(z) − P − Σ(z) + 2P + − Σ(z)] = [P + Σ(z) − P − Σ(z) + Σ(z) − 2P − ], or

W (z)[2P + − 2P − Σ(z)] = [2P + Σ(z) − 2P − ].

Cancellation yields W (z)[P + − P − Σ(z)] = [P + Σ(z) − P − ].

(7.11)

Let us show that operator [P + − P − Σ(z)] is invertible. We choose φ ∈ E such that

Then (7.11) implies

[P + − P − Σ(z)]φ = 0.

(7.12)

[P + Σ(z) − P − ]φ = 0.

(7.13)

We apply P + to both sides of (7.12) and obtain P + [P + − P − Σ(z)]φ = 0, or P + φ = 0. Similarly, we apply P − to both sides of (7.13) and get that P − φ = 0. Thus φ = 0 and operator [P + − P − Σ(z)] is invertible. Using this we obtain W (z) = [P + Σ(z) − P − ][P + − P − Σ(z)]−1 ,

that proves the theorem.

Corollary 7.2.5. Let the operator-valued function W (z) belong to the class Ω0 (R, J) (resp. Ω1 (R, J), Ω01 (R, J)), P + = 1/2(I + J), P − = 1/2(I − J). Then there exists an operator-valued function Σ(z) from Ω0 (R, J) (resp. Ω1 (R, J), Ω01 (R, J)) class such that W (z) = [P + Σ(z) − P − ][P + − P − Σ(z)]−1 . Corollary 7.2.5 can be proved in exactly the same way as Theorem 7.2.4.

7.3 Multiplication Theorems for Ω(R, J) classes In this section we state and prove multiplication theorems for the operator-valued functions of Ω(R, J) class. Let A1 K1 J A2 K2 J Θ1 = and Θ2 = H+1 ⊂ H1 ⊂ H−1 E H+2 ⊂ H2 ⊂ H−2 E be two L-systems with ker(K1 ) = ker(K2 ) = {0}. Let H+ = H+1 ⊕ H+2 ,

H = H1 ⊕ H2 ,

H− = H−1 ⊕ H−2 ,

7.3. Multiplication Theorems for Ω(R, J) classes

215

and Pk : H → Hk , Pk+ : H+ → H+k , and Pk− : H− → H−k (k = 1, 2) denote the set of orthoprojections. In the space H we introduce an operator A˙ = A˙ 1 ⊕ A˙ 2 ,

(7.14)

where A˙ 1 ⊂ T1 ⊂ A1 , A˙ 2 ⊂ T2 ⊂ A2 are correspondent elements of Θ1 and Θ2 , respectively. Moreover, H+1 = Dom(A˙ ∗1 ) and H+2 = Dom(A˙ ∗2 ). Consequently, A˙ ∗ = A˙ ∗1 ⊕ A˙ ∗2 , and H+ = Dom(A∗ ) = Dom(A˙ ∗1 ) ⊕ Dom(A˙ ∗2 ). Now deﬁne A : H+ → H− A = A1 P1+ + A2 P2+ + 2iK1 JK2∗ P2+ .

(7.15)

Then the adjoint A∗ : H+ → H− is given by A∗ = A∗1 P1+ + A∗2 P2+ − 2iK2 JK1∗ P1+ .

(7.16)

˙ Put Both operators A and A∗ are extensions of the operator A. K = K1 + K 2 .

(7.17)

Clearly, K ∈ [E, H− ], ker(K) = {0}, and K∗ ∈ [H+ , E] is of the form K∗ = K1∗ P1+ + K2∗ P2+ .

(7.18)

In addition A − A∗ = 2iKJK∗. From (7.15)–(7.18) it follows that the resolvents (A−zI)−1 and (A∗ − z¯I)−1 are deﬁned for z ∈ ρ(T1 )∩ρ(T2 ) on the linear manifold H + Ran(K) and take the form (A − zI)−1 =(A1 − zI)−1 P1− + (A2 − zI)−1 P2− − 2i(A1 − zI)−1 K1 JK2∗ (A2 − zI)−1 P2− , (A∗ − z¯I)−1 =(A∗1 − z¯I)−1 P1− + (A∗2 − z¯I)−1 P2− + 2i(A∗2 − zI)−1 K2 JK1∗ (A1 − zI)−1 P1− .

(7.19)

(7.20)

In particular, (A−zI)−1 H ⊂ H, (A∗ − z¯I)−1 H ⊂ H. Let T and T∗ be quasi-kernels of A and A∗ , respectively, i.e., 8 9 Dom(T ) = f ∈ H+ : Af ∈ H , T = A Dom(T ), 8 9 Dom(T∗ ) = f ∈ H+ : A∗ f ∈ H , T∗ = A∗ Dom(T∗ ). Then for z ∈ ρ(T1 ) ∩ ρ(T2 ) operators T and T∗ have the resolvent (T − zI)−1 = (A − zI)−1 H, (T∗ − z¯I)−1 = (A∗ − z¯I)−1 H. ˙ So, the operator A is a quasi-selfThus T∗ = T ∗ . Obviously, T ⊃ A˙ and T ∗ ⊃ A. adjoint bi-extension of T with the range property (R), therefore T belongs to the

216

Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

˙ (see Deﬁnitions 4.2.2 and 4.2.3). It follows from Theorem 4.2.9 and class R(A) Theorem 4.3.2 that Re A is a t-self-adjoint bi-extension of A˙ in (7.14), i.e., the ˙ Let us set quasi-kernel B of Re A is a self-adjoint extension of A. ˙ = {g ∈ H+ : T g = T ∗ g}, Dom(A) ˙ via and deﬁne an operator A˙ on Dom(A) ˙ = Ag, Ag

˙ g ∈ Dom(A).

(7.21)

Clearly, A˙ is a maximal symmetric part of T and T ∗ and by constructions we have A˙ ⊂ A˙ ⊂ T and A˙ ⊂ A˙ ⊂ T ∗ . We are going to show that ˙ = {g ∈ H+ : g = g1 + g2 , g1 ∈ Dom(T ∗ ), Dom(A) 1

g2 ∈ Dom(T2 ) and K1∗ g1 + K2∗ g2 = 0}.

(7.22)

˙ where g1 ∈ H+1 , g2 ∈ H+2 . We Let us pick an element g = g1 + g2 ∈ Dom(A), are going to show that g ﬁts the description in (7.22). It follows from (7.15) and (7.16) that A1 g1 + A2 g2 + 2iK1 JK2∗ g2 = A∗1 g1 + A∗2 g2 − 2iK2 JK1∗ g1 ∈ H. Moreover,

A1 g1 + 2iK1JK2∗ g2 = A∗1 g1 ∈ H−1 , A2 g2 = A∗2 g2 − 2iK2 JK1∗ g1 ∈ H−2 .

(7.23)

(7.24)

From (7.24) we get A1 g1 − A∗1 g1 = −2iK1JK2∗ g2 , K1 JK1∗ g1 = −K1 JK2∗ g2 ,

K1 J(K1∗ g1 + K2∗ g2 ) = 0.

1 (A1 − A∗1 )g1 = −K1 JK2∗ g2 , 2i K1 JK1∗ g1 + K1 JK2∗ g2 = 0,

Since the operator K1 is invertible, we obtain K1∗ g1 + K2∗ g2 = 0, or K1∗ g1 = −K2∗ g2 . Also, (7.23) and (7.24) imply A2 g2 − A∗1 g1 ∈ H that yields g1 ∈ Dom(T1∗ ) and g2 ∈ Dom(T2 ). Let us show now that if g1 ∈ Dom(T1∗ ), g2 ∈ Dom(T2 ) and K1∗ g1 + K2∗ g2 = 0, ˙ We have then g = g1 + g2 belongs to Dom(A). Ag = A1 g1 + A2 g2 + 2iK1 JK2∗ g2 = A1 g1 + T2 g2 − 2iK1 JK1∗ g1 = A1 g1 + T2 g2 − A1 g1 + A∗1 g1 = T2 g2 + T ∗ g1 ∈ H.

˙ Let Therefore, Ag = A∗ g belongs to H or T g = T ∗ g, implying that g ∈ Dom(A). ∗ ∗ ∗ ˙ ˙ ˙ ˙ A be an adjoint to opewrator A. Then Dom(A ) ⊂ Dom(A ) = H+ . Let us set H+ = Dom(A˙ ∗ ),

7.3. Multiplication Theorems for Ω(R, J) classes

217

and construct a new rigged Hilbert space H+ ⊂ H ⊂ H− .

(7.25)

It is easy to see that the following inclusions take place: H+ → H+ ⊂H ⊂ H− → H− ∩ H− Let us denote by γ an embedding operator acting from H+ into H+ : γ : H+ → H+ ,

γf = f, ∀f ∈ H+ .

(7.26)

Let us deﬁne an adjoint operator γ ∗ as γ ∗ : H− → H− We have γ ∗ h = h for all h ∈ H. Indeed, for all f ∈ H+ , h ∈ H, (f, h) = (γf, h) = (f, γ ∗ h). Let the operator A ∈ [H+ , H− ] be deﬁned as A = γ ∗ A H+ .

(7.27)

Since for all f, g ∈ H+ , (Af, g) = (γ ∗ Af, g) = (Af, γg) = (Af, g) = (f, A∗ g) = (f, γ ∗ A∗ g) = (f, A∗ g), we get

A∗ = γ ∗ A∗ H+ .

Let f ∈ Dom(T ). Then Af ∈ H and Af = γ ∗ Af = γ ∗ T f = T f ∈ H. Thus, A ⊃ T . Similarly, A∗ ⊃ T ∗ . Let us show now that operator A deﬁned by (7.27) can be included as a state-space operator into an L-system Θ. Deﬁne K ∈ [E, H− ] as K = γ ∗ K, (7.28) where K is given by (7.17). Then, clearly, K ∗ = K∗ H+ . Furthermore, 1 1 1 (A − A∗ ) = (γ ∗ A − γ ∗ A∗ ) = γ ∗ (A − A∗ ) = γ ∗ KJK∗ = KJK ∗ . 2i 2i 2i ˙ A∗ ⊃ T ∗ ⊃ A, ˙ ImA = KJK ∗ , where K is deﬁned by (7.28). Thus, A ⊃ T ⊃ A, Moreover, since Re A is a t-self-adjoint bi-extension of A˙ with the quasi-kernel B = B ∗ , the operator Re A = γ ∗ Re A is a t-self-adjoint bi-extension of A˙ with the ˙ Let us same quasi-kernel B, i.e., A is a (∗)-extension of T and hence T ∈ Λ(A). show additionally that (A − zI)−1 K = (A − zI)−1 K,

(7.29)

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Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

and

˙ (A − zI)−1 KE = Nz (A)

(7.30)

˙ is the deﬁciency subspace of A. ˙ Equality (7.29) follows for z ∈ ρ(T ), where Nz (A) from (A − zI)(A − z)−1 K = γ ∗ (A − zI)(A − zI)−1 K = γ ∗ K = K. In order to −1 ˙ prove (7.30) suppose f ∈ H and (f, (A−zI) Kh) for all h ∈ E. Then there exists ∗ ∗ g ∈ Dom(T ) such that f = (T − z¯I)g. Then by (7.20) we have f = (A∗1 − z¯I)P1+ g + (A∗2 − z¯I)P2+ g − 2iK2 JK1∗ P1+ g. Further using (7.19) and (7.29) we get 0 = ((T ∗ − z¯I)g, (A − zI)−1 Kh) = ((T ∗ − z¯I)g, (A − zI)−1 Kh) = ((A∗1 − z¯I)P1+ g, (A1 − zI)−1 P1− Kh) + 2i((A∗1 − z¯I)P1+ g, (A1 − zI)−1 K1 JK2∗ (A2 − zI)−1 P2− Kh) + ((A∗2 − z¯I)P2+ g − 2iK2 JK1∗ P1+ g, (A2 − zI)−1 P2− Kh) = (P1+ g, P1− Kh − 2iK1 JK2∗ (A − zI)−1 P2− Kh) + (P2+ g, P2− Kh) − 2(iK2 JK1∗ P1+ g, (A2 − zI)−1 P1− Kh) = (g, Kh) = (K∗ g, h)E . Hence K∗ g = 0 and therefore, Im Ag = KJK∗ g = 0. Since g ∈ Dom(T ∗ ), we ˙ and f = (A˙ − z¯I)g. Thus (A − zI)−1 KE ⊇ get T g − T ∗ g = 0, i.e., g ∈ Dom(A) ˙ On the other hand if g ∈ Dom(A), ˙ then Im Ag = Im Ag = 0. It follows Nz (A). that K1 JK∗ g = K2 JK∗ g = 0. Since ker(K1 ) = {0} and ker(K2 ) = {0}, we get K∗ g = 0. Hence ((A˙ − z¯I)g, (A − zI)−1 Kh) = (A − zI)−1 Kh) = (g, Kh) = (K∗ g, h)E = 0, ˙ and thus (7.30) holds. for all h ∈ E. This means that (A − zI)−1 KE ⊆ Nz (A) ∗ From Theorem 4.3.2, the equality Im A = KJK , and equality (7.30) we get that Ran(K) = Ran(Im A) and the system A K J Θ= , (7.31) H+ ⊂ H ⊂ H− E is an L-system. Let now WΘ1 (z) = I − 2iK1∗ (A1 − zI)−1 K1 J, WΘ2 (z) = I − 2iK2∗ (A2 − zI)−1 K2 J, be transfer operator-valued functions of the L-systems Θ1 and Θ2 , respectively. We introduce a new auxiliary system A K J θ= , H+ ⊂ H ⊂ H− E

7.3. Multiplication Theorems for Ω(R, J) classes

219

where all its components are described above. Let us note that θ does not exactly satisfy the Deﬁnition 6.3.4 of the L-system because operator A˙ is not the maximal ˙ System θ will, however, suﬃce symmetric part of T and T ∗ and thus T ∈ / Λ(A). for our purposes. Let Wθ (z) = I − 2iK∗ (A − zI)−1 KJ,

z ∈ ρ(T1 ) ∩ ρ(T2 ),

be the transfer operator-valued function of the system θ. This implies Wθ (z) = I − 2iK∗ (A − zI)−1 KJ = I − 2iK∗ P1− (A1 − zI)−1 P1− KJ − 2iK∗ P2− (A2 − zI)−1 P2− KJ + (2i)2 K∗ P1− (A1 − zI)−1 K1 JK2∗ (A2 − zI)−1 P2− KJ = I + 2iK1∗(A1 − zI)−1 K1 J − 2iK2∗ (A2 − zI)−1 K2 J + (2i)2 K1∗ (A1 − zI)−1 K1 JK2∗ (A2 − zI)−1 K2 J = (I − 2iK1∗ (A1 − zI)−1 K1 J)(I − 2iK2∗ (A2 − zI)−1 K2 J) = WΘ1 (z) · WΘ2 (z). We have just shown that Wθ (z) = WΘ1 (z) · WΘ2 (z). Let now

WΘ (z) = I − 2iK ∗ (A − zI)−1 KJ,

z ∈ ρ(T1 ) ∩ ρ(T2 ),

be the transfer function of the L-system Θ deﬁned by (7.31). We will show now that WΘ (z) = Wθ (z) = WΘ1 (z) · WΘ2 (z). Equality (7.29) yields K ∗ (A − zI)−1 K = K∗ (A − zI)−1 K. Hence Wθ (z) = WΘ (z). Now we have WΘ (z) = WΘ1 (z) · WΘ2 (z). Deﬁnition 7.3.1. An L-system Θ of the form (7.31) is called a coupling of two L-systems (Θ = Θ1 · Θ2 ) A1 K1 J A2 K2 J Θ1 = and Θ2 = H+1 ⊂ H1 ⊂ H−1 E H+2 ⊂ H2 ⊂ H−2 E (7.32) with ker(K1 ) = ker(K2 ) = {0}, if operators A, K and rigged space H+ ⊂ H ⊂ H− are deﬁned by the formulas (7.27), (7.28) and (7.25), respectively. Theorem 7.3.2. Let an L-system Θ be the coupling of two L-systems Θ1 and Θ2 of the form (7.32). Then if z ∈ ρ(T1 ) ∩ ρ(T2 ), WΘ (z) = WΘ1 (z) · WΘ2 (z).

220

Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

The proof of Theorem 7.3.2 was constructively obtained above. We recall that if a function W (z) belongs to the class Ω(R, J), then according to Deﬁnition 7.2.1, by GW we denote the domain in the lower half-plane where W (z) is holomorphic. Theorem 7.3.3. Let operator-valued functions W1 (z) and W2 (z) in the Hilbert space E belong to the class Ω(R, J) and GW1 ∩ GW2 = ∅. Then the operator-valued function W (z) = W1 (z) · W2 (z) is deﬁned on GW1 ∩ GW2 and belongs to the class Ω(R, J). Proof. Since the operator-valued functions W1 (z) and W2 (z) belong to the class Ω(R, J), then there exist two L-system Θ1 and Θ2 such that W1 (z) = WΘ1 (z) and W2 (z) = WΘ2 (z) for z ∈ ρ(T1 ) ∩ ρ(T2 ). We note that, in view of Remark 6.5.3, the set ρ(T1 ) ∩ ρ(T2 ) is a non-empty set in C containing some neighborhood of the point (−i). Thus, we can say that W (z) is deﬁned on some domain of the complex plane. Let system Θ be the coupling of Θ1 and Θ2 . Then according to Theorem 7.3.2, WΘ (z) = WΘ1 (z) · WΘ2 (z) = W1 (z) · W2 (z) = W (z). Thus, for the function W (z) there exists an L-system Θ with W (z) = WΘ (z) for z ∈ ρ(T1 ) ∩ ρ(T2 ). Therefore, W (z) = W1 (z) · W2 (z) ∈ Ω(R, J). The next theorem establishes a similar result for the class Ω0 (R, J). Theorem 7.3.4. Let operator-valued functions W1 (z) and W2 (z) in the Hilbert space E belong to the class Ω0 (R, J) and GW1 ∩ GW2 = ∅. Then the operator-valued function W (z) = W1 (z) · W2 (z) is deﬁned on GW1 ∩ GW2 and belongs to the class Ω0 (R, J). Proof. According to Theorem 7.3.3 we have W (z) deﬁned on some domain of the complex plane and belonging to Ω(R, J). Therefore, there exists an L-system A K J Θ= (7.33) H+ ⊂ H ⊂ H− E such that W (z) = WΘ (z) for z ∈ ρ(T ). Hence it would be enough to show that if ˙ A∗ ⊃ T ∗ ⊃ A˙ are correspondent elements of Θ then Dom(A) ˙ =H A ⊃ T ⊃ A, and Dom(T ) = Dom(T ∗ ). Since W1 (z) and W2 (z) both belong to the class Ω0 (R, J), then corresponding systems Θ1 and Θ2 have a property that Dom(A˙ 1 ) = H1 and Dom(A˙ 2 ) = H2 . Let operator A be deﬁned by (7.2) and Dom(A) = Dom(A˙ 1 ) ⊕ Dom(A˙ 2 ).

(7.34)

7.3. Multiplication Theorems for Ω(R, J) classes

221

Considering the closure of the equality (7.34) yields Dom(A) = H1 ⊕ H2 = H. As ˙ ⊂ H. Hence, Dom(A) ˙ = H. it was shown above, A ⊂ A˙ or Dom(A) ⊂ Dom(A) ˙ = H already implies It was shown in the proof of Theorem 7.1.5 that Dom(A) that Dom(T ) = Dom(T ∗ ). Thus, WΘ (z) = WΘ1 (z) · WΘ2 (z) belongs to the class Ω0 (R, J). Theorem 7.3.5. Let operator-valued functions W1 (z) and W2 (z) in the Hilbert space E belong to the class Ω1 (R, J) and GW1 ∩ GW2 = ∅. Then their product W (z) = W1 (z) · W2 (z) is deﬁned on GW1 ∩ GW2 and belongs to the class Ω1 (R, J). Proof. Using the same argument as in Theorem 7.3.3 we conclude that W (z) is deﬁned on some domain of the complex plane, belongs to Ω(R, J) class, and is realizable by an L-system Θ of the type (7.33) such that W (z) = WΘ (z) for z ∈ ˙ = H ρ(T ). It only remains to show that the L-system Θ has the property Dom(A) ∗ and Dom(T ) = Dom(T ). Since both W1 (z) and W2 (z) belong to the class Ω1 (R, J) then correspondent L-systems Θ1 and Θ2 possesses properties Dom(A˙ 1 ) = H, Dom(T1 ) = Dom(T1∗ ) and Dom(A˙ 2 ) = H, Dom(T2 ) = Dom(T2∗ ). Due to Theorem 7.3.3 Θ = Θ1 · Θ2 . In the proof of Theorem 7.1.8 we have shown that L-systems with the above condition have reduced form. Namely, T1 K1 J T2 K2 J Θ1 = and Θ2 = H+1 ⊂ H1 ⊂ H−1 E H+2 ⊂ H2 ⊂ H−2 E where Im Tj = Kj JKj∗ , (j = 1, 2). Consequently, the domain of the state-space operator of the system Θ = Θ1 · Θ2 is determined by the formula Dom(T ) = Dom(T1 ) ⊕ Dom(T2 ). Using the fact that Dom(T1 ) = Dom(T1∗ ) and Dom(T2 ) = Dom(T2∗ ) we conclude that Dom(T ) = Dom(T ∗ ). This implies (see the argument in the proof of Theorem ˙ = H. Thus, W (z) ∈ Ω1 (R, J). 7.1.5) that Dom(A) The next result for the class Ω01 (R, J) is not that straightforward and an additional condition is required. Theorem 7.3.6. Let operator-valued functions W1 (z) and W2 (z) in the Hilbert space E belong to the class Ω01 (R, J) and GW1 ∩ GW2 = ∅. Then the product W (z) = W1 (z) · W2 (z) is deﬁned on GW1 ∩ GW2 and belongs to the class Ω01 (R, J) if and only if the set , 8 D = g = g1 + g2 ∈ H1 ⊕ H2 , g1 ∈ Dom(T1∗ ), g2 ∈ Dom(T2 ), 9 (7.35) K1∗ g1 + K2∗ g2 = 0 ,

222

Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

is not dense in H = H1 ⊕ H2 . Here T1 , T2 , K1 , K2 , H1 and H2 are correspondent elements of the L-systems Θ1 and Θ2 related to the functions W1 (z) and W2 (z). Proof. Since Ω(R, J) is a union of three distinct classes Ω0 (R, J), Ω1 (R, J), and Ω01 (R, J), Theorem 7.3.3 guarantees that W (z) is deﬁned on some domain in C and belongs to one of the indicated subclasses. The set D deﬁned by (7.35) actually coincides with the domain of operator A˙ deﬁned in (7.21). Therefore, since D is ˙ = H and W (z) is certainly not in the not dense in H = H1 ⊕ H2 , then Dom(A) Ω0 (R, J) class. Let us assume that W (z) belongs to Ω1 (R, J). Then the L-system Θ = Θ1 ·Θ2 has a property H+ = Dom(T ) = Dom(T ∗ ). The operator T here is actually the quasi-kernel of the operator A of the L-system Θ. Hence, for all g ∈ H+ , g = g1 +g2 , g1 ∈ H+1 , g2 ∈ H+2 , A1 g1 + A2 g2 + 2iK1 JK2∗ g2 ∈ H,

A∗2 g2 + A∗2 g2 − 2iK2JK1∗ g2 ∈ H,

where H = H1 ⊕ H2 and all operators belong to the correspondent systems Θ1 and Θ2 . Since g2 is an arbitrary element of H+2 , then we can choose it equal to 0. Thus the ﬁrst relation yields g1 ∈ Dom(T1 ) for all g1 ∈ H+1 . Because g1 is arbitrary we have that Dom(T1 ) = H+1 = Dom(A˙ ∗1 ). Taking into account that W1 (z) ∈ Ω01 (R, J) we get a contradiction. Hence, the product of W1 (z) and W2 (z) under the assumption of the theorem belongs to the class Ω01 (R, J). Remark 7.3.7. It is not hard to show that if the set D in the statement of the Theorem 7.3.6 is dense in H1 ⊕ H2 , then W (z) = W1 (z) · W2 (z) belongs to the class Ω0 (R, J). The theorem below describes properties of the mixed products of two operator-valued functions of Ω(R, J) class. Theorem 7.3.8. Let operator-valued functions W1 (z) and W2 (z) in the Hilbert space E (dim E > 1) belong to the classes Ω0 (R, J) and Ω1 (R, J), respectively, and GW1 ∩GW2 = ∅. Then their product W (z) = W1 (z)·W2 (z) is deﬁned on GW1 ∩GW2 and belongs to the class Ω01 (R, J) if and only if the set D in (7.35) is not dense in H = H1 ⊕ H2 . We omit the proof of this theorem because it is similar to the one of the Theorem 7.3.6. As before we should note that if the set D in the statement of Theorem 7.3.8 is not dense in H1 ⊕ H2 , then W (z) = W1 (z) · W2 (z) belongs to the class Ω0 (R, J). Furthermore, Theorem 7.3.8 holds even if we consider product W (z) = W2 (z) · W1 (z).

7.3. Multiplication Theorems for Ω(R, J) classes

223

Theorem 7.3.9. Let operator-valued functions W1 (z) and W2 (z) in the Hilbert space E (dim E > 1) belong to the classes Ω0 (R, J) and Ω1 (R, J), respectively, and GW1 ∩ GW2 = ∅. Let also W1 (z) · W2 (z) = W2 (z) · W1 (z) = W (z).

(7.36)

Then operator-valued function W (z) is deﬁned on GW1 ∩ GW2 and belongs to the class Ω01 (R, J). Proof. Condition (7.36) implies that A = A1 P1+ + A2 P2+ + 2iK1 JK2∗ P2+ = A2 P2+ + A1 P1+ + 2iK2 JK1∗ P1+ . The obvious cancellation yields K1 JK2∗ P2+ = K2 JK1∗ P1+ . Left- and right-hand sides of this equality belong to H−1 and H−2 , respectively. Hence the equality may hold only if K1 JK2∗ P2+ = K2 JK1∗ P1+ = 0. Thus, A = A1 P1+ + A2 P2+ , and we are actually dealing with operator A of a block-diagonal structure. Now let f = f1 + f2 be an element of Dom(T ), then T f = Af = A1 f1 + A2 f2 ∈ H, but A2 f2 ∈ H, and therefore, A1 f1 ∈ H, or T f = T 1 f1 + T 2 f2 , Similarly,

T ∗ g = T1∗ g1 + T2∗ g2 ,

where g = g1 + g2 is an element of Dom(T ∗ ). In other words we have just shown that f ∈ Dom(T ) implies f1 ∈ Dom(T1 ) and f2 ∈ Dom(T2 ), where f = f1 + f2 . Conversely, if f1 ∈ Dom(T1 ) and f2 ∈ Dom(T2 ), then f = f1 + f2 ∈ Dom(T ), i.e., Dom(T ) = Dom(T1 ) ⊕ Dom(T2 ). Similarly shown,

Dom(T ∗ ) = Dom(T1∗ ) ⊕ Dom(T2∗ ).

But since Dom(T1 ) = Dom(T1∗ ) and Dom(T2 ) = Dom(T2∗ ), then Dom(T ) = Dom(T ∗ ). It is also not hard to see that, under these circumstances, ˙ = Dom(A˙ 1 ) ⊕ Dom(A˙ 2 ). Dom(A) ˙ = H. It follows then that W (z) belongs Hence, if Dom(A˙ 2 ) = H2 , then Dom(A) to the class Ω01 (R, J).

224

Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

7.4 Boundary triplets and self-adjoint bi-extensions Let A˙ be a closed densely deﬁned symmetric operator in H with equal deﬁciency numbers. Deﬁnition 7.4.1. The triplet Π = {N , Γ1 , Γ0 } is called a boundary triplet for A˙ ∗ if N is a Hilbert space and Γ0 , Γ1 are bounded linear operators from the Hilbert space H+ = Dom(A˙ ∗ ) (with the inner product (2.6)) into N such that the mapping ; < Γ := Γ0 , Γ1 : H+ → N ⊕ N , is surjective and the abstract Green’s identity A˙ ∗ f, g − f, A˙ ∗ g = (Γ1 f, Γ0 g)N − (Γ0 f, Γ1 g)N ,

(7.37)

holds for all f, g ∈ H+ . A boundary triplet for A˙ ∗ exists since the deﬁciency numbers are assumed to be equal [164], [142], [143]. It follows from Deﬁnition 7.4.1 (see [101], [103]) that the operators Dom(Ak ) := Ker Γk ,

Ak := A˙ ∗ Dom(Ak ),

(k = 0, 1),

(7.38)

˙ Moreover, they are transversal, i.e., are self-adjoint extensions of A. Dom(A˙ ∗ ) = Dom(A0 ) + Dom(A1 ). Notice that if Π = {N , Γ1 , Γ0 } is a boundary triplet for A˙ ∗ , then Π = {N , −Γ0 , Γ1 } is the boundary triplet for A˙ ∗ too. Let N be a Hilbert space whose dimension is equal to the deﬁciency number ˙ We choose a self-adjoint extension A of A. ˙ of A. Deﬁnition 7.4.2. The function Γ(z) ∈ [N , H] is called the γ-ﬁeld, corresponding to A if 1. the operator Γ(z) isomorphically maps N onto Nz for all z ∈ ρ(A), where ˙ Nz are the defect subspaces of A, 2. for every z, ζ ∈ ρ(A) the identity Γ(z) = Γ(ζ) + (z − ζ)(A − zI)−1 Γ(ζ)

(7.39)

holds. Due to (7.39) the γ-ﬁeld Γ(z) is a holomorphic operator-valued function of z ˆ in ρ(A). Two γ-ﬁelds Γ(z) and Γ(z) corresponding to the same self-adjoint extenˆ X, ˆ where X ˆ is an isomorphism sion A are connected by the relation Γ(z) = Γ(z) ˆ of Hilbert spaces N and N [175].

7.4. Boundary triplets and self-adjoint bi-extensions

225

The γ-ﬁeld corresponding to A can be constructed as follows: ﬁx ζ0 ∈ ρ(A) and let Γ(0) ∈ [N , H] be a bijection of N and Nζ0 . Then clearly, the function Γ(z) = (A − ζ0 I)(A − zI)−1 Γ(0) = Γ(0) + (z − ζ0 )(A − zI)−1 Γ(0) , z ∈ ρ(A) is a γ-ﬁeld corresponding to A. Let Π = {N , Γ1 , Γ0 } be a boundary triplet for A˙ ∗ . Deﬁne an operator-valued function −1 γ0 (z) = (Γ0 Nz ) , z ∈ ρ(A0 ). (7.40) It can be derived from (7.37) and (7.40) (see [101], [103]) that γ0 (z) = (A0 − ζI)(A0 − zI)−1 γ0 (ζ), γ0∗ (¯ z ) = Γ1 (A0 − zI)−1 , z, ζ ∈ ρ(A0 ).

(7.41)

Thus, the function γ0 forms the γ-ﬁeld, corresponding to A0 . Deﬁnition 7.4.3. Let Π = {N , Γ1 , Γ0 } be a boundary triplet for A˙ ∗ . An operatorvalued function M0 (z) ∈ [N , N ] deﬁned by the relation M0 (z) := Γ1 γ0 (z),

z ∈ ρ(A0 ),

(7.42)

is called the Weyl-Titchmarsh function corresponding to the boundary triplet Π. Notice that (7.42) implies the relations M0 (z)Γ0 fz = Γ1 fz , fz ∈ Nz , z ∈ ρ(A0 ). From (7.41) one can also get M0 (¯ z ) = M0∗ (z),

¯ ∗ (ζ)γ0 (z), M0 (z) − M0∗ (ζ) = (z − ζ)γ 0

z, ζ ∈ ρ(A0 ).

Similarly, one deﬁnes another Weyl-Titchmarsh function related to the boundary triplet Π = {N , Γ1 , Γ0 } or with Π = {N , −Γ0 , Γ1 }. Set γ1 (z) := (Γ1 Nz )

−1

, z ∈ ρ(A1 )

and M1 (z) = −Γ0 γ1 (z).

(7.43)

The function M1 is connected with the function M0 via M1 (z) = −M0−1 (z), Im z = 0.

(7.44)

Now we are going to establish connections between self-adjoint bi-extensions and boundary triplets. The proposition below immediately follows from Deﬁnition 7.4.1.

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Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

Proposition 7.4.4. Let A˙ be a closed densely deﬁned symmetric operator with equal deﬁciency indices in the Hilbert space H. Suppose N is a Hilbert space, Γ0 , Γ1 ∈ ; < [H+ , N ], and the operator Γ0 , Γ1 ∈ [H+ , N ⊕N ] is surjective. Then the following statements are equivalent: (i) Π = {N , Γ1 , Γ0 } is the boundary triplet for A˙ ∗ ; (ii) the sesquilinear form w0 (f, g) := (A˙ ∗ f, g) + (Γ0 f, Γ1 g)N ,

f, g ∈ H+ = Dom(A˙ ∗ )

is Hermitian, i.e., w0 (f, g) = w0 (g, f ); (iii) the sesquilinear form w1 (f, g) := (A˙ ∗ f, g) − (Γ1 f, Γ0 g)N ,

f, g ∈ H+ = Dom(A˙ ∗ )

is Hermitian. The following theorem sets up the connection between boundary triplets and t-self-adjoint bi-extensions. Theorem 7.4.5. Let A˙ be a closed densely-deﬁned symmetric operator with equal deﬁciency numbers in the Hilbert space H. Consider the rigged Hilbert space H+ ⊂ ˙ H ⊂ H− generated by A. 1. Let Π = {N , Γ1 , Γ0 } for A˙ ∗ be a boundary triplet for A˙ ∗ . Deﬁne operators A0 and A1 , A0 := A˙ ∗ + Γ× A1 := A˙ ∗ − Γ× (7.45) 1 Γ0 , 0 Γ1 , × where Γ× 0 and Γ1 ∈ [N , H− ] are the adjoint operators to Γ0 and Γ1 , respectively. Then:

˙ (i) A0 and A1 belong to [H+ , H− ] and are t-self-adjoint bi-extensions of A. Moreover, A0 ⊃ A0 , A1 ⊃ A1 , (ii) the Weyl-Titchmarsh function deﬁned by (7.42) is given by M0 (z) = Γ1 (A0 − zI)−1 Γ× 1 ,

(7.46)

(iii) the Weyl-Titchmarsh function deﬁned by (7.43) is given by M1 (z) = Γ0 (A1 − zI)−1 Γ× 0,

z ∈ ρ(A1 ).

(7.47)

˙ then 2. If A0 is a t-self-adjoint bi-extension of a self-adjoint extension A0 of A, ∗ ∗ ˙ ˙ there exists a boundary triplet Π = {N , Γ1 , Γ0 } for A such that A ker Γ0 = A0 and A0 = A˙ ∗ + Γ× 1 Γ0 .

7.4. Boundary triplets and self-adjoint bi-extensions

227

Proof. 1. The statement (i) follows from Proposition 7.4.4 and relations (7.38). To prove (ii) we let fz ∈ Nz , Im z = 0. Then × (A0 − zI)fz = (A˙ ∗ − zI)fz + Γ× 1 Γ0 fz = Γ1 Γ0 fz .

Let h = Γ0 fz . Then by (7.40) we have fz = γ0 (z)h and (A0 − zI)γ0 (z)h = Γ× 1 h. Consequently, γ0 (z) = (A0 −zI)−1 Γ× 1 . Now from (7.42) we get (7.46). Let us prove (iii). From (7.45) we obtain (A1 − zI)fz = −Γ× 0 Γ 1 fz ,

f z ∈ Nz ,

z ∈ ρ(A1 ).

It follows that Ran(Γ× 0 ) ⊂ Ran(A1 − zI) for all z ∈ ρ(A1 ). Since (A0 − zI)−1 Γ× 1 Γ0 fz = fz , for all fz ∈ Nz , z ∈ ρ(A0 ), we get Γ0 (A0 − zI)−1 Γ× 1 Γ0 fz = Γ0 fz . The equalities ˙ z , ker(Γ0 ) = Dom(A0 ), and Ran(Γ0 ) = N imply H+ = Dom(A0 )+N Γ0 (A0 − z)−1 Γ× 1 = I,

z ∈ ρ(A0 ).

Similarly one obtains (A1 − zI)−1 Γ× 0 Γ1 fz = −fz for all fz ∈ Nz , z ∈ ρ(A1 ), and Γ1 (A1 − z)−1 Γ× 0 = −I,

z ∈ ρ(A1 ).

Notice that for z ∈ ρ(A0 ) ∩ ρ(A1 ) we have −1 × M1 (z)M0 (z) = Γ0 (A1 − z)−1 Γ× Γ1 = −Γ0 (A0 − z)−1 Γ× 0 Γ1 (A0 − z) 1 = −I.

Similarly, M0 (z)M1 (z) = −I. 2. Since A0 is a t-self-adjoint bi-extension of A˙ containing A0 as a quasikernel, the operator A0 is deﬁned by means of a self-adjoint extension of A1 transversal to A0 via formula (see Theorem 3.4.9) A0 = A˙ ∗ − R−1 A˙ ∗ PA0 A1 = A˙ ∗ − R−1 A1 PA0 A1 ,

(7.48)

where PA0 A1 is a projector in H+ onto (I + U1 )Ni corresponding to the decompositions of the form (3.35) and (3.36) with U = −U or ˙ + U1 )Ni , H+ = Dom(A0 )+(I ˙ ⊕ (I + U1 )Ni . Let PA1 A0 be a projector in H+ onto and Dom(A1 ) = Dom(A) (I + U0 )Ni corresponding to the decomposition ˙ + U0 )Ni , H+ = Dom(A1 )+(I

228

Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

˙ ⊕ (I + U0 )Ni . Deﬁne and Dom(A0 ) = Dom(A) A1 = A˙ ∗ − R−1 A˙ ∗ PA1 A0 = A˙ ∗ − R−1 A0 PA1 A0 .

(7.49)

Then A1 is a t-self-adjoint bi-extension of A˙ containing A1 as a quasi-kernel. The connection with A0 is given by the relation (3.31), i.e., (A1 f, g) = (A˙ ∗ f, g) + (f, A˙ ∗ g) − (A0 f, g),

f, g ∈ H+ ,

which is equivalent to the equality (A0 − A˙ ∗ )∗ = A˙ ∗ − A1 ,

(7.50)

where (A0 − A˙ ∗ )∗ ∈ [H+ , H− ]. Let subspaces Φ0 ⊂ H− , Φ1 ⊂ H− be deﬁned as Φ0 := Ran(A0 − A˙ ∗ ),

Φ1 := Ran(A1 − A˙ ∗ ).

From (3.34) we get Φ0 = R−1 (I − U1 )Ni ,

Φ1 = R−1 (I − U0 )Ni .

Then dim Φ0 = dim Ni . Choose a Hilbert space N with dim N = dim Φ0 . Let K1 ∈ [N , H− ] be a bijection onto Φ0 . It follows from (7.50)that the relation K0 K1× f = (A˙ ∗ − A1 )f,

f ∈ H+ ,

(7.51)

deﬁnes K0 ∈ [N , H− ], which is a bijection onto Φ1 . Finally let Γ0 := K0× ,

Γ1 = K1× .

Then Γ0 , Γ1 ∈ [H+ , N ] and A0 = A∗ + Γ× 1 Γ0 ,

A1 = A∗ − Γ× 0 Γ1 .

(7.52)

Because A0 and A1 are mutually transversal, the operator ; < Γ0 , Γ1 : H+ → N ⊕ N is a surjection. Thus, Π = {N , Γ1 , Γ0 } is a boundary triplet for A˙ ∗ and A˙ ∗ ker Γ0 = A0 .

7.5 The Krein-Langer Q-functions and their realizations Let A˙ be a densely-deﬁned closed symmetric operator in a Hilbert space H with ﬁnite equal deﬁciency indices and let again N be a Hilbert space whose dimension ˙ Let us also choose a self-adjoint extension is equal to the deﬁciency number of A. ˙ A of A.

7.5. The Krein-Langer Q-functions and their realizations

229

Deﬁnition 7.5.1. Let Γ(z) ∈ [N , H] be a γ-ﬁeld corresponding to A. An operatorvalued function Q(z) ∈ [N , N ]) with the property Q(z) − Q∗ (ζ) = (z − ζ)Γ∗ (ζ)Γ(z),

z, ζ ∈ ρ(A)

(7.53)

is called the Kre˘ın-Langer Q-function of A˙ corresponding to the γ-ﬁeld Γ(z). It follows from (7.53) that Q(z) = C − iIm ζ0 Γ∗ζ0 Γζ0 + (z − ζ¯0 )Γ∗ζ0 Γz , where C = Re Q(ζ0 ) ∈ [N , N ] is a self-adjoint operator. Thus, the Q-function is deﬁned up to the bounded self-adjoint term in N and is a Herglotz-Nevanlinna function. Moreover, for every z, Im z = 0 the operator −i Im z (Q(z) − Q∗ (z)) is positive deﬁnite. Hence, −Q−1 (z), (Im z = 0) is a Herglotz-Nevanlinna function too. From (7.39) we get Q(z) = C − iIm ζ0 Γ∗ζ0 Γζ0 + (z − ζ¯0 )Γ∗ζ0 Γζ0 + (z − ζ0 )(A − zI)−1 Γζ0 . (7.54) The γ-ﬁelds and corresponding Q-functions can be deﬁned by means of the resol˙ Indeed, consider the rigged Hilbert space vents of t-self-adjoint bi-extensions of A. ˙ Let A be a self-adjoint extension H+ ⊂ H ⊂ H− generated by the operator A. ˙ Choose a Hilbert of A˙ and let A be an arbitrary t-self-adjoint bi-extension of A. space N with dim N = dim Ni and an arbitrary operator K ∈ [N , H− ] such that Kmaps N isomorphically onto the subspace Ran(A − A˙ ∗ ). Put Γ(z) := (A − zI)−1 K,

z ∈ ρ(A).

(7.55)

Then clearly Γ(z) maps N isomorphically onto Nz for each z ∈ ρ(A), satisﬁes (7.39), and Γ∗ (z) = K × (A − z¯I)−1 , where K × ∈ [H+ , N ] is the adjoint operator. Deﬁne a [N , N ]-valued function Q(z) := K × Γ(z) = K × (A − zI)−1 K,

z ∈ ρ(A).

(7.56)

Then ¯ −1 K Q(z) − Q∗ (ζ) = K × (A − I)−1 − (A − ζI) ¯ × (A − ζI) ¯ −1 (A − zI)−1 K = (z − ζ)Γ ¯ ∗ (ζ)Γ(z). = (z − ζ)K Thus, the functions deﬁned by (7.55) and (7.56) form a γ-ﬁeld, corresponding to A and a Q-function. Clearly, the function C + K × (A − z)−1 K with an arbitrary self-adjoint operator C ∈ [N , N ] is also a Q-function, corresponding to the γ-ﬁeld (7.55).

230

Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

˙ Γ(z) ∈ [N , H] be the γTheorem 7.5.2. Let A0 be a self-adjoint extension of A, ﬁeld corresponding to A0 , and let Q(z) be the Q-function of A˙ corresponding to the γ-ﬁeld Γ(z). Then there exists a boundary triplet Π = {N , Γ1 , Γ0 } of A˙ ∗ whose Weyl-Titchmarsh function M0 coincides with Q. Proof. Let ˙ ⊕ (I + U0 )Ni , Dom(A0 ) = Dom(A) where U0 isometrically maps Ni onto N−i . The operator Γ(i) isomorphically maps N onto Ni . Hence one may consider Γ(i) as an element of [N , Ni ]. Let Γ∗ (i) ∈ [Ni , N ] be the adjoint of Γ(i). Deﬁne a unitary operator U0 ∈ [Ni , Ni ] by −1 U0 := − iI + Γ∗−1 (i)Re Q(i)Γ−1 (i) iI − Γ∗−1 (i)Re Q(i)Γ−1 (i) . Set U1 = U0 U0 and deﬁne a self-adjoint extension A1 of A˙ via ˙ ⊕ (I + U1 )Ni , Dom(A1 ) = Dom(A) We have

A1 := A˙ ∗ |Dom(A1 ).

−1 U0 − U1 = 2iU0 iI − Γ∗−1 (i)Re Q(i)Γ−1 (i) .

(7.57)

It follows that the operator U1 −U0 is an isomorphism of Ni and N−i . Therefore, the self-adjoint extensions A0 and A1 are transversal. Let t-self-adjoint bi-extensions A0 and A1 be deﬁned by (7.48) and (7.49). Then A0 ⊃ A0 and A1 ⊃ A1 . Set K1 = (A0 − iI)Γ(i),

Γ1 = K1× = Γ× (i)(A0 + iI) ∈ [H+ , N ].

Here Γ× (i) ∈ [H− , N ] is the adjoint operator for Γ(i) ∈ [N , H+ ]. For z ∈ ρ(A0 ) one has (A0 − zI)−1 K1 = (A0 − z)−1 (A0 − iI)Γ(i) = Γ(i) + (z − i)(A0 − zI)−1 Γ(i). It follows that

Γ(z) = (A0 − zI)−1 K1 .

(7.58)

Deﬁne the operator K0 ∈ [N , H+ ] by (7.51) and put Γ0 = K0× . As it has been proved above, the collection Π = {N , Γ1 , Γ0 } is a boundary triplet for A˙ ∗ and × −1 M0 (z) = Γ1 (A0 − zI)−1 Γ× K1 , 1 = K1 (A0 − zI)

is the Weyl-Titchmarsh function corresponding to Π. Let us show that M0 (z) = Q(z) for z ∈ ρ(A0 ). Notice that from (7.53) and (7.58) we get Q(z) = Q∗ (i) + (z + i)Γ∗ (i)Γ(z) = Q∗ (i) + (z + i)K1× (A0 + iI)−1 (A0 − zI)−1 K1 = Q∗ (i) + K1× (A0 − zI)−1 K1 − K1× (A0 + iI)−1 K1 .

7.5. The Krein-Langer Q-functions and their realizations

231

If z = i, then 2iIm Q(i) = K1× (A0 − iI)−1 K1 − K1× (A + iI)−1 K1 . Therefore Q(z) = M0 (z) + Re Q(i) − Re K1× (A0 − iI)−1 K1 .

(7.59)

Furthermore, Re K1× (A0 − iI)−1 K1 h, g N = (A0 (A − iI)−1 K1 h, (A − iI)−1 K1 g) = (A0 Γ(i)h, Γ(i)g), h, g ∈ N . By simple calculations we get PA0 A1 ϕ = (I + U1 )(U0 − U1 )−1 U0 ϕ,

ϕ ∈ Ni .

Hence for ϕ, ψ ∈ Ni , (A0 ϕ, ψ) = (iϕ − R−1 A1 PA0 A1 ϕ, ψ) = i(ϕ, ψ) − (A1 PA0 A1 ϕ, ψ)+ = i(ϕ, ψ) − i((I − U1 )(U0 − U1 )−1 U0 ϕ, ψ)+ = i(ϕ, ψ) − i((U0 − U1 )−1 U0 ϕ, ψ)+ = i(ϕ, ψ) − 2i((U0 − U1 )−1 U0 ϕ, ψ) = (if − 2i(U0 − U1 )−1 U0 ϕ, ψ). It follows that for h, g ∈ N , (A0 Γ(i)h, Γ(i)g) = (iΓ∗ (i)Γ(i)h − 2iΓ∗ (i))(U0 − U1 )−1 U0 Γ(i)h, g)N , and (7.57) yields Re Q(i) = iΓ∗ (i)Γ(i)f − 2iΓ∗ (i))(U0 − U1 )−1 U0 Γ(i). Taking into account (7.59), we obtain Q(z) = M0 (z), z ∈ ρ(A0 ).

The next statement immediately follows from Theorems 7.4.5, 7.5.2, and relations (7.43), (7.47), and (7.44). Theorem 7.5.3. Let Γ(z) ∈ [N , H] be a γ-ﬁeld corresponding to a self-adjoint extension A of A˙ and let Q(z) be a Q-function corresponding to Γ(z). Then there exist a t-self-adjoint bi-extension A of A˙ and an operator K ∈ [N , H− ] which isomorphically maps N onto Ran(A − A˙ ∗ ) such that Γ(z) = (A − zI)−1 K,

Q(z) = K × (A − zI)−1 K,

z ∈ ρ(A).

In addition, the function −Q−1 (z) takes the form −Q−1 (z) = K where:

×

(A − zI)−1 K ,

Im z = 0,

232

Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

1. A is a self-adjoint bi-extension of A˙ deﬁned as A = A˙ ∗ + (A˙ ∗ − A)∗ , 2. the quasi-kernel of A is a self-adjoint extension of A˙ transversal to A, 3. the operator K is deﬁned by the relation K K × = A˙ ∗ − A . Remark 7.5.4. Theorem 7.5.3 above implies that a Krein-Langer Q-function corresponding to a self-adjoint extension A of A˙ can be realized as a transfer function of an impedance system Δ of the form (6.49). This system Δ contains the state space operator A, E = N , and operator K is the one described in the statement of Theorem 7.5.3. It also follows from Theorem 7.1.4 that any Krein-Langer Q-function Q(z) as well as −Q−1 (z) belongs to the class N0 (R). Consequently, by Theorem 7.1.5, both Q(z) and −Q−1 (z) can be realized as impedance functions of L-systems. We can also conclude that if an invertible V (z) ∈ N0 (R), then −V −1 (z) ∈ N0 (R). Let Π = {N , Γ1 , Γ0 } be a boundary triplet for A˙ ∗ and let C = C ∗ ∈ [N , N ]. Then ΠC = {N , Γ1 + CΓ0 , Γ0 } is also a boundary triplet for A˙ ∗ . With ΠC are associated two self-adjoint bi-extensions × × A0C = A˙ ∗ + (Γ× 1 + Γ0 C)Γ0 = A0 + Γ0 CΓ0 ,

and × A1C = A˙ ∗ − Γ× 0 (Γ1 + CΓ0 ) = A1 − Γ0 CΓ0 ,

where A0 and A1 are given by (7.52). Note that A0C ⊃ A0 and A1C ⊃ A1C , where Dom(A1C ) = {f ∈ Dom(A˙ ∗ ) : Γ1 f + CΓ0 f = 0},

A1C = A˙ ∗ Dom(A1C ).

The Weyl-Titchmarsh function MC (z) = (Γ1 + CΓ0 )γ0 (z) corresponding to ΠC is of the form MC (z) = M0 (z) + C. On the other hand by Theorem 7.4.5 (see (7.46)) we have × MC (z) = (Γ1 + CΓ0 )(A0C − zI)−1 (Γ× 1 + Γ0 C) −1 × = (Γ1 + CΓ0 )(A0 + Γ× (Γ1 + Γ× 0 CΓ0 − zI) 0 C).

Thus, for all z ∈ ρ(A0 ) we obtain × × −1 Γ1 (A0 − zI)−1 Γ× (Γ× 1 + C = (Γ1 + CΓ0 )(A0 + Γ0 CΓ0 − zI) 1 + Γ0 C).

7.6. Examples

233

7.6 Examples Example. This example is to illustrate the realization in N0 (R) class. Let Tx =

1 dx , i dt

(7.60)

, , with Dom(T ) = x(t) , x(t) − abs. continuous, x (t) ∈ L2[0,l] , x(0) = 0 be the differential operator in H = L2[0,l] (l > 0). Obviously, T ∗x =

1 dx , i dt

, , x(t) , x(t) − abs. continuous, x (t) ∈ L2[0,l] , x(l) = 0 ˙ joint. Consider a symmetric operator A, with Dom(T ∗ ) =

is its ad-

˙ = 1 dx , Ax i dt , , ˙ Dom(A) = x(t) , x(t) − abs. continuous, x (t) ∈ L2[0,l] , x(0) = x(l) = 0 , (7.61) and its adjoint A˙ ∗ , 1 dx A˙ ∗ x = , i dt , , Dom(A˙ ∗ ) = x(t) , x(t) − abs. continuous, x (t) ∈ L2[0,l] .

(7.62)

Then H+ = Dom(A˙ ∗ ) = W21 is a Sobolev space with scalar product (x, y)+ =

l

x(t)y(t) dt + 0

l

x (t)y (t) dt.

0

Construct rigged Hilbert space W21 ⊂ L2[0,l] ⊂ (W21 )− and consider operators Ax =

1 dx + ix(0) [δ(t − l) − δ(t)] , i dt

A∗ x =

1 dx + ix(l) [δ(t − l) − δ(t)] , (7.63) i dt

where x(t) ∈ W21 , δ(t), δ(t − l) are delta-functions and elements of (W21 )− that generate functionals by the formulas (x, δ(t)) = x(0) and (x, δ(t − l)) = x(l). It is ˙ A∗ ⊃ T ∗ ⊃ A, ˙ and easy to see that A ⊃ T ⊃ A, ⎛ 1 dx Θ1 = ⎝

i dt

+ ix(0)[δ(t − l) − δ(t)]

W21 ⊂ L2[0,l] ⊂ (W21 )−

K

⎞ −1 ⎠ C

(J = −1),

234

Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

is an L-system where 1 Kc = c · √ [δ(t − l) − δ(t)], (c ∈ C), 2 1 1 K ∗ x = x, √ [δ(t − l) − δ(t)] = √ [x(l) − x(0)], 2 2

(7.64)

and x(t) ∈ W21 . Also

1 1 Im A = − ·, √ [δ(t − l) − δ(t)] √ [δ(t − l) − δ(t)]. 2 2

The transfer function of this system can be found as WΘ1 (z) = I − 2iK ∗ (A − zI)−1 KJ = eizl . Consider the following Herglotz-Nevanlinna function (hyperbolic tangent) i V (z) = −i tanh zl . 2 Obviously this function can be realized as i i i e 2 zl − e− 2 zl eizl − 1 V (z) = −i tanh zl = −i i zl = −i i 2 eizl + 1 e 2 + e− 2 zl = i [WΘ1 (z) + I]−1 [WΘ1 (z) − I] J Now let us consider another L-system ⎛ 1 dx + ix(l)[δ(t − l) − δ(t)] i dt Θ2 = ⎝ W21 ⊂ L2[0,l] ⊂ (W21 )−

(J = −1).

K

1 C

⎞ ⎠

(J = 1),

(7.65)

whose state-space operator is A∗ and J = 1. Similar reasoning conﬁrms that WΘ2 (z) = e−izl ,

but V (z) = −i tanh

i zl 2

(7.66) −1

= i [WΘ2 (z) + I]

[WΘ2 (z) − I] J.

This shows that V (z) ∈ N0 (R) is the impedance function of the L-system Θ2 in (7.65) with the transfer function WΘ2 (z) of the form (7.66). Example. Consider a bounded linear operator in C2 deﬁned by i i T = . (7.67) −i 1

7.6. Examples

235

Let x and ϕ be the elements of C2 such that x1 1 x= and ϕ = . x2 0 Obviously,

−i T = −i ∗

i . 1

It is clear that Dom(T ) = Dom(T ∗ ) = C2 . Let J = 1. Now we can ﬁnd T − T∗ 1 0 Im T = = . 0 0 2i and show that ϕ above is the only channel vector such that 1 0 x1 Im T x = = (x, ϕ)ϕ. 0 0 x2 Consider a symmetric operator A˙ of the form 8 9 x1 ˙ = 0 i ˙ = x ∈ C2 | (x, ϕ) = 0 . Ax , Dom(A) −i 1 x2

(7.68)

Obviously, A˙ is the maximal common symmetric part of T and T ∗ . Thus, operator T can be included in the system T K 1 Θ= , (7.69) C2 C with c Kc = c ϕ = , 0

∗

K x = (x, ϕ) = x1 ,

c ∈ C,

x=

x1 x2

∈ C2 ,

Then WΘ (z) is represented by the formula WΘ (z) =

z 2 + (1 − i)z − 1 − i . z 2 − (1 + i)z − 1 + i

The impedance function VΘ (z) is a Herglotz-Nevanlinna function VΘ (z) =

z2

1−z −z−1

and hence belongs to the class N1 (R) according to Theorem 7.1.7.

(7.70)

236

Chapter 7. Classes of realizable Herglotz-Nevanlinna functions

Example. In order to illustrate the realization in N01 (R) class we will use Examples 7.6 and 7.6. Consider an L-system ⎛ ⎞ A K J ⎠ Θ=⎝ W21 ⊕ C2 ⊂ L2[0,l] ⊕ C2 ⊂ (W21 )− ⊕ C2 C2 where A is a diagonal block-matrix A=

A1 0

0 T

,

with A1 =

1 dx + ix(0) [δ(t − l) − δ(t)] , i dt

of the form (7.63) from Example 7.6, and T of the form (7.67) from Example 7.6. The corresponding symmetric operator A˙ in this case takes the form A˙ 1 0 A˙ = , 0 A˙ 2 ˙ = where A˙ 1 and A˙ 2 are deﬁned by (7.62) and (7.68), respectively. Clearly, Dom(A) 2 2 2 1 2 L[0,l] ⊕ C . Operator K ∈ [C , (W2 )− ⊕ C ] here is deﬁned as a diagonal operator block-matrix K1 0 K= , 0 K2 where operators K1 and K2 are from Examples 7.6 and 7.6, respectively. Moreover, K ∗ ∈ [W21 ⊕ C2 , C2 ] and J are deﬁned by ∗ K1 0 −1 0 K∗ = , J = . 0 K2∗ 0 1 We ﬁnd that Im A f =

Im A1 0

0 Im T

f1 −(f1 , g) g = = KJK ∗ , f2 (f2 , ϕ) ϕ

f ∈ W21 ⊕ C2 ,

1 where g = and ϕ = are the channel vectors from Examples 0 7.6 and 7.6, respectively. It can be easily shown that ⎛ ⎞ eizl 0 ⎜ ⎟ WΘ (z) = ⎝ ⎠ z 2 +(1−i)z−1−1 0 z 2 −(1+i)z−1+i √1 [δ(t − l) − δ(t)] 2

7.6. Examples and

237 ⎛ VΘ (z) = ⎝

−i tanh

i

2 zl

0

0 1−z z 2 −z−1

⎞ ⎠.

According to Theorem 7.1.10, the function VΘ (z) above belongs to the class N01 (R). Example. This example illustrates multiplication Theorem 7.3.8. Let the L-system Θ1 be given by (7.65) with all the components constructed in Example 7.6. Its transfer function WΘ1 (z) is presented in (7.66). Clearly, by construction WΘ1 (z) ∈ Ω0 (R, 1). Similarly, let Θ2 be the L-system of the form (7.69) with the transfer function WΘ2 (z) given by (7.70). It follows then from derivations in Example 7.6 that WΘ2 (z) ∈ Ω1 (R, 1). Consider W (z) = WΘ1 (z) · WΘ2 (z) = (e−izl ) ·

z 2 + (1 − i)z − 1 − i . z 2 − (1 + i)z − 1 + i

As we have already mentioned in Section 7.2, the class Ω01 (R, J) does not exist for the case of scalar functions when dim E = 1. Thus the function W (z) above can only belong to either Ω0 (R, 1) or Ω1 (R, 1). The latter, however, is impossible because of Theorem 7.3.6 and Remark 7.3.7. Hence, W (z) ∈ Ω0 (R, 1).

Chapter 8

Normalized L-Systems In this chapter we consider special types of L-systems and study the properties of their transfer functions. In the ﬁrst two sections we will use the notion of an auxiliary canonical system to prove a theorem about the constant J-unitary factor. The theorem states that if an operator-valued function W (z) belongs to the class Ω0 (R, J) described in Section 7.2, then for an arbitrary J-unitary operator B the functions W (z)B and BW (z) again belong to the same class Ω0 (R, J). Consequently, they can be realized as transfer functions of the same type of Lsystem. We will construct this new realizing system and show that it contains the same unbounded operator T but diﬀerent channel operators K. In the remainder of the chapter we deal with specially constructed L-systems, so called normalized L-systems. It will be shown that under certain conditions on the state-space operator of such an L-system Θ, one can always ﬁnd another L system 1 Θ so that the two transfer functions are related with a property WΘ (λ) = WΘ λ .

8.1 Auxiliary canonical system Let H be a separable Hilbert space and let A˙ be a closed symmetric operator ˙ = H. We also assume that A˙ has ﬁnite whose domain is dense in H, i.e., Dom(A) and equal deﬁciency indices. ˙ deﬁned on Dom(T ) with the Now let T be an operator of the class Λ(A) resolvent set ρ(T ). Consider HT = Dom(T ) as a Hilbert space HT with the inner product (u, v)T := (u, v) + (T u, T v), u, v ∈ HT . Let HT ⊂ H ⊂ HT be the rigged Hilbert space. Since T is a bounded operator from HT into H, the adjoint operator T × is bounded from H into HT and satisﬁes the condition (T u, f ) = (u, T × f ) for all u ∈ HT and all f ∈ H. Let T ∗ be the adjoint of T in H. Then obviously T × ⊃ T ∗ . The resolvent Rz (T ) = (T − zI)−1 of Y. Arlinskii et al., Conservative Realizations of Herglotz–Nevanlinna Functions, Operator Theory: Advances and Applications, DOI 10.1007/978-3-7643-9996-2_8, © Springer Basel AG 2011

239

240

Chapter 8. Normalized L-Systems

T is a bounded operator from H onto Dom(T ) = HT . It follows that the resolvent Rz¯(T ∗ ) = (T ∗ −¯ zI)−1 of T ∗ has the continuation Rz¯(T ∗ ) on HT and maps HT onto H. Moreover, Rz¯(T ∗ ) = (T × − z¯I)−1 (see Section 6.2). We know that according ˙ Dom(A) ˙ = ker(T − T × ). Deﬁne to the deﬁnition of the class Λ(A), Φ := Ran(T − T × )

(the closure in HT ).

(8.1)

Thus Φ ∩ H = {0}. Then ˙ = ker(T − T × ) = {g ∈ HT : (g, ϕ) = 0 Dom(A)

for all

ϕ ∈ Φ} .

˙ then T ⊃ A, ˙ T ∗ ⊃ A. ˙ Let as above Nz = ker(A˙ ∗ − zI) be the Since T ∈ Λ(A), ˙ Then the following lemma takes place. defect subspaces of A. Lemma 8.1.1. If z is a regular point of the operator T ∗ , then (T × − zI)−1 Φ = Nz .

(8.2)

˙ we have Proof. For an arbitrary g ∈ Dom(A) ((A˙ − z¯I)g, (T × − zI)−1 f ) = (g, f ). Hence for f = (1/2i)(T − T × )h, h ∈ Dom(T ) and the fact that T g = T ∗ g, we have (T − T × ) (A˙ − z¯I)g, (T × − zI)−1 f = g, h = 0. (8.3) 2i ˙ If φz ∈ Nz , then for a g ∈ Dom(A), × (T − zI)φz , g = (φz , (T − z¯I)g) = (φz , (A˙ − z¯I)g) = 0. Relation (8.3) and the inequality c(z)f ≤ (T ∗ − zI)f HT ≤ d(z)f ,

c(z) > 0, d(z) > 0

completes the proof.

According to Remark 4.3.4 our assumptions on operator A˙ imply that there exists a self-adjoint extension A of A˙ in H such that (see (4.52)) Dom(T ) + Dom(A) = Dom(A˙ ∗ ).

(8.4)

Following the procedure from Section 2.2 we construct an operator-generated rig˙ As usual we denote by H+ the Hilbert space Dom(A˙ ∗ ) ging using the operator A. equipped by the inner product (u, v)+ = (u, v) + (A˙ ∗ u, A˙ ∗ v).

8.1. Auxiliary canonical system

241

Then, according to (2.9), a (+)-orthogonal decomposition holds: ˙ ⊕ Ni ⊕ N−i . H+ = Dom(A) By the von Neumann formula (1.15) ˙ ⊕ (I + U)Ni , A = A˙ ∗ Dom(A), Dom(A) = Dom(A) where U is an isometry from Ni onto N−i . Since A satisﬁes condition (8.4), the direct decomposition ˙ + U)Ni H+ = HT +(I (8.5) holds. Let HT ∗ = Dom(T ∗ ), (u, v)T ∗ = (u, v) + (T ∗ u, T ∗ ), u, v ∈ Dom(T ∗ ) and let HT ∗ ⊂ H ⊂ HT ∗ be the corresponding rigging. The operator T : H → HT ∗ is deﬁned as the adjoint to T ∗ : (T ∗ v, f ) = (v, (T ∗ )× f ) = (v, T f ). It is easy to see that T ⊂ T . Note that the subspace in HT ∗ , Φ∗ := Ran(T − T ∗ ) ( the closure is taken in HT ∗ ),

(8.6)

satisﬁes the condition Φ∗ ∩ H = {0}. The Hilbert spaces HT and HT ∗ are subspaces of the Hilbert space H+ and one can easily show that if (8.4) holds, then Dom(T ∗ ) + Dom(A) = Dom(A˙ ∗ ) and by (4.54) ˙ + U)Ni . H+ = HT ∗ +(I (8.7) We denote by P and P∗ the skew projections in H+ onto HT and HT ∗ with respect to the decompositions (8.5) and (8.7), respectively. Let H+ ⊂ H ⊂ H− be the rigged Hilbert triplet and let P × : HT → H− and P∗× : HT ∗ → H− be the adjoint operators. These operators are deﬁned by the relations

and

(Pu, ϕ) = (u, P × ϕ), u ∈ H+ , ϕ ∈ HT

(8.8)

(P∗ v, ψ) = (v, P∗× ψ), u ∈ H+ , ψ ∈ HT ∗ .

(8.9)

Observe that in view of (8.5) and (8.7) the operators P and P∗ are isomorphisms from HT ∗ onto HT and HT onto HT ∗ , respectively. Moreover the relations PP∗ f = f, f ∈ HT , P∗ Pg = g, g ∈ HT ∗ hold, and

P × Φ = P∗× Φ∗ = Ψ,

(8.10)

242

Chapter 8. Normalized L-Systems

where Ψ := {ϕ ∈ H− : (ϕ, h) = 0 for all h ∈ Dom(A)} . It is easy to see that if A˙ has ﬁnite deﬁciency indices (n, n), then Φ is an ndimensional subspace. In this case, by direct check one obtains that ker(P × Φ) = {0}, ker(P∗× Φ∗ ) = {0}, and Ψ = P × Φ is n-dimensional as well. Theorem 8.1.2. Let the operator AP : H+ → H− be deﬁned by the formula AP = A˙ ∗ + P × (T × − A˙ ∗ )(I − P).

(8.11)

Then its adjoint A∗P : H+ → H− acts by the rule A∗P = A˙ ∗ + P∗× (T − A˙ ∗ )(I − P∗ ).

(8.12)

Moreover, T ⊂ AP , T ∗ ⊂ A∗P and the following relations hold: AP = A˙ ∗ + P∗× (T − A˙ ∗ ), AP − A∗P = P × (T − T × )P,

A∗P = A˙ ∗ + P × (T × − A˙ ∗ ), AP − A∗P = P∗× (T − T ∗ )P∗ ,

(AP − zI)−1 P∗× ϕ = (T − zI)−1 ϕ, (A∗P

−1

×

×

ϕ ∈ Φ∗ ,

(8.13)

−1

− z¯I) P ϕ = (T − z¯I) ϕ, ϕ ∈ Φ, (AP − zI)−1 ψ, ψ = ψ, (A∗P − z¯I)−1 ψ , ψ ∈ Ψ,

z ∈ ρ(T ).

Proof. Let AP be an operator deﬁned by (8.11). Then for f ∈ Dom(T ) = HT we get AP f = T f, i.e., T ⊂ AP . Since A˙ ∗ (I + U)Ni is a symmetric operator, we have (A˙ ∗ (I − P)u, (I − P)v) = ((I − P)u, A˙ ∗ (I − P)v), u, v ∈ H+ . Therefore for all u, v ∈ H+ , (AP u, v)−(u, AP v) = (T Pu + A˙ ∗ (I − P)u, Pv + (I − P)v) − (Pu + (I − P)u, T Pv + A˙ ∗ (I − P)v) + ((I − P)u, T Pv) − (T Pu, (I − P)v) − (A˙ ∗ (I − P)u, Pv) + (Pu, A˙ ∗ (I − P)v) = ((T − T × )Pu, Pv) = (P × (T − T × )Pu, v). Thus, AP − A∗P = P × (T − T × )P. It follows that A∗P = AP − P × (T − T × )P = A˙ ∗ + P × (T × − A˙ ∗ )(I − P) − P × (T − T × )P = A˙ ∗ + P × (T × − A˙ ∗ ). Further we show that P × (T × − A˙ ∗ )(I − P) = P∗× (T − A˙ ∗ ),

(8.14)

8.1. Auxiliary canonical system

243

holds for f ∈ HT . Let f ∈ (I + U)Ni and g ∈ H+ , then Pg − P∗ g = −(I − P)g + (I − P∗ )g ∈ (I + U)Ni . Therefore, (A˙ ∗ f, Pg − P∗ g) = (Af, Pg − P∗ g) = (f, A(Pg − P∗ g)) = (f, A˙ ∗ (Pg − P∗ g)) = (f, T Pg − T ∗ P∗ g). Now we get ((T × − A˙ ∗ )f, Pg) = ((T − A˙ ∗ )f, P∗ g). Consequently P × (T × − A˙ ∗ )f = P∗× (T − A˙ ∗ )f, f ∈ (I + U)Ni . From (8.11) and (8.14) we obtain (8.12). Let z ∈ ρ(T ∗ ) and ϕz ∈ Nz . Then (A∗P − zI)ϕz = P × (T × − A˙ ∗ )ϕz = P × (T × − zI)ϕz . Now from (8.2) we obtain the relation (A∗P − zI)−1 P × ϕ = (T × − zI)−1 ϕ for ϕ ∈ Φ. ˙ onto H+ Theorem 8.1.3. The resolvents (AP − zI)−1 , (A∗P − z¯I)−1 map H+Ψ × for all z ∈ ρ(T ). The real part Re AP = (AP + AP )/2 satisﬁes the condition : where A : is a self-adjoint extension of A˙ transversal to A. The resolvent Re AP ⊃ A, ˙ onto H+ for all z, Im z = 0. (Re AP − zI)−1 maps H+Ψ ˙ z = H+ . This decomposition and the Proof. If we let z ∈ ρ(T ∗ ), then HT ∗ +N relations (T × − zI)Nz = Φ, P∗ Nz = (I + U)Ni imply (T × − A˙ ∗ )H+ = (T × − A˙ ∗ )(I + U)Ni = Φ. −1 Let (T × − A˙ ∗ )−1 = (T × − A˙ ∗ ) (I + U)Ni . From (8.11) and (8.13) we get 1 × ∗ × × ∗ ˙ ˙ Re AP = A + P (T − A )(I − P) + (T − T )P . 2 Let : = {u ∈ H+ : Re AP u ∈ H} , Dom(A) Then : = Dom(A)

: = Re AP u, u ∈ Dom(A). : Au

1 × ∗ −1 × ˙ I − (T − A ) (T − T ) HT . 2

: +(I : is a symmetric extension of A˙ ˙ + U)Ni = H+ . Since A It follows that Dom(A) : the last equality implies the self-adjointness of A.

244

Chapter 8. Normalized L-Systems

Remark 8.1.4. The projections P and P∗ coincide with projections PT A and PT ∗ A deﬁned in Theorem 4.4.3. Moreover, it follows from Theorems 8.1.2 and 8.1.3 that the (∗)-extension AP in (8.11) and its adjoint A∗P in (8.12) coincide with operators A and its adjoint A∗ given by (4.60). Deﬁnition 8.1.5. The collection T Θ0 = HT ⊂ H ⊂ HT

K0

J E

(8.15)

is called the auxiliary rigged canonical system if E is a ﬁnite-dimensional Hilbert space, J is an operator in E such that J = J ∗ = J −1 , K0 is a bounded linear operator from E into HT such that Ker K0 = {0}, and the identity holds: T −T × = K0 JK0× . Here K0× : HT → E is the adjoint to K0 operator. ˙ and let Θ0 be an auxiliary rigged canonical Assume that operator T ∈ Λ(A) system. Suppose that A is a self-adjoint extension of A˙ satisfying the condition (8.4). If P is a corresponding projection determined by (8.5), then AP P × K0 J ΘP = (8.16) H+ ⊂ H ⊂ H− E is the L-system with the state-space operator AP , since by Theorem 8.1.2 we have AP − A∗P = P × (T − T × )P = P × K0 JK0× P. The operator-valued function WΘP (z) = I − 2iK0×P(AP − zI)−1 P × K0 J, is a transfer function of the system ΘP . Observe that WΘ∗ P (z) = I + 2iJK0× P(A∗P − z¯I)−1 P × K0 . It follows from Theorem 8.1.2 that WΘ∗ P (z) = I + 2iJK0× P(T × − z¯I)−1 K0 .

(8.17)

Remark 8.1.6. In a similar manner one can introduce an auxiliary system based upon operator T ∗ as T∗ K∗,0 −J Θ∗,0 = , (8.18) HT ∗ ⊂ H ⊂ HT ∗ E where K∗,0 is a bounded linear operator from E into HT ∗ with ker(K∗,0 ) = {0} × × and such that T ∗ − T = K∗,0 JK∗,0 . Here K∗,0 : HT ∗ → E is adjoint to the K∗,0 operator. If P∗ is the corresponding projection determined by (8.7), then AP∗ P∗× K∗,0 −J ΘP∗ = H+ ⊂ H ⊂ H− E is an L-system similar to ΘP in (8.16).

8.2. Constant J-unitary factor

245

8.2 Constant J-unitary factor In this section we study the change of a given L-system when its transfer function is multiplied by a constant J-unitary factor. Theorem 8.2.1. Let Θ0 be an auxiliary rigged canonical system with the state space operator T and let A1 and A2 be two self-adjoint extensions of A˙ satisfying the condition of transversality (8.4). If P1 and P2 are two corresponding to A1 and A2 skew projections deﬁned by decomposition (8.5) and ΘP1 , ΘP2 are two L-systems of the form (8.16) with the state-space operators AP1 and AP2 , then for all z ∈ ρ(T ) the identity WΘP2 (z) = WΘP1 (z)B holds, where B is a J-unitary operator acting in the Hilbert space E. ˙ be the orthogonal complement of Dom(A) ˙ in the Proof. Let MT = HT Dom(A) × Hilbert space HT . Then K0 f = 0 for all f ∈ MT \ {0}. Let us write −1 ×[−1] K0 = K0× MT . (8.19) For all f ∈ MT we have from (8.17) WΘ∗ Pm (z)JK0× f = JK0× f + Pm (T × − z¯I)−1 (T − T × )f = JK0× Pm (T ∗ − z¯I)−1 (T − z¯I)f,

(m = 1, 2).

From (8.10) we obtain ×[−1]

(T ∗ − z¯I)−1 (T − z¯I)f = P∗1 K0 and

JWΘ∗ P1 (z)JK0× f,

×[−1]

WΘ∗ P2 (z)JK0× f = JK0× P2 P∗1 K0

Set

JWΘ∗ P1 (z)JK0× f.

×[−1]

B = JK0× P2 P∗1 K0

J.

(8.20)

The operator B is obviously a bounded operator in E (dim E < ∞). Thus, WΘ∗ P2 (z) = BWΘ∗ P1 (z). If h ∈ E, then using Theorem 8.1.2 we get ×[−1]

×[−1]

(JBh, Bh)E = (K0× P2 P∗1 K0 Jh, JK0× P2 P∗1 K0 ×[−1] ×[−1] = (T − T × )P2 P∗1 K0 Jh, P2 P∗1 K0 Jh ×[−1] ×[−1] = (AP2 − A∗P2 )P∗1 K0 Jh, P∗1 K0 Jh ×[−1]

×[−1]

Jh)E

×[−1]

×[−1]

= (P∗1 K0 Jh, T ∗ P∗1 K0 Jh) − (T ∗ P∗1 K0 Jh, P∗1 K0 Jh) ×[−1] ×[−1] = (T − T ∗ )P∗1 K0 Jh, P∗1 K0 Jh ×[−1] ×[−1] ×[−1] ×[−1] = (AP1 − A∗P1 )K0 Jh, K0 Jh = (K0 JK0× K0 Jh, K0 Jh) = (Jh, h)E .

246

Chapter 8. Normalized L-Systems

This implies the identity B ∗ JB = J. Since Ran(B) = E, the operator B is a Junitary operator in E. The operator B = B ∗ is J-unitary as well and WΘP2 (z) = WΘP1 (z)B for all z ∈ ρ(T ). Notice that it follows from (6.45) and (8.13) that ¯ −1 K0 , z, ζ ∈ ρ(T ). WΘP (z)JWΘ∗ P (ζ) − J = 2i(ζ¯ − z)K0∗ (T − zI)−1 (T × − ζ) Hence,

WΘP1 (z)JWΘ∗ P1 (ζ) = WΘP2 (z)JWΘ∗ P2 (ζ).

If ρ(T ) contains points ζ0 and ζ¯0 , then applying relation (6.46) yields WΘP2 (z) = WΘP1 (z)JWΘ∗ P1 (ζ0 )WΘ∗−1 (ζ0 )J P 2

= WΘP1 (z)JWΘ∗ P1 (ζ0 )JWΘP2 (ζ¯0 ). Put

B = JWΘ∗ P1 (ζ0 )JWΘP2 (ζ¯0 ).

Then WΘP2 (z) = WΘP1 (z)B for all z ∈ ρ(T ). Since WΘP1 (ζ0 )BJB ∗ WΘ∗ P1 (ζ0 ) = WΘP1 (ζ0 )JWΘ∗ P1 (ζ0 ) and WΘP1 (ζ0 ) has bounded inverse, we get that B is J-unitary in E. Now we will establish an important inverse result. Theorem 8.2.2. Let E be a ﬁnite-dimensional Hilbert space and let J be a selfadjoint and unitary operator in E. If the operator-valued function W (z) in E belongs to the class Ω0 (R, J) and B is a J-unitary operator in E, then the function WB (z) = W (z)B also belongs to the class Ω0 (R, J). Proof. According to the deﬁnition of the class Ω0 (R, J) (see Section 7.2) and Theorem 7.2.2 there exists an L-system A K J Θ= (8.21) H+ ⊂ H ⊂ H− E whose transfer operator-valued function coincides with W (z). This system can be used to generate the auxiliary system T K0 J Θ0 = (8.22) HT ⊂ H ⊂ HT E in the following way. Let us set HT = Dom(T ) where T is taken from the system Θ in (8.21). Let also P be a skew projector determined by the decomposition (8.5).

8.2. Constant J-unitary factor

247

In this case the subspace Φ deﬁned in (8.1) is ﬁnite-dimensional and hence, as we mentioned it in the previous section, the projection operator P × Φ is invertible, i.e., (P × Φ)−1 exists.Thus we can deﬁne operator K0 ∈ [E, HT ] by the formula K0 = (P × Φ)−1 K,

(8.23)

where K is an operator from the system Θ in (8.21). Consequently, K = P × K0 , A = AP of the form (8.11), and our L-system Θ in (8.21) and auxiliary system Θ0 in (8.22) can be related as in (8.16). One can also show that if K0× ∈ [HT , E], then K0× = K ∗ HT . Observe that ˙ = H+ ˙ Ker (A − A∗ ) Dom(A) HT + and P is the projector on HT with respect to this decomposition. Let B be a J-unitary operator in E and let ×[−1] ∗−1 × Dom(A1 ) = I − K0 B K0 P Dom(T ∗ ), A1 = A˙ ∗ Dom(A1 ), (8.24) ×[−1] where K0 is deﬁned by (8.19) and A˙ is from the system Θ in (8.21). Let us ×[−1] ∗−1 × show that A1 is symmetric. For u = g − K0 B K0 Pg, g ∈ Dom(T ∗ ) we have ×[−1] ∗−1 × ×[−1] ∗−1 × (A˙ ∗ u, u) − (u, A˙ ∗ u) = (T ∗ g − T K0 B K0 Pg, g − K0 B K0 Pg) ×[−1]

×[−1]

− (g − K0 B ∗−1 K0× Pg, T ∗g − T K0 B ∗−1 K0× Pg) = ((A∗ − A)g, g) ×[−1] ∗−1 × ×[−1] ∗−1 × + (A − A∗ )K0 B K0 Pg, K0 B K0 Pg = −2i(P ×K0 JK0× Pg, g) ×[−1]

+ 2i(P × K0 JK0× K0

×[−1]

B ∗−1 K0× Pg, K0

B ∗−1 K0× Pg)

= −2i(P ×K0 JK0× Pg, g) + 2i(P × K0 B −1 JB ∗−1 K0× Pg, g) = 0. Thus, A1 is a symmetric operator. Let dim E = n, then dim Ran(K) = ∗ ˙ dim Ran(K 0 ) = dim Ran(A − A ) = n. Also the defect numbers of A are (n, n) ˙ = n. Therefore, A1 is a self-adjoint extension of A. ˙ and dim Dom(A1 )/Dom(A) ˙ Thus Moreover, Dom(A1 ) ∩ Dom(T ∗ ) = Dom(A). Dom(T ) + Dom(A1 ) = Dom(T ∗ ) + Dom(A1 ) = Dom(A˙ ∗ ) = H+ . This means that A1 is transversal to T . Let P1 and P∗1 be the corresponding skew projections onto Dom(T ) and Dom(T ∗ ) according to (8.5) and (8.7) written for A1 , respectively. From (8.24) we obtain the equality K0× P1 g = B ∗−1 K0× Pg for all g ∈ Dom(T ∗ ). Therefore, using (8.10) we get K0× P1 P∗ f = B ∗−1 K0× f

for all f ∈ Dom(T ).

248

Chapter 8. Normalized L-Systems

We deﬁne operator AP1 by means of P1 and via formula (8.11). If K1 = P1× K0 , then the L-system AP 1 K1 J Θ1 = H+ ⊂ H ⊂ H− E has the transfer function WΘ1 (z) = I − 2iK1× (AP1 − zI)−1 K1 J and WΘ∗ 1 (z) = I + 2iJK0×P1 (T × − zI)−1 K0 . Using the proof of Theorem 8.2.1 we get the equality WΘ1 (z) = W (z)B.

The same result as in Theorem 8.2.2 holds for BW (z). The next theorem follows from Theorem 8.2.2. Theorem 8.2.3. Let Θ be an L-system of the form (6.31) deﬁned by (T − zI)x = KJϕ− , ϕ+ = ϕ− − 2iK ∗ x,

(8.25)

with the transfer function WΘ (z). If B is a J-unitary operator in the input-output space E of Θ, then the function WΘB (z) = WΘ (z)B is the transfer function of the L-system ΘB of the form (T − zI)x = KB Jϕ− , (8.26) ∗ ϕ+ = ϕ− − 2iKB x, where

KB = P1× (P × Φ)−1 K = ((K ∗ MT )−1 JBJ(K ∗ MT )−1 P∗−1 )× (P × Φ)−1 K.

(8.27)

Here the operators P, P∗ , P × , P1 , and the set Φ are deﬁned in the proof of Theorem 8.2.2 by relations (8.5), (8.7), (8.8), (8.9), and (8.1), respectively. Proof. In order to prove the theorem we follow the main steps of the proof of Theorem 8.2.2. First we build an auxiliary system Θ0 based on the L-system Θ in (8.25) and then construct the projection P1 . Using the same operator T as in (8.25) we form a new L-system (8.26) as prescribed in the proof of Theorem 8.2.2. This L-system has a new channel operator KB . In order to obtain representation (8.27) of KB we utilize the formula (8.20). Theorem 8.2.3 above shows that if one multiplies the transfer function of a given L-system by a J-unitary constant factor, then the result is a transfer function of another L-system with the same operator T and a new operator KB determined by (8.27). Now let us consider an L-system A K J Θ= , H+ ⊂ H ⊂ H− E

8.2. Constant J-unitary factor

249

˙ invertible operator K, and of the form (8.16) with operator T of the class Λ(A), ˙ = H. From Section 8.1 we recall that for a system of the form (8.16) Dom(A) there is a skew projection operator P determined by a decomposition ˙ + U)Ni H+ = Dom(T )+(I that maps H+ onto Dom(T ). Therefore we can follow the argument of the proof of Theorem 8.2.2 to obtain the auxiliary system (8.22) with the invertible channel operator K0 deﬁned by (8.23) such that K = P × K0 . After that we form the system ΘP of the form (8.16) with operator AP deﬁned by (8.11). Since Im AP = P × K0 JK0× P = KJK ∗ = Im A, we can apply the uniqueness Theorem 4.3.9 to conclude that A = AP . Similarly, we can use the reasoning in Remark 8.1.6 and represent the elements of Θ in terms of the skew projector P∗ deﬁned by (8.7). This yields A∗ = AP∗ , K = P∗× K∗,0 , where K∗,0 is a bounded invertible linear operator from E into HT ∗ deﬁned in (8.18). Theorem 8.2.4. Let T ∈ Λ, (−1) ∈ ρ(T ), A be a (∗)-extension of T . Let S be the fractional-linear transformation of T given by S = (I − T )(I + T )−1 . If

Θ=

A H+ ⊂ H ⊂ H−

K

is an L-system with ker(K) = {0}, then √ S 2(I + A)−1 K Θ = H

(8.28) J E

−J E

(8.29)

is an L-system as well and the transfer functions WΘ and WΘ are connected by the relation 1−z WΘ (z) = WΘ (−1)WΘ , z ∈ ρ(T ), z = −1. (8.30) 1+z Proof. It is easy to see that 2(I +T )−1 −I = (I −T )(I +T )−1 . Taking into account (8.28) we get S = 2(I + T )−1 − I, and (S − λI)

−1

−1 1 1−λ = − (I + T ) T − I , λ 1+λ

implying that Im S = 2Im (I + T )−1 = 2Im (I + A)−1 = −2(I + A)−1 Im A(I + A∗ )−1 = −2(I + A)−1 KJK ∗ (I + A∗ )−1 .

250

Chapter 8. Normalized L-Systems

˜ ˜ ∗ , where Hence, Im S = K(−J) K ˜ = K

√

2(I + A)−1 K,

˜ ∈ [E, H]. K

The transfer function WΘ is of the form WΘ (λ) = I + 4iK ∗ (I + A∗ )−1 (S − λI)−1 (I + A)−1 KJ = WΘ (λ) = I − 2i(z + 1)K ∗ (I + A∗ )−1 (A − zI)−1 KJ, Let z= Then λ=

λ ∈ ρ(S).

1−λ . 1+λ

1−z 1 , = z + 1. 1+z 1+λ

For z ∈ ρ(T ) one has (z + 1)(I + A∗ )−1 (A − zI)−1 = 2i((I + A∗ )−1 KJK ∗ (A − zI)−1 + (A − zI)−1 − (I + A∗ )−1 . Further JWΘ∗ (−1)JW + 2iK ∗ (I + A∗ )−1 KJ)(I − 2iK ∗ (A − zI)−1 KJ) Θ (z) = (I ∗ −1 ∗ −1 = I − 2iK (A − zI) − (I + A∗ )−1 + 2i(I + A∗ )−1 KJ KJK (A − zI) 1−z = I − 2i(z + 1)K ∗ (I + A∗ )−1 (A − zI)−1 KJ = WΘ . 1+z Since WΘ−1 (−1) = JWΘ∗ (−1)J, we get (8.37).

Note that since (−1) ∈ ρ(T ) then by properties (6.46) the operator WΘT (−1) is J-unitary.

8.3 The Donoghue transform and impedance functions of scattering L-systems Let operator-function V (z) act on a ﬁnite-dimensional Hilbert space E. If α ∈ [E, E] is a self-adjoint operator, then the formula [V (z)]α = e−iα [cos α + (sin α)V (z)][sin α − (cos α)V (z)]−1 eiα , deﬁnes the Donoghue transform of the operator-function V (z).

(8.31)

8.3. The Donoghue transform

251

Theorem 8.3.1. Let the scattering L-systems Θ and Θα with transfer functions WΘ (z) and WΘα (z) be such that WΘα (z) = WΘ (z)(−e2iα ),

(8.32)

where α ∈ [E, E] is a self-adjoint operator. Then the impedance functions VΘα (z) and VΘ (z) are connected by the Donoghue transform (8.31). Proof. Formula (8.32) implies −WΘα (z)e−2iα = WΘ (z) = [I − i VΘ (z)][I + i VΘ (z)]−1 , that is equivalent to (eiα WΘα (z)e−iα ) · e−iα [−I − i VΘ (z)] = eiα [I − i VΘ (z)]. Dividing both sides by i yields (eiα WΘα (z)e−iα )[ie−iα − e−iα VΘ (z)] = (−i)eiα − eiα VΘ (z). Let W = eiα WΘα (z)e−iα . Then we apply Euler’s formula and get W[i(cos α − i sin α) − (cos α − i sin α) VΘ (z)] = (−i)(cos α + i sin α) − (cos α + i sin α) VΘ (z). Pulling out i and regrouping yields W[(sin α + i cos α) − (cos α − i sin α) VΘ (z)] = (sin α − i cos α) − (cos α + i sin α) VΘ (z). Now we multiply both sides from the right by (sin α − cos α VΘ (z))−1 and get W + iW(cos α + sin α VΘ (z))(sin α − cos α VΘ (z))−1 = I − i(cos α + sin α VΘ (z))(sin α − cos α VΘ (z))−1 . Let [VΘ (z)]α be the Donoghue transform (8.31) of the function VΘ (z). Then W + iW[VΘ (z)]α = I − i[VΘ (z)]α . The latter gives us W = eiα WΘα (z)e−iα = [I − i[VΘ (z)]α ][I + i[VΘ (z)]α ]−1 , or

eiα WΘα (z)e−iα [I + i[VΘ (z)]α ] = I − i[VΘ (z)]α .

Distributing and multiplying both sides from the left by e−iα and from the right by eiα we obtain WΘα (z) + iWΘα (z)e−iα [VΘ (z)]α eiα = I − ie−iα [VΘ (z)]α eiα .

252

Chapter 8. Normalized L-Systems

Since WΘα (z) = [I − iVΘα (z)][I + iVΘα (z)]−1 is equivalent to WΘα (z) + iWΘα (z)VΘα (z) = I − iVΘα (z), the last statement implies that [VΘ (z)]α = (cos α + sin α VΘ (z))(sin α − cos α VΘ (z))−1 = VΘα (z),

which completes the proof.

Theorem 8.3.2. If a function V (z) belongs to the class N0 (R), then its Donoghue transform [V (z)]α also belongs to N0 (R). Moreover, both V (z) and [V (z)]α can be realized as impedance functions of minimal scattering systems Θ and Θα with the same operator T . Proof. Since V (z) ∈ N0 (R), then it can be realized as the impedance function of a minimal scattering system Θ with the transfer function WΘ (z) and properties described in Theorem 7.1.5. Consider the function Wα (z) = [I − i[V (z)]α ][I + i[V (z)]α ]−1 . Reversing the argument of the proof of Theorem 8.3.1 we obtain that Wα (z) = WΘ (z)(−e2iα ). Taking into account that for a self-adjoint operator α ∈ [E, E], the operator B = (−e2iα ) is unitary, we apply Theorem 8.2.3 utilizing the last formula. This theorem gives us the existence and description of the L-system Θα = ΘB with the desired properties. Since both L-systems Θ and Θα share not only operator T but ˙ then by Theorem 7.1.4 we have that also densely-deﬁned symmetric operator A, [V (z)]α = VΘα belongs to the class N0 (R).

8.4 Normalized (∗)-extensions and normalized L-systems ˙ where A˙ is a densely-deﬁned closed symmetric Theorem 8.4.1. Let T ∈ Λ(A), operator. Suppose 0 ∈ ρ(T ). Then there is a unique (∗)-extension A0 of T such that Re A0 is a t-self-adjoint bi-extension of A˙ whose quasi-kernel is the operator −1 Aˆ0 = Re (T −1 ) . Proof. The operator Re (T −1 ) is invertible since the equality Re (T −1 )f = 0 yields 1 1 A˙ ∗ Re (T −1 )f = T T −1f + T ∗ (T −1 )∗ f = f = 0. 2 2 Since the operator Re (T −1 ) is a bounded self-adjoint extension of non-denselydeﬁned bounded symmetric operator A˙ −1 , the operator −1 Aˆ0 = Re (T −1 )

8.4. Normalized (∗)-extensions and normalized L-systems

253

˙ Hence, 0 ∈ ρ(Aˆ0 ). Because 0 ∈ is a self-adjoint extension of the operator A. ∗ ˆ ρ(T ) ∩ ρ(T ) ∩ ρ(A0 ), the direct decompositions ˙ 0, H+ = Dom(T ∗ )+N

˙ 0, H+ = Dom(T )+N

˙ 0 H+ = Dom(Aˆ0 )+N

hold, where N0 = ker(A˙ ∗ ) and H+ = Dom(A˙ ∗ ). Let us deﬁne one more self-adjoint extension A˜0 of A˙ via ˙ +N ˙ 0, Dom(A˜0 ) = Dom(A) Then, clearly,

A˜0 = A˙ ∗ Dom(A˜0 ).

Dom(A˙ ∗ ) = Dom(Aˆ0 ) + Dom(A˜0 ).

˜ admits the von Neumann representation The domain Dom(A) ˙ ⊕ (I − U0 )Ni , Dom(A˜0 ) = Dom(A) where U0 is an isometry from Ni onto N−i . Thus, ˙ − U0 )Ni , H+ = Dom(T )+(I ∗ ˙ H+ = Dom(T )+(I − U0 )Ni , ˙ − U0 )Ni . H+ = Dom(Aˆ0 )+(I

(8.33) (8.34) (8.35)

Let PT A˜0 , PT ∗ A˜0 , and PAˆ0 A˜0 be skew projections in H+ onto Dom(T ), Dom(T ∗ ), and Dom(Aˆ0 ), corresponding to decompositions (8.33), (8.34), (8.35), respectively. Now deﬁne a (∗)-extension of T generated by A˜0 (see Theorem 4.4.3 and relations (4.60)) by the formula A0 = A˙ ∗ − R−1 A˙ ∗ (I − PT A˜0 ). Then A0 is a (∗)-extension of T . The adjoint A∗0 is given by A∗0 = A˙ ∗ − R−1 A˙ ∗ (I − PT ∗ A˜0 ). Let us show that the quasi-kernel of Re A0 coincides with the operator Aˆ0 . Observe that ker(Im A0 ) = Dom(A˜0 ) (see Theorem 4.4.3). Clearly, ˙ ker(Im (T −1 )) = Ran(A). Hence, Ran(Im (T −1 )) = N0 . It follows that for all f ∈ H one has (Im A0 )(T −1 − T ∗−1 )f = 0. Further, for f ∈ H we have ∗−1 ∗ ∗ (A∗0 T0−1 + A0 T ∗−1 )f = (A0 − (A0 − A∗0 ))A−1 f 0 f + (A0 + (A0 − A0 )A0

= 2f + 2i(Im A0 )(T ∗−1 − T −1 )f = 2f, (Re A0 )(Re (T −1 ))f = (Re A0 )Re (A−1 0 )f = = f.

1 2I + (A∗0 T0−1 + A0 T ∗−1 f 4

254

Chapter 8. Normalized L-Systems

Thus, (Re A0 )−1 f = (Re (T −1 ))f = Aˆ−1 0 f, i.e., the quasi-kernel of Re A0 is the operator Aˆ0 . Observe that by Theorems 3.4.9 and 4.4.4 the operator Re A0 is given by Re A0 = A˙ ∗ − R−1 A˙ ∗ (I − PAˆ0 A˜0 ). Uniqueness of (∗)-extension A0 of T with quasi-kernel Aˆ0 follows from Theorem 4.4.6. ˙ (0 ∈ ρ(T )) with quasi-kernel A (∗)-extension A0 of an operator T ∈ Λ(A) −1 −1 Re (T ) is called normalized at point zero. Similarly one can construct a (∗)-extension Aλ0 normalized at a real point λ0 of an operator T ∈ Λ having a real regular point λ0 = 0. Namely, such a (∗)-extension is generated via Theorem 4.4.3 and formulas (4.60) by a self-adjoint extension A˜λ0 of A˙ of the form

˙ +N ˙ λ0 . Dom(A˜λ0 ) = Dom(A) −1 The quasi-kernel of Re Aλ0 coincides with Aˆλ0 = Re ((T − λ0 I)−1 ) . Proposition 8.4.2. Let T ∈ Λ, 0 ∈ ρ(T ), and let A be a (∗)-extension of T . Let also A K J Θ= H+ ⊂ H ⊂ H− E be an L-system. Then WΘ (0) = I if and only if A is normalized at point zero. Proof. We have WΘ (0) = I − 2iK ∗ A−1 KJ. Then WΘ (0) = I ⇐⇒ K ∗ A−1 KJ = 0. Since A−1 KE = N0 , we get WΘ (0) = I ⇐⇒ K ∗ N0 = 0. ˜ where a self-adjoint extension Now the relation ker(K ∗ ) = ker(Im A) = Dom(A), ˜ A generates A, yields ˜ = Dom(A) ˙ +N ˙ 0. K ∗ N0 = 0 ⇐⇒ Dom(A) Therefore, WΘ (0) = I if and only if A = A0 , where A0 is normalized at point zero. ˙ 0 ∈ ρ(T ), A˙ be a densely-deﬁned closed symmetric Deﬁnition 8.4.3. Let T ∈ Λ(A), operator with deﬁciency numbers (n, n), and let A0 be a normalized at point zero (∗)-extension of T . An L-system A0 K J Θ0 = , H+ ⊂ H ⊂ H− Cn with ker(K) = {0} is called a normalized at point zero L-system .

8.4. Normalized (∗)-extensions and normalized L-systems

255

Proposition 8.4.2 shows that an L-system Θ is normalized at point zero if and only if WΘ (0) = I. Since the quasi-kernel Aˆ0 of the real part Re A0 has bounded inverse, the impedance function VΘ0 (z) = K ∗ (Re A0 − zI)−1 K of a normalized at zero L-system is holomorphic at zero. Moreover, VΘ0 (0) = K ∗ (Re A0 )−1 K = 0. Similarly, a normalized at real point λ0 L-system is characterized by the condition WΘ (λ0 ) = I and is given by Aλ0 K J Θλ0 = , H+ ⊂ H ⊂ H− Cn where Aλ0 is a (∗)-extension normalized at λ0 . In addition VΘλ0 is holomorphic at λ0 and VΘλ0 (λ0 ) = 0. ˙ 0 ∈ ρ(T ) and A˙ be a densely-deﬁned closed symTheorem 8.4.4. Let T ∈ Λ(A), metric operator. Let A be a (∗)-extension of T . If A K J Θ= , H+ ⊂ H ⊂ H− E is an L-system with ker(K) = {0}, then −1 T A∗−1 K Θ = H

−J E

,

(8.36)

is an L-system as well and the transfer functions WΘ and WΘ are connected by the relation 1 WΘ (z) = WΘ WΘ (0), z ∈ ρ(T ) \ {0}. (8.37) z 1 Thus, if Θ0 is normalized at zero, then WΘ0 (z) = WΘ0 z , z ∈ ρ(T ) \ {0}. Proof. The relation A − A∗ = 2iKJK ∗ implies T −1 − T ∗−1 = A−1 − A∗−1 = −2iA∗−1 KJK ∗ A−1 . Hence Θ of the form (8.36) is an L-system. The transfer function WΘ (z) is of the form WΘ (z) = I + 2iK ∗A−1 (T −1 − zI)−1 A∗−1 KJ −1 2i 1 = I − K∗ A − I A∗−1 KJ. z z For z ∈ ρ(T ) one has z(A − zI)−1 A∗−1 = (A − zI)−1 − A∗−1 + 2i(A − zI)−1 KJK ∗ A∗−1 . Further WΘ (z)JWΘ∗ (0)J = (I − 2iK ∗ (A − zI)−1 KJ)(I + 2iK ∗ A∗−1 KJ) = I − 2iK ∗ (A − zI)−1 − A∗−1 + 2i(A − zI)−1 KJK ∗ A∗−1 KJ 1 = I − 2izK ∗(A − zI)−1 A∗−1 KJ = WΘ . z

256

Chapter 8. Normalized L-Systems

Since WΘ−1 (0) = JWΘ∗ (0)J, we get (8.37). If Θ0 is a normalized at zero L-system, then by Proposition 8.4.2 we get for the transfer function of the corresponding L-system Θ0 the relation WΘ0 z1 = WΘ0 (z), z ∈ ρ(T ), z = 0. One can easily see that if −1 T Θ = H

K

is an L-system with ker(K) = {0}, then A Θ= H+ ⊂ H ⊂ H−

J , E

A∗−1 K

−J , E

is an L-system too. Similarly, if Aλ0 is (∗)-extension normalized at real point λ0 and Aλ0 K J Θλ0 = , H+ ⊂ H ⊂ H− E is an L-system, then transfer function WΘλ of the system 0

Θλ0

(T − λ0 I)−1 = H

(A∗ − λ0 I)−1 K

is connected with WΘ0 by the relation 1 WΘλ = WΘλ0 (z), 0 z − λ0

−J E

,

z ∈ ρ(T ) \ {λ0 }.

8.5 Realizations of eizl and eil/z as transfer functions of L-systems Example. We consider the construction that we have used in Example 7.6. Let Tx =

1 dx , i dt

, , Dom(T ) = x(t) , x(t) − abs. cont., x (t) ∈ L2[0,l] , x(0) = 0 ,

˙ A, and K be given by formulas (7.61), (7.63), and (7.64), respecand operators A, tively. Let also W21 ⊂ L2[0,l] ⊂ (W21 )− , be the rigged Hilbert space constructed in Example 7.6. With these elements we form a normalized at point zero L-system with J = −1, ⎛ ⎞ A K −1 ⎠, Θ=⎝ (8.38) 1 2 1 W2 ⊂ L[0,l] ⊂ (W2 )− C

8.5. Realizations of eizl and eil/z as transfer functions of L-systems

257

As it was shown in Example 7.6 the transfer function of system (8.38) is WΘ (z) = I − 2iK ∗ (A − zI)−1 KJ = eizl .

(8.39)

Now let us construct L-system Θ of the form (8.36). It is easy to see that T

−1

g=i

t

g(u) du,

T

−1∗

0

and hence

g = −i

l

g(u) du, t

T −1 − T −1∗ g= 2i

l

g(u) du = (g, h)h, 0

√ where h ≡ −i/ 2. Deﬁning K via the formula K ∗ x = (x, h), we form Θ =

T −1 L2[0,l]

Using direct computations we obtain i √i t (T −1 − zI)−1 h = e 2z , z

K

1 . C

i (T −1 − zI)−1 h, h = 2i[e z l − 1],

and i WΘ (z) = I − 2iK ∗ (T −1 − zI)−1 K J = 1 − 2i (T −1 − zI)−1 h, h = e z l . Example. In this example we will show a simple illustration of a normalized at point ξ L-system. Based on an operator T from the previous example we introduce a new operator 1 dx T (ξ) x = − ξx, i dt with Dom(T (ξ) ) = Dom(T ) and any real number ξ. Following the steps of Example 8.5 we obtain 1 dx , i dt , , Dom(T (ξ)∗ ) = x(t) , x(t) − abs. cont., x (t) ∈ L2[0,l] , x(l) = 0 . T (ξ)∗ x − ξx =

Also a symmetric operator A˙ (ξ) is given by 1 dx A˙ (ξ) x = − ξx, i dt , , Dom(A˙ (ξ) ) = x(t) , x(t) − abs. cont., x (t) ∈ L2[0,l] , x(0) = x(l) = 0 ,

258

Chapter 8. Normalized L-Systems

and , 1 dx , A˙ (ξ)∗ x = − ξx, Dom(A˙ (ξ)∗ ) = x(t) , x(t) − abs. cont., x (t) ∈ L2[0,l] . i dt The operator T (ξ) has inverse T (ξ)

−1

t

g = ieiξt

e−iξu g(u) du.

0

Evaluating the imaginary component of T (ξ)

−1

,

l −1∗ − T (ξ) iξt g=e e−iξu g(u) du = (g, h)h, 2i 0 √ we obtain h = ieiξt / 2. Using the rigged Hilbert space triplet T (ξ)

−1

W21 ⊂ L2[0,l] ⊂ (W21 )− , we deﬁne operators 1 dx + ix(0) eiξl δ(t − l) − δ(t) , i dt 1 dx ∗ A x= + ie−iξl x(l) eiξl δ(t − l) − δ(t) , i dt Ax =

and identify A as a (∗)-extension normalized at point ξ. Then we construct a normalized at point ξ L-system ⎛ ⎞ A K (ξ) −1 ⎠ , (J = −1), Θ(ξ) = ⎝ (8.40) W12 ⊂ L2[0,l] ⊂ (W21 )− C where 1 K (ξ)c = c · √ [eiξl δ(t − l) − δ(t)], (c ∈ C), 2 ∗ 1 iξl 1 (ξ) K x = x, √ [e δ(t − l) − δ(t)] = √ [e−iξl x(l) − x(0)], 2 2 √ ˆ of A is h ˆ = (eiξl δ(t − l) − δ(t))/ 2. Using and x(t) ∈ W21 . The channel vector h standard steps one ﬁnds ∗

WΘ(ξ) (z) = −2iK (ξ) (A − zI)−1 K (ξ) J ˆ h ˆ = ei(z−ξ)l . = 1 − 2i (A − zI)−1 h,

(8.41)

8.5. Realizations of eizl and eil/z as transfer functions of L-systems

259

Now let us construct L-system Θ(ξ) . Following the lead of Example 8.5 we have

Θ

where

(ξ)

⎛ (ξ) −1 T =⎝ L2[0,l] ∗

K (ξ) x = (x, h),

K (ξ)

1

⎞ ⎠,

C x ∈ L2[0,l] .

Using direct computations one conﬁrms that −1 il −1 1 WΘ(ξ) (z) = 1 − i T (ξ) − zI h, h = e z−ξ = WΘ(ξ) ) z−ξ Examples 8.5-8.5 can also be used to illustrate Theorems 8.2.1-8.2.3 about constant J-unitary factor. Indeed, according to formula (8.41), the transfer function WΘ(ξ) (z) of L-system Θ(ξ) in (8.40) is represented by WΘ(ξ) (z) = ei(z−ξ)l = eizl · e−iξl . By (8.39), WΘ (z) = eizl is the transfer function of L-system Θ in (8.38) while B = e−iξl (with real ξ) plays the role of the constant J-unitary factor.

Chapter 9

Canonical L-systems with Contractive and Accretive Operators In this chapter the Kre˘ın classical theorem is extended to the case of quasiself-adjoint contractive extensions of symmetric contractions, and their complete parametrization is given. On its basis we present the solution of the Phillips-Kato restricted extension problem on existence and description of all proper maximal accretive and sectorial extensions of a densely-deﬁned non-negative symmetric operator. The criterion in terms of the impedance function of an L-system for the state-space operator to be a contraction (or so-called θ-co-sectorial contraction) is obtained. We establish the conditions for a given Stieltjes and inverse Stieltjes function to be realized as an impedance function of some L-system. We also verify when the state-space operator of an L-system is maximal accretive or sectorial. The connections between the Friedrichs and Kre˘ın-von Neumann extensions and impedance function of L-systems are provided.

9.1 Contractive extensions and their block-matrix forms We will keep the following notations. By PH we always denote the orthogonal projection in a Hilbert space H onto its subspace H. By IH we denote the restriction of the identity operator I in H onto a proper subspace H of H. For a given contraction T ∈ [H1 , H2 ] the operator DT = (I − T ∗ T )1/2 ∈ [H1 , H1 ] Y. Arlinskii et al., Conservative Realizations of Herglotz–Nevanlinna Functions, Operator Theory: Advances and Applications, DOI 10.1007/978-3-7643-9996-2_9, © Springer Basel AG 2011

(9.1) 261

262 Chapter 9. Canonical L-systems with Contractive and Accretive Operators is called the defect operator of T . W write DT = Ran(DT ).

(9.2)

It is well known [243] that the defect operators satisfy the following commutation relation: T DT = DT ∗ T . If B and C are two bounded self-adjoint operators acting on H, then the notation B ≥ C means that the operator B − C is non-negative. The statement below is well known (see [13], [177]). Lemma 9.1.1. Let B be a non-negative self-adjoint operator in a Hilbert space H and let B [−1] be its Moore-Penrose inverse. Then Ran(B B and

1/2

[−1/2]

2

)= g∈H: 2

g =

sup f ∈Dom(B) 2

sup f ∈Dom(B)

lim (B − λI) λ↑0

−1

g, g =

|(f, g)| , (Bf, f )

|(f, g)| <∞ , (Bf, f ) g ∈ Ran(B

1/2

(9.3) ),

B [−1/2] g2 , g ∈ Ran(B 1/2 ), +∞, g ∈ H \ Ran(B 1/2 ).

(9.4)

Proof. Since H = ker B ⊕ Ran(B) and the Moore-Penrose inverse B [−1] is the inverse to the operator B Ran(B), we may assume that ker B = {0}. Let u ∈ Dom(B 1/2 ). Since Dom(B) is dense in Dom(B 1/2 ) (with respect to the graph norm of the operator B 1/2 ) and Dom(B 1/2 ) is dense in H, we get sup

, , ,(f, B 1/2 u),2

f ∈Dom(B)

(Bf, f )

=

, 1/2 , ,(B f, u),2

sup f ∈Dom(B)

B 1/2 f

= u2 .

Conversely, let 2

sup f ∈Dom(B)

|(f, g)| ≤ C(g), (Bf, f )

where C(g) < ∞. Then |(ϕ, g)|2 ≤ C(g)B 1/2 ϕ2

for all ϕ ∈ Dom(B 1/2 ).

Consequently, |(B −1/2 ψ, g)|2 ≤ C(g)ψ2 for all ψ ∈ Dom(B −1/2 ). By the Riesz theorem, we have g ∈ Dom(B −1/2 ) = Ran(B 1/2 ). Suppose that (B−λI)−1/2 g2 ≤ C(g) for λ < 0. Then , , 2 ,((B − λI)1/2 f, (B − λI)−1/2 g),2 (Bf, f ) − λf 2 |(f, g)| = ≤ C(g). (Bf, f ) (Bf, f ) (Bf, f )

9.1. Contractive extensions and their block-matrix forms

263

Letting λ ↑ 0 we get 2

|(f, g)| ≤ C(g) for all f ∈ Dom(B). (Bf, f ) It follows from (9.3) that g ∈ Ran(B 1/2 ). Let E(t), t ≥ 0 be the resolution of the identity for B. Then for each h ∈ Ran(B) and all λ < 0 we have 2 t1/2 − 1 d(E(t)h, h) (t − λ)1/2 0 ∞ ∞ d(E(t)h, h) d(E(t)h, h) = λ2 ≤ λ2 = λ2 B −1 h2 . 2 1/2 1/2 2 t (t − λ)((t + (t − λ) ) 0 0 ( 1/2 ( ∞ (B (B − λI)−1/2 h − h(2 =

Hence, lim B 1/2 (B − λI)−1/2 h = h, λ↑0

for all h ∈ Ran(B). Since the operator B 1/2 (B − λI)−1/2 is a contraction for λ < 0 and the linear manifold Ran(B) is dense in H , we get lim B 1/2 (B − λI)−1/2 u = u, λ↑0

for all u ∈ H. In particular, for all ϕ ∈ Dom(B 1/2 ), lim(B − λI)−1/2 B 1/2 ϕ = lim B 1/2 (B − λI)−1/2 ϕ = ϕ. λ↑0

λ↑0

Thus, relations (9.4) are proved.

The following lemma is useful for the parametrization of contractions in a block form. Lemma 9.1.2. Let H, H, M, and N be Hilbert spaces, and operators F ∈ [H, H], M ∈ [M, DF ∗ ], K ∈ [DF , N], and X ∈ [DM , DK ∗ ] be contractions. Then the operator G deﬁned by G = KF M + DK ∗ XDM ∈ [M, N],

(9.5)

satisﬁes the identity 2

2

h2 − Gh2 = DF M h + DX DM h2 + (DK F M − K ∗ XDM ) h , (9.6) for all h ∈ M. In particular, G is a contraction.

264 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Proof. From the deﬁnition of G in (9.5) one obtains h2 − Gh2 = h2 − (KF M + DK ∗ XDM ) h2

(9.7)

= h2 − KF M h2 − DK ∗ XDM h2 − 2Re (KF M h, DK ∗ XDM h) . The relation K ∗ DK ∗ = DK K ∗ gives (KF M h, DK ∗ XDM h) = (DK F M h, K ∗ XDM h) . The deﬁnition of DK ∗ shows that −KF M h2 = DK F M h2 − F M h2 , and, likewise, −DK ∗ XDM h2 = −XDM h2 + K ∗ XDM h2 . Now the right-hand side of (9.7) becomes h2 − F M h2 − XDM h2 + DK F M h2 + K ∗ XDM h2 − 2Re (DK F M h, K ∗ XDM h) = h2 − F M h2 − XDM h2 + (DK F M − K ∗ XDM ) h2 . Finally, observe that

DF M h2 = M h2 − F M h2 , DX DM h2 = h2 − M h2 − XDM h2 .

Hence the proof of (9.6) is complete.

M. Kre˘ın has introduced and studied in [172] the following operator transformation. Let H be a bounded non-negative self-adjoint operator in H and let L be a subspace in H. He has proved that the set of all bounded self-adjoint operators C in H such that 0 ≤ C ≤ H, Ran(C) ⊂ L, has a maximal element

1

1

H L = H 2 PΩ H 2 ,

(9.8)

where PΩ is the orthogonal projection in H onto the subspace 1

Ω = {f ∈ H : H 2 f ∈ L}. In addition (HL f, f ) =

inf

ϕ∈HL

((H(f + ϕ), f + ϕ)) , 1

(9.9) 1

for all f ∈ H. It follows from (9.8) that Ran((HL ) 2 ) = Ran(H 2 ) ∩ L and hence HL = 0 ⇐⇒ Ran(H 1/2 ) ∩ L = {0}.

(9.10)

9.1. Contractive extensions and their block-matrix forms

265

The operator HL is called the shorted operator. Let H and H be two Hilbert spaces. Suppose that G is a subspace of H and ˙ A : G → H is a contraction. The operator A deﬁned on H is called a contractive extension of A˙ if A ⊃ A˙ and A ≤ 1. Consider A˙ as an operator from [G, H ]. Then A˙ has the adjoint A˙ ∗ ∈ [H , G]. Let N = H G. Theorem 9.1.3. The formula ˙ G + D ˙ ∗ KPN , A = AP A

(9.11)

establishes a one-to-one correspondence between all contractive extensions A ∈ [H, H ] of A˙ and all contractions K ∈ [N, DA˙ ∗ ]. Proof. Let operator A be of the form (9.11), where K ∈ [N, DA˙ ∗ ] is a contraction. Then A∗ = A˙ ∗ + K ∗ DA˙ ∗ . It follows that for all f ∈ H , A∗ f 2 = A˙ ∗ f 2 + K ∗ DA˙ ∗ f 2 ≤ A˙ ∗ f 2 + DA˙ ∗ f 2 = f 2 . The norms above and below are understood in the sense of the corresponding Hilbert space. Thus A∗ is a contraction. Hence A is a contraction as well. Moreover ˙ A G = A. ˙ then its adjoint A∗ : H → H Conversely, if A is a contractive extension of A, is a contraction. Because A˙ ⊂ A, we get PG A∗ = A˙ ∗ . Therefore the operator A∗ takes the form A∗ = A˙ ∗ + L, where the range of the operator L is contained in N. It follows that A∗ f 2 = A˙ ∗ f 2 + Lf 2 for all f ∈ H . Since A∗ is a contraction, we obtain Lf 2 ≤ f 2 − A˙ ∗ f 2 , f ∈ H . By Theorem 2.1.2 we get L∗ = DA˙ ∗ K, where K : N → DA˙ ∗ is a contraction.

˙ G + D ˙ ∗ KPN with a contraction K ∈ [N, D ˙ ∗ ] As a consequence for A = AP A A one has from (9.6) the following relations: DA f 2 = (DA˙ PG − A˙ ∗ KPN )f 2 + DK PN f 2 , f ∈ H, DA∗ g2 = DK ∗ DA˙ ∗ g2 , g ∈ H . Because A˙ ∗ DA˙ ∗ ⊂ DA˙ , the ﬁrst relation in (9.12) gives inf DA (f + ϕ)2 = DK PN f 2

ϕ∈G

for all f ∈ H.

(9.12)

266 Chapter 9. Canonical L-systems with Contractive and Accretive Operators 2 This means that the shorted operator (DA )N of the form (9.8)-(9.9) has a property 2 2 (DA )N = DK PN ,

Ran(DA ) ∩ N = Ran(DK ).

(9.13)

The second equality in (9.12) yields Ran(DA∗ ) = DA˙ ∗ Ran(DK ∗ ).

(9.14)

Suppose now that the Hilbert space H is decomposed as H = G ⊕ M. Then A˙ = A + C, where A = PG A˙ ∈ [G, G ] and C = PM A˙ ∈ [G, M]. We can rewrite A˙ in the block-matrix form A ˙ A= . (9.15) C Since A˙ is a contraction, we have Ag2 + Cg2 ≤ g2 for all g ∈ G. It follows that C = KDA , (9.16) where K ∈ [DA , M] is a contraction. Because A˙ ∗ = A∗ PG + DA KPM , we get from (9.6) the relation DA˙ ∗ f 2 = (DA∗ PG − AK∗ PM )f 2 + DK∗ PM f 2 , f ∈ H . This yields the relations 2 2 (DA ˙ ∗ )M = DK∗ PM ,

(9.17)

Ran(DA˙ ∗ ) ∩ M = Ran(DK∗ ).

(9.18)

2 Here and below (DA ˙ ∗ )M is the shorted operator of the form (9.8)–(9.9). All bounded extensions A of A˙ also have the block-matrix form A B G G A= : → . C D N M

Theorem 9.1.4. The formula A D A∗ N G G A= : → , KDA −KA∗ N + DK∗ XDN N M

(9.19)

establishes a bijective correspondence between all contractive extensions A of the contraction A˙ = A + KDA and all pairs N ∈ [N, DA∗ ], X ∈ [DN , DK∗ ] of contractive operators.

9.1. Contractive extensions and their block-matrix forms

267

Proof. It follows from (9.16) that A˙ ∗ = A∗ PG + DA K∗ PM . Therefore using (9.6) for all f ∈ H we get DA˙ ∗ f 2 = DA∗ PG f − AK∗ PM f 2 + DK∗ PM f 2 . Thus, DA˙ ∗ f 2 = DA∗ PG f − AK∗ PM f 2 + DK∗ PM f 2 , f ∈ H .

(9.20)

In view of the equality ADA = DA∗ A we get that ADA ⊂ DA∗ and since Ran(K∗ ) ⊂ DA , (9.20) yields 8 9 inf DA˙ ∗ (f − ϕ)2 , ϕ ∈ G = DK∗ PM f 2 , f ∈ H . (9.21) Let K = DA˙ ∗ G and L = DA˙ ∗ K. Observe that (see [172]) L = {f ∈ DA˙ ∗ : DA˙ ∗ f ∈ M} . From (9.20) and (9.21) we get the equalities PK DA˙ ∗ f 2 = DA∗ PG f − AK∗ PM f 2 , PL DA˙ ∗ f 2 = DK∗ PM f 2 , f ∈ H . In particular,

(9.22)

PK DA˙ ∗ ϕ2 = DA∗ ϕ2 , ϕ ∈ G .

It follows from (9.22) that there are unitary operators U ∈ [K, DA∗ ] and Z ∈ [L, DK∗ ] such that UPK DA˙ ∗ f = DA∗ PG f − AK∗ PM f, ZPL DA˙ ∗ f = DK∗ PM f, f ∈ H .

(9.23)

Obviously, U ∗ ∈ [DA∗ , K], Z ∗ ∈ [L, DK∗ ] and U ∗ = U −1 , Z ∗ = Z −1 . Then from (9.23) we have DA˙ ∗ = U ∗ (DA∗ PG − AK∗ PM ) + Z ∗ DK∗ PM , and

DA˙ ∗ = (DA∗ − KA∗ ) UPK + DK∗ ZPL .

(9.24)

Let K be an arbitrary bounded operator from N into DA˙ ∗ . Then K = PK K + PL K. Let N = UPK K and Y = ZPL K. Clearly, K is a contraction if and only if = N +Y ∈ [M, DA∗ ⊕ DK∗ ] is a contraction and K is a contraction the operator K if and only if Y = XDN , where X ∈ [DN , DK∗ ] is a contraction. Further, for a contraction K ∈ [N, DA˙ ∗ ] from (9.24) and for all h ∈ N we get DA˙ ∗ Kh = (DA∗ − KA∗ ) N h + DK∗ XDN h.

(9.25)

˙ G + D ˙ ∗ KPN . Then (9.11) and (9.25) yield (9.19). Let A = AP A

268 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Let the unitary operators U and Z be deﬁned by (9.23). Deﬁne the unitary operator U ∈ [DA˙ ∗ , DA∗ ⊕ DK∗ ] by the operator matrix U 0 K DA∗ U= : → . 0 Z L DK∗ Then the parameters K ∈ [N, DA˙ ∗ ] in (9.11) and N DA∗ :N→ XDN DK∗ in (9.19) are connected by the relation N U ∗N K K = U∗ = : N → . XDN Z ∗ XDN L

(9.26)

Suppose that G ⊂ H, G ⊂ H , A˙ : G → H , B : G → H, and the operators A˙ and B form a dual pair, i.e., ˙ h)H = (f, Bh)H (Af,

for all f ∈ G, g ∈ G .

(9.27) ;

<

˙ B if A ⊃ The operator A ∈ [H, H ] is called an extension of the dual pair A, ∗ A˙ and A ⊃ B. Theorem 9.1.4 enables us to give the; block-matrix form of all < ˙ B . Put contractive extensions of the dual pair of contractions A, B = PG B, Then B ∈ [G , G], C ∈ [G , N], B=

C = PN B. B , C

C = MDB , and M ∈ [DB , N] is a contraction. In addition, in view of (9.27) one has B ∗ = A. Thus DB = DA∗ . Theorem 9.1.5. The formula A D A∗ M ∗ G G A= : → , KDA −KA∗ M∗ + DK∗ XDM∗ N M

(9.28)

establishes a bijective correspondence between all contractive extensions A of the ; < ˙ B and all contractions X ∈ [DM∗ , DK∗ ]. dual pair of contractions A, Proof. Let A be given by (9.28), then by Theorem 9.1.4 the operator A is a ˙ Since A∗ f = (A∗ + MDA∗ )f = Bf for all f ∈ G , contractive extension of A. ∗ the operator A is a contractive ; < extension of B. Conversely, let A be a contractive ˙ B . Since A is a contractive extension of A, ˙ it is of extension of a dual pair A, ∗ ∗ the form (9.19) and since A G = B, we get N DA∗ = PN B = C = MDA∗ . So, N ∗ = M and hence N = M∗ . This completes the proof.

9.2. Quasi-self-adjoint contractive extensions of symmetric contractions

269

9.2 Quasi-self-adjoint contractive extensions of symmetric contractions Deﬁnition 9.2.1. Let α ∈ [0, π/2) and let S(α) := {z ∈ C : | arg z| ≤ α} be a sector on the complex plane C with the vertex at the origin and the semiangle α. A linear operator T in a Hilbert space H is called sectorial with vertex at z = 0 and the semi-angle α (α-sectorial) if |Im (T f, f )| ≤ (tan α) Re (T f, f ),

(9.29)

for all f ∈ Dom(T ). Deﬁnition 9.2.2. Let α ∈ (0, π/2). We say that a bounded operator T ∈ [H, H] belongs to the class CH (α) if T sin α ± iI cos α ≤ 1.

(9.30)

It is clear that CH (π/2) is the set of all linear contractions in H, the class CH (α) is a convex and closed (with respect to the strong operator topology) set, which is the intersection of two closed operator balls corresponding to ±. Moreover, by virtue of (9.30) one immediately concludes that T ∈ CH (α) ⇐⇒ −T ∈ CH (α) ⇐⇒ T ∗ ∈ CH (α), and that condition (9.30) is equivalent to (tan α) (f 2 − T f 2) ≥ 2|Im (T f, f )|,

f ∈ H.

(9.31)

Together with Deﬁnition 9.2.1, it also proves that T ∈ CH (α) is equivalent to the statement that (I − T ∗ )(I + T ) is a bounded α-sectorial operator. According to (9.31) it is natural to identify CH (0) with the set of all self-adjoint contractions. If α ∈ (0, π/2), then the members of the class CH (α) are also referred to as α-cosectorial contractions. Let A˙ be a non-densely-deﬁned symmetric contraction in a Hilbert space H. An operator T ∈ [H, H] is called a quasi-self-adjoint contractive extension of A˙ or ˙ T ∗ ⊃ A, ˙ and T ≤ 1. If T = T ∗ , then T is called a selfqsc-extension if T ⊃ A, adjoint contractive extension of A˙ or sc-extension. Below we provide a description ˙ of all qsc-extensions of operator A. Theorem 9.2.3. Let A˙ be a closed symmetric contraction in H = H0 ⊕ N with ˙ = H0 and let A˙ be decomposed as in (9.15). Then: Dom(A) (i) the formula A T = KDA

D A K∗ ∗ −KAK + DK∗ XDK∗

H0 H0 : → , N N

(9.32)

270 Chapter 9. Canonical L-systems with Contractive and Accretive Operators sets a one-to-one correspondence between all qsc-extensions T of the symmetric contraction A˙ = A + KDA and all contractions X in the subspace DK∗ ; (ii) T is a self-adjoint contractive extension of A˙ if and only if X in (9.32) is a self-adjoint contraction in DK∗ ; (iii) T in (9.32) belongs to the class CH (α) if and only if X belongs to the class CDK∗ (α), α ∈ (0, π/2). Proof. A˙ is a symmetric contraction, we apply Theorem 9.1.4 to the dual ; Because < ˙ ˙ pair A, A . In that case M = K and hence (9.28) takes the form (9.32). Clearly, T is an sc-extension if and only if X ∈ [DK∗ , DK∗ ] is a self-adjoint contraction. It follows from (9.32) that 2 DA − DA K∗ KDA −ADA K∗ − DA K∗ D I − T ∗T = , 2 2 ∗ ∗ −KDA A − D ∗ KDA DK ∗ − KA K − D D i.e.,

f f ∗ (I − T T ) , h h = DK (DA f − AK∗ h) − K∗ XDK∗ h2 + DX DK∗ h2 ≥ 0,

(9.33)

˙ h ∈ N. It follows from (9.33) that T ∈ CH (α) is for all f ∈ H0 = Dom(A), equivalent to |(Im XDK∗ h, DK∗ h)| tan α ≤ DK (DA f − AK∗ h) − K∗ XDK∗ h2 + DX DK∗ h2 2

(9.34)

which holds for all f ∈ H0 , h ∈ N. Due to the inclusions Ran(X) ⊂ DK∗ ,

K∗ DK∗ ⊂ DK ⊆ DA ,

ADA ⊂ DA ,

one can choose a sequence {fn }∞ n=1 ⊂ DA such that for a given h ∈ N the equality lim DK DA fn = DK AK∗ h + K∗ XDK∗ h,

n→∞

(9.35)

holds for all f ∈ H0 , h ∈ N. In view of (9.35) the condition (9.34) is equivalent to |(Im Xh, h)| ≤ for all h ∈ DK∗ .

tan α DX h2 ⇐⇒ X ∈ CDK∗ (α). 2

(9.36)

Deﬁne two operators Aμ and AM , corresponding to X = −IDK∗ and X = IDK∗ in (9.32), respectively: A DA K ∗ Aμ = , (9.37) 2 KDA −KAK∗ − DK ∗

9.2. Quasi-self-adjoint contractive extensions of symmetric contractions AM =

A KDA

DA K ∗ 2 −KAK∗ + DK ∗

271

.

(9.38)

Then as a consequence of Theorem 9.2.3 we get the following result. Theorem 9.2.4. The class of all sc-extensions of A˙ forms an operator interval [Aμ , AM ] with endpoints given by (9.37) and (9.38). Proposition 9.2.5. The operators Aμ and AM possess the following properties for all f ∈ H: inf

˙ ϕ∈Dom(A)

((I + Aμ )(f + ϕ), f + ϕ) =

inf

˙ ϕ∈Dom(A)

((I − AM )(f + ϕ), f + ϕ) = 0.

Proof. From (9.37) for f = x + h, x ∈ H0 , h ∈ N we have ((I + Aμ )(x + h), x + h) = ((IH0 + A)x, x) + 2Re (DA K∗ h, x) − (AK∗ h, K∗ h) + K∗ h2 = (IH0 + A)1/2 x2 + 2Re ((IH0 + A)1/2 x, (IH0 − A)1/2 K∗ h) + (IH0 − A)1/2 K∗ h2 = (IH0 + A)1/2 x + (IH0 − A)1/2 K∗ h2 . Hence, if ϕ ∈ H0 , then ((I + Aμ )(f + ϕ), f + ϕ) = (IH0 + A)1/2 (x + ϕ) + (IH0 − A)1/2 K∗ h2 . Since Ran(K∗ ) ⊆ DA , one can ﬁnd a sequence {ϕn } ⊂ H0 such that lim (IH0 + A)1/2 ϕn = −(IH0 + A)1/2 x − (IH0 − A)1/2 K∗ h.

n→∞

It follows that inf

˙ ϕ∈Dom(A)

inf

˙ ϕ∈Dom(A)

((I + Aμ )(f + ϕ), f + ϕ) = 0 for any f ∈ H. Similarly

((I − AM )(f + ϕ), f + ϕ) = 0, f ∈ H.

Thus the sc-extensions Aμ and AM coincide with extreme contractive selfadjoint extensions of a symmetric contraction A˙ discovered by M. Kre˘ın. Following Kre˘in’s notations we call the sc-extensions Aμ and AM of a symmetric contraction A˙ the rigid and the soft extensions, respectively. Hence for the extreme scextensions Aμ and AM of A˙ from Proposition 9.2.5 one has (I + Aμ )N = (I − AM )N = 0, where (I + Aμ )N and (I − AM )N are the shorted operators of the form (9.8)-(9.9). These equalities together with (9.10) yield the next statement. Proposition 9.2.6. Let A be an sc-extension of A˙ and let E(x) be the resolution of identity for A. Then:

272 Chapter 9. Canonical L-systems with Contractive and Accretive Operators 1. A = Aμ if and only if 1 −1

d(E(x)ϕ, ϕ) = +∞, 1+x

for all ϕ ∈ N \ {0};

(9.39)

for all ϕ ∈ N \ {0}.

(9.40)

2. A = AM if and only if 1 −1

d(E(x)ϕ, ϕ) = +∞, 1−x

It follows from (9.37) and (9.38) that Aμ + AM AM − Aμ A DA K ∗ 0 0 2 = , = = DK (9.41) ∗ PN . 2 KDA −KAK∗ 0 DK ∗ 2 2 Applying (9.32) and (9.41) and performing straightforward calculations we obtain the following result. Theorem 9.2.7. The formula T =

1 1 (AM + Aμ ) + (AM − Aμ )1/2 X(AM − Aμ )1/2 , 2 2

(9.42)

establishes a one-to-one correspondence between the set of all qsc-extensions T of A˙ and the set of contractions X on the subspace N0 = Ran(AM − Aμ ). Moreover, (i) the operator T is an sc-extension of A˙ if and only if X is a self-adjoint contraction in N0 , (ii) T ∈ CH (α) if and only if X ∈ CN0 (α). Thus, the set of all qsc-extensions of symmetric contraction A˙ forms the operator ball AM + Aμ AM − Aμ B , (9.43) 2 2 with the center (AM + Aμ )/2 and the left and right radii (AM − Aμ )/2. ˙ The Deﬁnition 9.2.8. Let T be a qsc-extension of a symmetric contraction A. operator T is said to be extremal qsc-extension if T is an extreme point (see [118]) of the operator ball B in (9.43). It is easy to see that an operator T is an extremal qsc-extension if and only if the corresponding to T contraction X ∈ [DK∗ , DK∗ ] in representation (9.32) or in representation (9.42) is an isometry or co-isometry. From (9.33), arguing as in

9.2. Quasi-self-adjoint contractive extensions of symmetric contractions

273

the proof of Theorem 9.2.3, we immediately obtain the relations for the shorted operators (DT2 )N and (DT2 ∗ )N , 2 2 (DT2 )N = DK∗ DX DK∗ PN = 12 (AM − Aμ )1/2 DX (AM − Aμ )1/2 ,

1 2 2 1/2 (DT2 ∗ )N = DK∗ DX (AM − Aμ )1/2 DX . ∗ D K ∗ PN = ∗ (AM − Aμ ) 2

(9.44)

Consequently, (DT2 )N = 0 ⇐⇒ (DT2 ∗ )N = 0 ⇐⇒

X is an isometry, X ∗ is an isometry,

and we arrive at the following result. Proposition 9.2.9. (1) The following statements are equivalent: ˙ (i) T is an extremal qsc-extension of symmetric contraction A; (ii) either (DT2 )N = 0 or (DT2 ∗ )N = 0; (iii) either Ran(DT ) ∩ N = {0} or Ran(DT ∗ ) ∩ N = {0}. (2) The sc-extension T is extremal if and only if the corresponding self-adjoint contraction X in [DK∗ , DK∗ ] is unitary, i.e., X = X ∗ = X −1 . (3) If T is an extremal qsc-extension and T ∈ CH (α) for some α < π/2, then T is an extremal sc-extension. Remark 9.2.10. If dim N < ∞, then (DT2 )N = 0 if and only if (DT2 ∗ )N = 0. In view of the relations 2 (I ± X ∗ )(I ± X) + DX = 2(I ± ReX),

(9.45)

for any contraction X, we get from (9.42) that there are no non-selfadjoint qscextensions T with the real parts Re T equal to Aμ or AM . Notice also, that if sc-extension A is deﬁned via (9.42) by means of a selfadjoint contraction X ∈ [N0 , N0 ], then from Proposition 9.2.5 we get 1 (AM − Aμ )1/2 (IN0 − X)(AM − Aμ )1/2 = AM − A, 2 1 = (AM − Aμ )1/2 (IN0 + X)(AM − Aμ )1/2 = A − Aμ . 2

(I − A)N = (I + A)N

(9.46)

From (9.46) and (9.8) we obtain A − Aμ = (I + A)1/2 PΩ+ (I + A)1/2 , AM − A = (I − A)1/2 PΩ− (I − A)1/2 , where Ω± := {f ∈ H : (I ± A)1/2 f ∈ N}.

(9.47)

274 Chapter 9. Canonical L-systems with Contractive and Accretive Operators ˙ Then Proposition 9.2.11. Let A be an sc-extension of A. Ran((I + Aμ )1/2 ) ⊆ Ran((I + A)1/2 ), Ran((I − AM )1/2 ) ⊆ Ran((I − A)1/2 ),

(9.48)

and the equalities (I + A)−1/2 (I + Aμ )1/2 h2 = h2 , h ∈ Ran((I + Aμ )1/2 ), (I − A)−1/2 (I − AM )1/2 g2 = g2 , g ∈ Ran((I − AM )1/2 ),

(9.49)

hold. Proof. Inclusions (9.48) follow from the inequalities I +Aμ ≤ I +A, I −AM ≤ I −A and Theorem 2.1.2. In order to prove (9.49) we use (9.47) which produces the relations I + Aμ = (I + A)1/2 (I − PΩ+ )(I + A)1/2 , I − AM = (I − A)1/2 (I − PΩ− )(I − A)1/2 . Hence

(I + Aμ )1/2 = V+ (I − PΩ+ )(I + A)1/2 , (I − AM )1/2 = V− (I − PΩ− )(I − A)1/2 ,

where V+ and V− are isometries, which map Ran(I − PΩ+ ) and Ran(I − PΩ− ) onto Ran((I + Aμ )1/2 ) and

Ran((I − AM )1/2 ),

respectively. It follows that (I + Aμ )1/2 = (I + A)1/2 V+∗ ,

(I − AM )1/2 = (I − A)1/2 V−∗ ,

These relations yield (9.49).

Let us give other formulas for the operators Aμ , AM , (Aμ + AM )/2, and (AM − Aμ )/2. We introduce the following subspaces of DA˙ ∗ : ˙ LA˙ = DA˙ ∗ Dom(A),

L0 = DA˙ ∗ LA˙ ,

˙ Clearly, where H0 = Dom(A). −1 L0 = D A ˙ ∗ (N) = {f ∈ DA˙ ∗ : DA˙ ∗ f ∈ N},

and by (9.14) one has

2 (DA ˙ ∗ )N = DA˙ ∗ PL0 DA˙ ∗ .

(9.50)

Deﬁne the operator KA˙ via relation ˙ ˙, KA˙ DA˙ ∗ fA = PN Af A

˙ fA˙ ∈ Dom(A).

(9.51)

By (9.15) we have A˙ ∗ H0 = PH0 A˙ = A and the operator KA˙ in (9.51) is a ∗ contraction in [LA˙ , N]. Let KA ˙ ∈ [N, LA˙ ] be the adjoint to KA˙ operator.

9.2. Quasi-self-adjoint contractive extensions of symmetric contractions

275

Proposition 9.2.12. Let T0 = (Aμ + AM )/2. Then the following relations hold: ∗ ˙∗ T0 = APH0 + DA˙ ∗ KA ˙ PN = A + KA PLA˙ DA˙ ∗ ,

(9.52)

AM − Aμ 2 2 2 2 = (DA ˙ ∗ )N = (DT0 )N = DK∗ PN = DK ∗˙ PN , A 2

(9.53)

∗ Aμ = APH0 + DA˙ ∗ (KA PN − PL0 DA˙ ∗ ), ∗ AM = APH0 + DA˙ ∗ (KA PN + PL0 DA˙ ∗ ).

(9.54)

Proof. By (9.11)

T0 = APH0 + DA˙ ∗ KT∗0 PN ,

with some contraction KT0 ∈ [N, DA˙ ∗ ]. On the other hand the operator ∗ K D A∗ :N→ 0 DK∗ is related to T0 via formula (9.32). The connection is given by (9.26). Therefore KT0 = U0∗ K∗ ∈ [N, LA˙ ]. 2 2 Since T0 is self-adjoint, we get T0 = A˙ ∗ + KT∗0 DA˙ ∗ , DK = DK ∗ , and T0 f = Af T0 ∗ ∗ 2 2 for all f ∈ H0 . Hence KT0 DA˙ ∗ f = PN Af . It follows KT0 = KA˙ , DK . ∗ = DK T ˙ A

0

2 2 2 Applying (9.17) yields (DA ˙ ∗ )N = DK∗ PN while (9.13) implies that (DT0 )N = 2 DK ∗ PN . Relations (9.54) follow from (9.52), (9.53), and (9.50). This completes ˙ A the proof.

˙ = H0 in a Hilbert Theorem 9.2.13. Let A˙ be a symmetric contraction with Dom(A) space H. Then the following conditions are equivalent: (i) the operator A˙ admits a unique qsc-extension; (ii) Aμ = AM ; (iii) Ran(DA˙ ∗ ) ∩ N = {0} ; (iv) the operator K∗ is isometric; (v) sup ϕ∈H0

, ,2 , ˙ , ,(Aϕ, h), DA˙ ϕ2

=∞

for all

h ∈ N \ {0}.

(9.55)

Proof. By Theorem 9.2.3 (i) ⇐⇒ (ii) ⇐⇒ (iv). The equivalence of conditions (iii) and (iv) follows from (9.10) and (9.17). According to Lemma 9.1.1 condition (9.55) is equivalent to A˙ ∗ N ∩ Ran(DA˙ ) = {0}.

276 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Since (see (9.15)-(9.16)) PN A˙ = KDA , we have DA f 2 = DK DA f 2 , f ∈ ˙ = H0 , A˙ ∗ h = DA K∗ h, h ∈ N. By the Douglas Theorem 2.1.2 we get Dom(A) Ran(DA˙ ) = Ran(DA DK ).

(9.56)

Therefore A˙ ∗ N ∩ Ran(DA˙ ) = {0} if and only if R(K∗ ) ∩ Ran(DK ) = {0}. Since K∗ DK∗ = DK K∗ , we have that Ran(K∗ ) ∩ Ran(DK ) = {0} if and only if the operator K∗ is isometric. Corollary 9.2.14. If AM = Aμ , then a symmetric contraction A˙ does not admit non-self-adjoint qsc-extensions. The proof of the corollary follows from Theorem 9.2.7 and Theorem 9.2.13. Corollary 9.2.15. If AM = Aμ , then for any α ∈ (0, π/2) there is a non-self-adjoint ˙ α-co-sectorial qsc-extension of the symmetric contraction A. Proof. If in the formula (9.42) one sets X = i tan

α I N0 , 2

then by direct computations it is conﬁrmed that X sin α ± i cos α I = 1. This equality implies the α-co-sectorialilty of X. Then by Theorem 9.2.7 the corresponding T belongs to CH (α). Theorem 9.2.16. Let A˙ = A + KDA be a symmetric contraction of the form (9.15) ˙ Then the written for the decomposition H = H0 ⊕ N, where H0 = Dom(A). following conditions are equivalent: (i) Ran(AM − Aμ ) = N; (ii) K < 1; (iii) Ran(DA˙ ∗ ) ⊃ N; (iv) Ran(DA˙ ) = Ran(DA ); (v) for all h ∈ N holds sup ϕ∈H0

, ,2 , ˙ , ,(Aϕ, h), DA˙ ϕ2

< ∞.

(9.57)

Proof. By (9.17), (9.18), and (9.56) conditions (i), (ii), and (iii) are equivalent ˙ It follows from (9.3) that for every non-densely-deﬁned symmetric contraction A. condition (9.57) is equivalent to A˙ ∗ N ⊂ Ran(DA˙ ). By the Douglas Theorem 2.1.2 the latter is equivalent to ˙ PN Aϕ ≤ γDA˙ ϕ

for all

ϕ ∈ H0 ,

9.2. Quasi-self-adjoint contractive extensions of symmetric contractions

277

where γ > 0. Since PN A˙ = KDA the last inequality is equivalent to Kϕ2 ≤

γ2 ϕ2 , ϕ ∈ DA ⇐⇒ K < 1. 1 + γ2

Let b be a real number, −1 < b < 1. Consider the function z → wb =

z−b 1 − bz

and corresponding operator fractional-linear transformation ˙ = (A˙ − bI)(I − bA) ˙ −1 . A˙ → wb (A)

(9.58)

˙ is also symmetric Proposition 9.2.17. Let A˙ be symmetric contraction. Then wb (A) contraction and ˙ μ = wb (Aμ ), (wb (A))

˙ M = wb (AM ), (wb (A))

i.e., the operators wb (Aμ ) and wb (Aμ ) are the rigid and the soft sc-extensions of ˙ respectively. Moreover, the function wb (A), A → wb (A) = (A − bI)(I − bA)−1 bijectively maps the operator interval [Aμ , AM ] onto the operator interval [wb (Aμ ), wb (AM )]. ˙ = (I −bA)Dom( ˙ ˙ HDom(wb (A)) ˙ = (I −bA˙ ∗ )−1 N, Proof. Clearly, Dom(wb (A)) A), ˙ is symmetric contraction. Let us ﬁnd and wb (A) inf

˙ ψ∈Dom(wb (A))

((I + wb (Aμ )(f + ψ), f + ψ)

for f ∈ H. Since I + wb (Aμ ) = (1 − b)(I + Aμ )(I − bAμ )−1 = (1 − b)(I − bAμ )−1 (I − bAμ )(I + Aμ )(I − bAμ )−1 , we obtain for each h ∈ H that ((I + wb (Aμ )h, h) = (1 − b)(I − bAμ )1/2 (I + Aμ )1/2 (I − bAμ )−1 h2 . Therefore, ((I + wb (Aμ ))f + ψ, f + ψ) = (1 − b)(I − bAμ )1/2 (I + Aμ )1/2 ((I − bAμ )−1 f + (I − bAμ )−1 ψ)2 . ˙ = (I − bA) ˙ −1 Dom(wb (A)) ˙ = Dom(A) ˙ and Because (I − bAμ )−1 Dom(wb (A)) inf

˙ ϕ∈Dom(A)

((I + Aμ )(h + ϕ), h + ϕ) = 0,

278 Chapter 9. Canonical L-systems with Contractive and Accretive Operators for all h ∈ H, we get inf

((I + wb (Aμ ))(f + ψ), f + ψ) =

˙ ψ∈Dom(wb (A)) ×(I − bAμ )1/2 (I

+ Aμ )1/2 ((I − bAμ )−1 f +

inf

(1 − b)

˙ ψ∈Dom(wb (A)) (I − bAμ )−1 ψ)2

= 0.

Hence (wb (A))μ = wb (Aμ ). Similarly (wb (A))M = wb (AM ). Let A1 and A2 be two self-adjoint contractions. If A1 ≥ A2 , then bI − b2 A1 ≤ bI − b2 A2 . Then it follows that (bI − b2 A1 )−1 ≥ (bI − b2 A2 )−1 . This inequality yields wb (A1 ) ≥ wb (A2 ). Therefore, A ∈ [Aμ , AM ] ⇒ wb (A) ∈ [wb (Aμ ), wb (AM )]. Since the inverse mapping is given by C → (C + bI)(I + bC)−1 , we get that A ∈ [Aμ , AM ] if and only if wb (A) ∈ [wb (Aμ ), wb (AM )].

Let the Hilbert space H be decomposed as H = H1 ⊕ H2 and T ∈ [H, H] be decomposed accordingly: T11 T12 T = , Tij ∈ [Hi , Hj ]. T21 T22 Deﬁne the operator-valued functions VT (z) = T21 (T11 − zI)−1 T12 − T22 ,

WT (z) = −zI − VT (z),

z ∈ ρ(T11 ). (9.59)

By the Schur-Frobenius formula (see [157]) the resolvent (T − z)−1 of T can be rewritten in the block form (T11 −zI)−1 (I+T12 WT (z)−1 T21 (T11 −zI)−1 ) −(T11 −zI)−1 T12 WT−1 (z) , (9.60) −1 −1 −1 −WT (z)T21 (T11 −zI)

WT (z)

for z ∈ ρ(T ) ∩ ρ(T11 ). In particular, −1

PH2 (T − zI)−1 H2 = − (VT (z) + zI)

,

z ∈ ρ(T ) ∩ ρ(T11 ).

(9.61)

Following the deﬁnition from Section 6.5, we call a non-densely-deﬁned symmetric contraction A˙ by a prime contraction if there is no non-trivial subspace in ˙ which is invariant under A. ˙ Since A˙ is also symmetric, its primeness is Dom(A) ˙ equivalent to A being completely non-self-adjoint, i.e., to A˙ having no self-adjoint parts.

9.2. Quasi-self-adjoint contractive extensions of symmetric contractions

279

Lemma 9.2.18. Let the symmetric contraction A˙ = A + KDA in H = H0 ⊕ N, ˙ be decomposed as in (9.15) with K : DA → N. Then A˙ is prime if H0 = Dom(A), and only if the subspace 8 9 H0s := c.l.s. (A − zI)−1 K∗ N, z ∈ ρ(A) = c.l.s. { An K∗ N, n = 0, 1, . . . } (9.62) coincides with H0 . In this case, DA = H0 , K : H0 → N, and Af < f for all f ∈ H0 \{0}. Proof. Suppose that A˙ is prime. Then clearly ker DA = {0} or equivalently Af < f for all f ∈ H0 \ {0}, so that DA = H0 and K : H0 → N. Observe, that the subspace H0s in (9.62) and hence also H0 H0s , is invariant under A = A∗ . Then the subspace H0 H0s is also invariant under DA . Moreover, H0 H0s = { f ∈ H0 : KAn f = 0, n = 0, 1, . . . } .

(9.63)

It follows that KDA f = 0 for all f ∈ H0 H0s . Hence, in view of (9.15) Af = Af ˙ for all f ∈ H0 H0s . This means that the subspace H0 H0s is invariant under A. Since A˙ is a prime, one concludes that H0s = H0 . Conversely, assume that H0s = H0 . Since Ran(K∗ ) ⊂ DA and DA is invariant under A, the deﬁnition of H0s in (9.62) shows that H0s ⊂ DA . Hence, the assumption implies that H0 = DA = Ran(DA ), so that ker DA = {0}. Now, suppose that 0 ⊂ H0 is a subspace which is invariant under A. ˙ Then for every f ∈ H 0 one has H ˙ ˙ 0 . Af = Af + KDA f ∈ H0 , so that KDA f = 0 for all f ∈ H0 and A H0 = A H Hence, H0 is invariant under A and DA . Moreover, since ker DA = {0} the image 0 is dense in H 0 . This implies that KH 0 = {0} and since An H 0 ⊂ H 0 one DA H n has KA H0 = {0} for all n = 0, 1, . . ., i.e., 0 ⊂ { f ∈ H0 : KAn f = 0, n = 0, 1, . . . } = H0 Hs = {0}, H 0

cf. (9.63). Therefore, A˙ is prime.

Let T be a qsc-extension of A˙ in the Hilbert space H = H0 ⊕ N with H0 = ˙ It is evident that the subspace Dom(A). 8 9 HT := c.l.s. (T − zI)−1 N : |z| > 1 = c.l.s. { T nN : n = 0, 1, 2, . . . } , (9.64) is invariant under T , and that the subspace HT := H HT , is invariant under T ∗ . Since N ⊂ HT , one obtains ˙ ⊂ ker(T − T ∗ ). HT ⊂ N⊥ = Dom(A) Therefore the restriction of T ∗ onto HT is a self-adjoint operator in HT . The restriction T HT (= PHT T HT ) is called the N-minimal part of T . Moreover, T

280 Chapter 9. Canonical L-systems with Contractive and Accretive Operators is said to be N-minimal if the equality H = HT holds. If T be a qsc-extension of ˙ then its adjoint T ∗ is also a qsc-extension of A˙ and one can associate with it A, the subspace HT ∗ and the corresponding N-minimal part of T ∗ . The next result shows that the N-minimal parts of T and T ∗ are qsc-extensions of the prime part ˙ Hs of A˙ in the same subspace H = H ∗ . A 0 T T Proposition 9.2.19. Let A˙ be a symmetric contraction in H = H0 ⊕ N with H0 = ˙ T be a qsc-extension of A˙ in H, and T ∗ be its adjoint. Then the subspaces Dom(A), HT , HT ∗ , and H0s of H = H0 ⊕ N as deﬁned in (9.64) and (9.62) are connected by (H :=) HT = HT ∗ = H0s ⊕ N.

(9.65)

In particular, the symmetric contraction A˙ is prime if and only if the qsc-extension T (or equivalently T ∗ ) of A˙ is N-minimal. Proof. It follows from the Schur-Frobenius formula (9.60) that (T − z)−1 N =

−(A − z)−1 DA K∗ N , N

|z| > 1,

which implies that 8 9 c.l.s. (T − zI)−1 N : |z| > 1 8 9 = c.l.s. (A − zI)−1 DA K∗ N : z ∈ ρ(A) ⊕ N 8 9 = c.l.s. DA (A − zI)−1 K∗ N : z ∈ ρ(A) ⊕ N. This shows that HT = DA H0s ⊕ N.

(9.66)

Since Ran(K∗ ) ⊂ DA and DA is invariant under A one has H0s ⊂ DA . In particular, H0s ∩ ker DA = {0}, which together with DA H0s ⊂ H0s implies that DA H0s = H0s . Hence, (9.66) implies the equality HT = H0s ⊕ N. It follows from (T ∗ − zI)−1 − (T − zI)−1 = (T − zI)−1 [T − T ∗ ](T ∗ − zI)−1 ,

|z| > 1,

and the inclusion Ran(T − T ∗ ) ⊂ N that (T ∗ − zI)−1 N ⊂ (T − zI)−1 N ⊂ HT ,

|z| > 1.

Therefore, HT ∗ ⊂ HT and the reverse inclusion follows via symmetry. This completes the proof of (9.65). The last statement is clear from (9.65).

9.3. Weyl-Titchmarsh functions of quasi-self-adjoint contractive extensions 281

9.3 The Weyl-Titchmarsh functions of quasi-self-adjoint contractive extensions Let G be a Hilbert space. The subclass of Herglotz-Nevanlinna [G, G]-valued functions V (z) holomorphic on the domain Ext[−1, 1] = C \ [−1, 1] is denoted by N(G, [−1, 1]). It is well known (see [89]) that every function V (z) in N(G, [−1, 1]) has an integral representation of the form 1 V (z) = Q + −1

dG(t) , t−z

where Q is a bounded self-adjoint operator on G and the [G, G]-valued function G(t) is non-decreasing, non-negative, normalized by G(−1 − 0) = 0, and has ﬁnite total variation concentrated on [−1, 1]. Clearly, V (∞) = s − lim V (z) = Q. The z→∞

next result is also well known, cf. [89]. Theorem 9.3.1. Let G be a Hilbert space and let V (z) ∈ N(G, [−1, 1]). Then there exist a Hilbert space H, a self-adjoint contraction B on H, and F ∈ [G, H], such that V (z) = V (∞) + F ∗ (B − zI)−1 F, z ∈ Ext[−1, 1]. (9.67) In what follows the subclass of functions V (z) in N(G, [−1, 1]) which have the limit values V (±1) in [G, G] will play a central role. In this case Theorem 9.3.1 can be completed as follows. Theorem 9.3.2. Let G be a Hilbert space and let V (z) ∈ N(G, [−1, 1]). If for all f ∈ G the limit values lim (V (x)f, f ) ,

x↑−1

lim (V (x)f, f ) x↓1

(9.68)

are ﬁnite, then there exist a Hilbert space H, a self-adjoint contraction B in H, and an operator G ∈ [G, DB ], such that 2 V (z) = V (∞) + G∗ DB (B − zI)−1 G,

z ∈ Ext[−1, 1],

(9.69)

where DB and DB are deﬁned by (9.1) and (9.2), respectively. Conversely, for every function V (z) of the form (9.69) the limit values (9.68) exist for all f ∈ G and are ﬁnite. Proof. By Theorem 9.3.1, V (z) has the representation (9.67), where B is a selfadjoint contraction in a Hilbert space H and F ∈ [G, H]. Since the limits in (9.68) exist for all f ∈ G, one concludes that Ran(F ) ⊂ Ran(I − B)1/2 ∩ Ran(I + B)1/2 .

282 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Consequently, Ran(F ) ⊂ Ran(DB ) and this implies (see Theorem 2.1.2) that F = DB G for some operator G ∈ [G, DB ].Taking into account (9.67) and the fact that DB commutes with B we get (9.69). Conversely, if V (z) is of the form (9.69), then the inclusions Ran(DB ) ⊂ Ran((I ± B)1/2 ) imply the existence of the limit values (9.68) for all f ∈ G (see (9.4)). It follows from Theorem 9.3.2 that V (−1) := s − lim V (x) = V (∞) + G∗ (I − B)G ∈ [G, G], x↑−1

V (1) := s − lim V (x) = V (∞) − G∗ (I + B)G ∈ [G, G], x↓1

so that V (−1) + V (1) = 2V (∞) − 2G∗ BG,

V (−1) − V (1) = 2G∗ G.

The operator T ∈ [H, H] is called a quasi-self-adjoint contraction (a qscoperator) if T ≤ 1, and ker(T − T ∗ ) = {0}. Let T be a qsc-operator and let G be a subspace of H such that G ⊇ Ran(T − T ∗ ). Then T is a qsc-extension of a non-densely-deﬁned contraction A˙ deﬁned as ˙ = H G, Dom(A)

˙ A˙ = T Dom(A).

˙ ⊂ ker(T − T ∗ ). Clearly, Dom(A) Let T be a qsc-operator in a separable Hilbert space H and let G be a subspace of H such that G ⊃ Ran(T − T ∗ ). The operator-valued function QT (z) = PG (T − zI)−1 G,

|z| > 1,

(9.70)

where PG is the orthogonal projection in H onto G, is said to be the WeylTitchmarsh function associated with T and the subspace G. Clearly, it has the limit value QT (∞) = 0 and the Weyl-Titchmarsh functions of T and T ∗ in G are connected by QT ∗ (z) = QT (¯ z )∗ , |z| > 1. If T is a self-adjoint contraction, then the Weyl-Titchmarsh function QT (z) in (9.70) is a Herglotz-Nevanlinna function of the class N(G, [−1, 1]). The next result contains some basic properties for the Weyl-Titchmarsh function QT of a qscoperator T as deﬁned in (9.70). Proposition 9.3.3. Let QT be Weyl-Titchmarsh function of a qsc-operator T as deﬁned in (9.70). Then:

9.3. Weyl-Titchmarsh functions of quasi-self-adjoint contractive extensions 283 (i) QT has the following asymptotic expansion 1 1 1 QT (z) = − IG + 2 F + o , z z z2

z → ∞,

(9.71)

where F = −PG T G; (ii) Q−1 T (z) ∈ [G, G] for all |z| > 1; −1 (iii) Q−1 T (z) has strong limit values QT (±1) that are −1 Q−1 T (−1) = lim QT (z), z→−1

−1 Q−1 T (1) = lim QT (z); z→1

(iv) for all f, g ∈ G the following inequality holds: , −1 , , Q (−1) + Q−1 (1) f, g ,2 T T −1 −1 ≤ Q−1 QT (−1) − Q−1 T (−1) − QT (1) f, f T (1) g, g ;

(9.72)

(v) the function −Q−1 T (z) − F − zIG is an operator-valued Herglotz-Nevanlinna function; (vi) the following conditions are equivalent: a) QT ∈ N(G, [−1, 1]), b) F = F ∗ , −1 c) Q−1 T (−1) ≥ 0 and QT (1) ≤ 0.

Moreover, if T is decomposed as in (9.32) with H0 = H G and A˙ = T H0 , then F = KAK∗ − DK∗ XDK∗ , ∗ ∗ Q−1 T (−1) = DK (X + IDK∗ )DK ,

−Q−1 T (z)

(9.73)

∗ ∗ Q−1 T (1) = DK (X − IDK∗ )DK ,

2

−1

− F − zIG = K(IH0 − A )(A − zIH0 )

∗

K .

(9.74) (9.75)

Proof. (i) Clearly, limz→∞ zQT (z)h = limz→∞ zPG (T −zI)−1 h = −h for all h ∈ G. Moreover, for all h ∈ G, lim z(IG + zQT (z))h = lim zPG T (T − zI)−1 h = −PG T h.

z→∞

z→∞

(9.76)

Hence, QT admits the asymptotic expansion (9.71). (ii) Let |z| > 1, f ∈ G, and ϕ = (T − zI)−1 f . Then f ≤ (1 + |z|)ϕ and , , |(QT (z)f, f )| = , (T − zI)−1 f, f , = |(ϕ, (T − zI)ϕ)| , , |z| − 1 = ,(ϕ, T ϕ) − z¯ϕ2 , ≥ f 2 . (|z| + 1)2

284 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Since |(QT (z)f, f )| = |(QT (z)∗ f, f )|, this implies that QT (z)f ≥

|z| − 1 f , (|z| + 1)2

QT (z)∗ f ≥

|z| − 1 f . (|z| + 1)2

Therefore, Q−1 T ∈ [G, G] for all |z| > 1. (iii) Decompose H = H0 ⊕ G and write T in block form as in (9.32), where 1/2 A = PH0 A˙ is a self-adjoint contraction in H0 , DA = IH0 − A2 , K ∈ [DA , G] is a contraction, and X is a contraction in the subspace DK∗ ⊂ G. The formula (9.73) for F is immediate from (9.32). Write Q−1 T (z) as in (9.61), Q−1 T (z) = −VT (z) − zIG ,

|z| > 1,

where VT (z) = K A + (A − zIH0 )−1 (IH0 − A2 ) K∗ − DK∗ XDK∗ .

(9.77)

This shows that the limit values Q−1 T (±1) exist and that they are given by (9.74). (iv) It follows from (9.74) that −1 Q−1 T (−1) + QT (1) = DK∗ XDK∗ , 2

−1 Q−1 T (−1) − QT (1) 2 = DK ∗. 2

(9.78)

Since X is a contraction in DK∗ we get (9.72). (v) It follows from (9.73) and (9.77) that (9.75) holds. Clearly, the function in (9.75) is a Herglotz-Nevanlinna function. The statement (vi) follows from (9.73) and (9.74). −1 Corollary 9.3.4. If G = Ran(Im T ), then the operator Q−1 T (−1)−QT (1) is positive deﬁnite. If, also, dim G < ∞, then there exist the limits QT (±1).

Proof. Let T be decomposed as in (9.32). If G = Ran(Im T ), then the operator Im T G is boundedly invertible in G. Since Im T G = DK∗ Im XDK∗ , it follows that Ran(DK∗ ) = G and Im X is boundedly invertible in G. Hence, from (9.78) we −1 get that the operator Q−1 T (−1) − QT (1) is positive deﬁnite and −1 −1 (Q−1 = T (−1) − QT (1))

1 −2 D ∗. 2 K

Suppose dim G < ∞. Since Im X is invertible, the relations (9.45) yield that the operators IG ±X and IG ±Re X are invertible. Now from (9.74) we get that QT (±1) exist and −1 −1 −1 −1 −1 −1 QT (−1) = DK DK∗ , QT (1) = DK D K∗ . ∗ (X + IG ) ∗ (X − IG )

(9.79)

9.3. Weyl-Titchmarsh functions of quasi-self-adjoint contractive extensions 285 Corollary 9.3.5. Let dim G < ∞ and let V (z) ∈ N(G, [−1, 1]), V (∞) = 0. If ker(Im V (z)) = {0} for some non-real z, then V (z) has the representation V (z) = K ∗ QB (z)K, where B is a self-adjoint contractive extension of a non-densely-deﬁned symmetric contraction in some Hilbert space H, and K ∈ [G, H] is an invertible operator. Hence, the limit values V −1 (±1) exist. Proof. By Theorem 9.3.1 the function V (z) has the representation V (z) = K ∗ (B − zI)−1 K, where B is a self-adjoint contraction in some Hilbert space H and K ∈ [G, H]. From the condition ker(Im V (z)) = {0} for some non-real z it follows that ker(K) = {0}. Let G := Ran(K) and let H0 := H G. Deﬁne the symmetric contraction A˙ := B H0 . Then B is an sc-extension of A˙ and V (z) = K ∗ QB (z)K. Now the statements of the corollary follow from Proposition 9.3.3.

Let Aμ and AM be the M. Kre˘in rigid and soft sc-extensions of a symmetric ˙ contraction A˙ deﬁned by (9.37) and (9.37), respectively and G = H Dom(A). Consider the Weyl-Titchmarsh functions of the operators Aμ and AM , QAμ (z) = PG (Aμ − zI)−1 G,

QAM (z) = PG (AM − zI)−1 G,

where z ∈ Ext[−1, 1]. Since Ran((IG + Aμ )1/2 ) ∩ G = Ran((IG − AM )1/2 ) ∩ G = {0}, using (9.3) we get the relations lim (QAμ (z)f, f ) = +∞,

z↑−1

lim(QAM (z)f, f ) = −∞, f ∈ G \ {0}. z↓1

Proposition 9.3.6. Let QT (z) be the Weyl-Titchmarsh function of some qsc-operator T . Suppose that lim inf |(QT (λ)f, f )| = ∞,

for all f ∈ G \ {0},

(9.80)

lim inf |(QT (λ)f, f )| = ∞,

for all f ∈ G \ {0}.

(9.81)

λ↑−1

or λ↓1

Then Q is a Herglotz-Nevanlinna function from the class N(G, [−1, 1]) and T = Aμ or T = AM .

286 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Proof. We have, (T − λI)−1 = (Re T − λI)−1/2 (I + iB)

−1

(Re T − λI)−1/2 , λ < −1,

where B = (Re T − λI)−1/2 Im T (Re T − λI)−1/2 , and −1

(T − λI)−1 = −(λI − Re T )−1/2 (I − iB)

(λI − Re T )−1/2 , λ > 1,

where B = (λI − Re T )−1/2 Im T (λI − Re T )−1/2 . This shows that for all f ∈ G and λ < −1, (QT (λ)f, f ) = (I + iB)−1 (Re T − λI)−1/2 f, (Re T − λI)−1/2 f . Since (I + iB)−1 ≤ 1, one obtains ( (2 ( ( |(QT (λ)f, f )| ≤ ((Re T − λI)−1/2 f ( . Now the assumption (9.80) implies that ( (2 ( ( lim inf ((Re T − λI)−1/2 f ( = ∞, λ↑−1

for all f ∈ G \ {0}.

˙ Thus Ran((I + Re T )1/2 ) ∩ G = {0} (see (9.3)). Since Re T is a sc-extension of A, one concludes that Re T = Aμ . Now one has Im T = 0 and T = Aμ . The proof of the other statement is similar.

9.4 Canonical L-systems with contractive state-space operators Let us consider a canonical L-system Θ of the type (5.6) with a bounded operator T acting on a Hilbert space H, T K J Θ= . (9.82) H E Here dim E = r < ∞, and we suppose that ker K = {0}. Then we have Im T = KJK ∗ and the transfer function of Θ is given by WΘ (z) = I − 2iK ∗ (T − zI)−1 KJ, while the impedance function of Θ is VΘ (z) = K ∗ (Re T − zI)−1 K = i(WΘ (z) + I)−1 (WΘ (z) − I)J.

(9.83)

Clearly, if T is a contraction, then the function VΘ (z) is holomorphic in Ext[−1, 1] and WΘ (z) is holomorphic outside of the closed unit disk.

9.4. Canonical L-systems with contractive state-space operators

287

Theorem 9.4.1. Let the state-space operator T of the L-system Θ of the form (9.82) be a contraction. Then the transfer function WΘ (z) and the impedance function VΘ (z) satisfy the following conditions: 1. the limit values VΘ (±1), WΘ (±1) exist; 2. the operators WΘ (±1) are J-unitary; 3. the operators VΘ (±1) are invertible; 4. the operator VΘ−1 (−1) − VΘ−1 (1) is invertible; 5. the operator KJ = [VΘ−1 (−1) − VΘ−1 (1)]−1/2 2iJ + [VΘ−1 (−1) + VΘ−1 (1)] × [VΘ−1 (−1) − VΘ−1 (1)]−1/2

(9.84)

is a contraction; 6. Re (i(WΘ (−1)JWΘ∗ (1) − J)) ≥ 0. Proof. Let T be a contraction in H. We set G = Ran(Im T ),

H0 = ker(Im T ),

˙ = T x for x ∈ Dom(A) ˙ = H0 . Clearly, the operator and introduce an operator Ax ˙ T is a qsc-extension of the non-densely-deﬁned symmetric contraction A. Using the deﬁnition of the Weyl-Titchmarsh function associated with T and Re T and the subspace G, we apply formula (9.70) and rewrite the functions WΘ (z) and VΘ (z) as follows: WΘ (z) = I − 2iK ∗ QT (z)KJ,

VΘ (z) = K ∗ QRe T (z)K.

From Proposition 9.3.3 and Corollary 9.3.4 we get that the limit values WΘ (±1), ˙ it takes the form (9.32) VΘ (±1) exist. Since T is a qsc-extension of A, A DA K ∗ H0 H0 T = : → . KDA −KAK∗ + DK∗ XDK∗ G G Then it follows from (9.79) that −1 −1 −1 WΘ (−1) = I − 2iK ∗ DK DK∗ KJ, ∗ (X + IG ) ∗ −1 −1 −1 WΘ (1) = I − 2iK DK∗ (X − IG ) DK∗ KJ, −1 −1 −1 VΘ (−1) = K ∗ DK DK∗ K, ∗ (IG + Re X) ∗ −1 −1 −1 VΘ (1) = K DK∗ (Re X − IG ) DK∗ K,

and

VΘ−1 (−1) = K −1 DK∗ (IG + Re X)DK∗ K ∗[−1] , VΘ−1 (1) = −K −1 DK∗ (IG − Re X)DK∗ K ∗[−1] ,

288 Chapter 9. Canonical L-systems with Contractive and Accretive Operators where K ∗[−1] ∈ [E, Ran(Im (T ))] is the Moore-Penrose inverse of K ∗ . Therefore 1 −1 2 ∗[−1] K −1 DK = VΘ (−1) − VΘ−1 (1) , ∗K 2 −1 1 −1 −1 −2 V (−1) − VΘ (1) = K ∗ DK ∗ K, 2 Θ 1 −1 K −1 DK∗ Re XDK∗ K ∗[−1] = VΘ (−1) + VΘ−1 (1) . 2 Since KJK ∗ = DK∗ Im XDK∗ , we have K −1 DK∗ Im XDK∗ K ∗[−1] = J. Hence, 1 −1 −1 −1 −1 X = Re X + iIm X = DK∗ K V (−1) + VΘ (1) + iJ K ∗ DK ∗. 2 Θ Furthermore, from 1/2 2 1 DK∗ K ∗[−1] f 2G = √ VΘ−1 (−1) − VΘ−1 (1) f E , 2

f ∈E

we get 1/2 1 DK∗ K ∗[−1] = √ U VΘ−1 (−1) − VΘ−1 (1) , 2

(9.85)

where U ∈ [E, G] and unitarily maps E onto G. Now −1 −1/2 1 −1 −1 −1 X = 2U VΘ (−1) − VΘ (1) V (−1) + VΘ (1) + iJ 2 Θ −1/2 ∗ × VΘ−1 (−1) − VΘ−1 (1) U = U KJ U ∗ . Since X ≤ 1, we get (9.84). From the relations (5.22) written for all real λ ∈ Ext[−1, 1] we obtain WΘ (λ)JWΘ∗ (λ) = J,

WΘ∗ (λ)JWΘ (λ) = J

and hence the operators WΘ (±1) are J-unitary. From the expressions for WΘ (±1) one can easily derive that −1 −1 −1 i(WΘ (−1)JWΘ∗ (1) − J) = 4K ∗ DK (IG − X ∗ )−1 DK ∗ (IG + X) ∗ K. 2 Since Re ((IG − X ∗ )(IG + X)) = DX , we have

Re (i(WΘ (−1)JWΘ∗ (1) − J)) ≥ 0. The next theorem is the converse statement.

9.4. Canonical L-systems with contractive state-space operators

289

Theorem 9.4.2. Let E be a ﬁnite-dimensional Hilbert space, J ∈ [E, E], J = J ∗ = J −1 and let an [E, E]-valued function W (z) belong to the class ΩJ . Suppose that the function V (z) = i(W (z) + I)−1 (W (z) − I)J, has a holomorphic continuation to a function from the class N(E, [−1, 1]) and ker Im V (z) = {0} for some non-real z. If the operator KJ = [V −1 (−1) − V −1 (1)]−1/2 2iJ + [V −1 (−1) − V −1 (1)] × [V −1 (−1) − V −1 (1)]−1/2 is a contraction, then there exists a minimal L-system Θ of the form (9.82) with contractive state-space operator T such that WΘ (z) = W (z). Moreover, T belongs to the class CH (α) for some α ∈ (0, π/2) if and only if the operator KJ ∈ CE (α). Proof. By Theorem 5.5.4 there exists a prime system of the form (9.82) whose transfer function WΘ (z) is equal to W (z) and V (z) = K ∗ (Re T − zI)−1 K for z ∈ Ext[−1, 1]. The operator Re T is a self-adjoint contraction. Set H0 = ker(K ∗ ), N = H H0 , and A˙ := T H0 . Then Re T is an sc-extension of A˙ and by Corollary 9.3.5, V (z) = K ∗ QRe T (z)K and there exist the limit values V (±1) of the function V . It follows from Corollary 9.3.4 that the operator V −1 (−1)−V −1 (1) is invertible. Represent Re T in the form (9.32) A D A K∗ H0 H0 Re T = : → , KDA −KAK∗ + DK∗ Y DK∗ N N where Y ∈ [N, N] is a self-adjoint contraction. Then the formulas V −1 (−1) = K −1 DK∗ (IN + Y )DK∗ K ∗−1 , V −1 (1) = −K −1 DK∗ (IN − Y )DK∗ K ∗−1 hold. Therefore, (9.85) holds with some unitary operator U ∈ [E, N]. Put −1 −1 −1 ∗ −1 Z = DK ∗ Im T DK∗ = DK∗ KJK DK∗

and let X = Y + iZ. Arguing in the reverse direction in the proof of Theorem 9.4.2 we get that the operator X is a contraction in N. It follows from Theorem ˙ Moreover, 9.2.3 that T is a qsc-extension of A. K ∈ CE (α) ⇐⇒ X ∈ CN (α) ⇐⇒ T ∈ CH (α).

Notice that the condition KJ ≤ 1 is equivalent to the condition Re (i(WΘ (−1)JWΘ∗ (1) − J)) ≥ 0, and the condition KJ ∈ CE (α) is equivalent to the condition of the operator i(WΘ (−1)JWΘ∗ (1) − J) being α-sectorial.

290 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Example. Let us consider an example that illustrates Theorem 9.4.2. Let α(x) be a non-decreasing function deﬁned on a closed interval [0, l]. We deﬁne an operator Tα by the formula l Tα f (x) = α(x)f (x) + i f (t) dt, x

where f (x) ∈

L2[0,l] ,

l > 0. It is easy to see that

l

1 Im Tα f = 2

f (t) dt = (f, g)g, 0

i Re Tα f = α(x)f (x) + 2

l

1 g(x) = √ 2

Θα =

Tα L2[0,l]

Kα

I C

, (9.86)

x

i f (t) dt − 2

f (t) dt. 0

x

Consider an L-system

,

where Kα c = cg, c ∈ C. Let us set α(x) ≡ 0 and let l T0 f (x) = i

f (t) dt,

T0∗ f (x)

x = −i

f (t) dt.

(9.87)

0

x

From (9.86), (9.87), Theorem 5.2.1, and the fact that a linear span of polynomials is dense in L2[0,l] we conclude that T0 is a prime operator. Thus, since J = 1, we can apply Theorem 9.4.2 and ﬁnd those values of l such that the number KJ in (9.84) does not exceed 1 in absolute value. Clearly (see [89], WΘ0 (z) = exp

il , z

VΘ0 (z) = i

WΘ0 (z) − 1 l = − tan . WΘ0 (z) + 1 2z

Note that the numbers l , (2n + 1)π

n∈Z

are the eigenvalues of the compact self-adjoint operator Re T0 [89]. Hence, Re T0 = l/π and Re T0 ≤ 1 if and only if l ≤ π. The operator Re T0 is a self-adjoint extension of a non-densely-deﬁned symmetric operator ⎧ ⎫ l l ⎨ ⎬ 2 ˙ ˙ Dom(A) = f (x) ∈ L[0,l] : f (x) dx = 0 , Af = i f (t) dt. (9.88) ⎩ ⎭ 0

x

9.4. Canonical L-systems with contractive state-space operators

291

Let l ≤ π. Then the operator A˙ is a contraction and Re T0 is its sc-extension. If l = π, then lim |VΘ0 (z)| = lim |VΘ0 (z)| = ∞. z↑−1

z↓1

According to Proposition 9.3.6 we get Aμ = AM = Re T0 . Thus in the case l = π the operator A˙ has a unique sc-extension. In particular, it follows that in this case T0 > 1. Let l < π. Then l l VΘ0 (1) = − tan , VΘ0 (−1) = tan . 2 2 We have V0−1 (−1) + V0−1 (1) = 0,

V0−1 (−1) − V0−1 (1) =

2 . tan 2l

It is worth mentioning that the Weyl-Titchmarsh function of Re T0 is 2 l QRe T0 (z) = − tan . l 2z Let Aμ and AM be extremal extensions of the operator A˙ given by (9.88). From relations (9.74) we get that Re T0 = and

Aμ + AM 2

l 1 1 l Aμ f = i f (t) dt − i+ f (t) dt, 2 tan 2 x 0 l l 1 1 l AM f = i f (t) dt − i− f (t) dt. 2 tan 2 x 0 l

For the operator KJ given by (9.84) we have KJ = [VΘ−1 (−1) − VΘ−1 (1)]−1/2 2iJ + [VΘ−1 (−1) + VΘ−1 (1)] 0 0 0 0 l × [VΘ−1 (−1) − VΘ−1 (1)]−1/2 = i tan . 0 0 2 Then KJ ≤ 1 ⇐⇒ l ≤

π . 2

Thus, the operator T0 is a contraction if and only if l ≤ π/2. By virtue of Theorem 9.4.2 the operator T0 on L2[0,l] is an α-co-sectorial if and only if 0 < l < π2 and in this case α = l.

292 Chapter 9. Canonical L-systems with Contractive and Accretive Operators

9.5 The restricted Phillips-Kato extension problem Let us remind readers that a symmetric operator B˙ is called non-negative if ˙ f ) ≥ 0, (Bf,

˙ ∀f ∈ Dom(B).

Let B˙ be a closed densely-deﬁned non-negative operator in a Hilbert space H and ˙ g), f, g ∈ let B˙ ∗ be its adjoint. Consider the sesquilinear form τB˙ [f, g] = (Bf, ˙ A sequence {fn } ⊂ Dom(B) ˙ is called τ ˙ -converging to the vector f ∈ H Dom(B). B if lim fn = f and lim τB˙ [fn − fm ] = 0. n→∞

n,m→∞

The form τB˙ is closable [163], i.e., there exists a minimal closed extension (the ˙ ·] the closure of closure) of τB˙ . Following the M. Kre˘ın notations we denote by B[·, ˙ ˙ ˙ ˙ Because τB˙ and by D[B] its domain. By deﬁnition B[u] = B[u, u] for all u ∈ D[B]. ˙ B[u, v] is closed, it possesses the property: if lim un = u

n→∞

and

˙ n − um ] = 0, lim B[u

n,m→∞

˙ − un ] = 0. then lim B[u n→∞

The Friedrichs extension BF of B˙ is deﬁned as a non-negative self-adjoint ˙ ·] by the First Representation Theorem operator associated with the form B[·, [163]: ˙ v] for all (BF u, v) = B[u,

u ∈ Dom(BF )

and for all

˙ v ∈ D[B].

It follows that ˙ ∩ Dom(B˙ ∗ ), BF = B˙ ∗ Dom(BF ). Dom(BF ) = D[B] The Friedrichs extension BF is a unique non-negative self-adjoint extension having ˙ Notice that by the Second Representation Theorem [163] one its domain in D[B]. has ˙ = D[BF ] = Dom(B 1/2 ), B[u, ˙ v] = (B 1/2 u, B 1/2 v), u, v ∈ D[B]. ˙ D[B] F F F Let B˙ be a non-negative closed densely-deﬁned symmetric operator. Consider the family of symmetric contractions ˙ ˙ −1 , a > 0, A˙ (a) = (aI − B)(aI + B) ˙ ˙ Notice that the orthogonal complement deﬁned on Dom(A˙ (a) ) = (aI + B)Dom( B). ˙ N(a) = H Dom(A˙ (a) ) coincides with the defect subspace N−a of the operator B. Let A˙ = A˙ (1) and let b = (1 − a)(a + 1)−1 . Then b ∈ (−1, 1) and (see (9.58)) ˙ −1 = wb (A). ˙ A˙ (a) = (A˙ − bIH )(I − bA)

9.5. The restricted Phillips-Kato extension problem

293

Clearly, there is a one-one correspondence given by the Cayley transform B = a(I − A(a) )(I + A(a) )−1 ,

A(a) = (aI − B)(aI + B)−1 ,

between all non-negative self-adjoint extensions B of the operator B˙ and all scextensions A(a) of A˙ (a) . The next result describe the sesquilinear form B[u, v] by means of the fractional-linear transformation A = (I − B)(I + B)−1 . Proposition 9.5.1. (1) Let B be a non-negative self-adjoint operator and let A = (I − B)(I + B)−1 be its Cayley transform. Then D[B] = Ran((I + A)1/2 ), B[u, v] = −(u, v) + 2 (I + A)−1/2 u, (I + A)−1/2 v ,

u, v ∈ D[B]. (9.89)

(2) Let B˙ be a closed densely-deﬁned non-negative symmetric operator and let ˙ ˙ −1 , A = B be its non-negative self-adjoint extension. If A˙ = (I − B)(I + B) −1 (I − B)(I + B) , then D[B] = Ran(I + Aμ )1/2 Ran(A − Aμ )1/2 .

(9.90)

Proof. (1). Since B = (I − A)(I + A)−1 , one obtains with f = (I + A)h, B[f ] = ((I − A)h, (I + A)h) = −(I + A)h2 + 2(I + A)1/2 h2 = −f 2 + 2(I + A)−1/2 f 2 . Now the closure procedure leads to (9.89). ˙ we get Aμ ≤ A ≤ AM . Hence I + A = (2) Since A is an sc-extension of A, I + Aμ + (A − Aμ ). Because I + Aμ and A − Aμ are non-negative self-adjoint operators, we get the equality [123]: Ran((I + A)1/2 ) = Ran((I + Aμ )1/2 ) + Ran((A − Aμ )1/2 ). ˙ and Ran(A−Aμ ) ⊆ N, Since Ran((I +Aμ )1/2 )∩N = {0}, where N = HDom(A), we get Ran((I + Aμ )1/2 ) ∩ Ran((A − Aμ )1/2 ) = {0}. Then we arrive at (9.90). We note that Ran(B 1/2 ) = Ran((I −A)1/2 ). Now let Aμ and AM be the rigid ˙ Then the operators and the soft extensions of A.

and

BF = (I − Aμ )(I + Aμ )−1 ,

(9.91)

BK = (I − AM )(I + AM )−1 ,

(9.92)

˙ It also follows from Proposition are non-negative self-adjoint extensions of B. 9.2.17 that (a) −1 BF = a(I − A(a) , μ )(I + Aμ )

(a)

(a)

BK = a(I − AM )(I + AM )−1 .

294 Chapter 9. Canonical L-systems with Contractive and Accretive Operators (a)

(a)

Since, the operators Aμ and AM possess the properties (a)

1/2 Ran((I + A(a) ) ∩ N−a = Ran((I − AM )1/2 ) ∩ N−a = {0}, μ )

we get the following result. Proposition 9.5.2. Let B be a non-negative self-adjoint extension of B˙ and let E(λ) be its resolution of identity. Then 1. B = BF if and only if at least for one a > 0 (then for all a > 0) the relation ∞ λ(dE(λ)ϕ, ϕ) = +∞,

(9.93)

0

holds for each ϕ ∈ N−a \ {0}; 2. B = BK if and only if at least for one a > 0 (then for all a > 0) the relation ∞ 0

(dE(λ)ϕ, ϕ) = +∞, λ

(9.94)

holds for each ϕ ∈ N−a \ {0}. Theorem 9.5.3. For a > 0, 8 9 inf (BF (h − ψ), h − ψ) + ah − ψ2 = 0, ∀h ∈ Dom(BF ), ˙ ψ∈Dom(B) 8 9 inf BK (g − ψ)2 + a(BK (g − ψ), g − ψ) = 0, ∀g ∈ Dom(BK ). ˙ ψ∈Dom(B)

Proof. By (9.91) we have (a)

(a)

Dom(BF ) " h = (I + Aμ )f, BF h = a(I − Aμ )f, (a) (a) Dom(BK ) " g = (I + AM )f, BK g = a(I − AM )f, f ∈ H. Then (BF (h − ψ), h − ψ) + ah − ψ2 = 2a((I + Aμ )(f − ϕ), f − ϕ), BK (g − ψ)2 + a(BK (g − ψ), g − ψ) = 2a2 ((I − AM )(f − ϕ), f − ϕ), ˙ where ψ = (I + A)ϕ, ϕ ∈ Dom(A˙ (a) ). The statement follows from Propositions 9.2.5 and 9.2.17. Thus, the self-adjoint extension BF given by (9.91) coincides with the Fried˙ In the sequel we will call the operator BK deﬁned in (9.92) richs extension of B. ˙ As we already know (Theorem 9.2.4), the the Kre˘ın-von Neumann extension of B. set of all sc-extensions of a symmetric contraction A˙ forms the operator interval [Aμ , AM ]. This result yields the following theorem established by M. Kre˘in [172].

9.5. The restricted Phillips-Kato extension problem

295

Theorem 9.5.4. The following conditions are equivalent: ˙ (i) the non-negative self-adjoint operator B is the extension of B, (ii)

(BF + aI)−1 ≤ (B + aI)−1 ≤ (BK + aI)−1 , for some (then for all) positive numbers a,

(iii) BF ≤ B ≤ BK in the sense of quadratic forms, i.e., ˙ ⊆ D[B] ⊆ D[BK ], B[u] ≥ BK [u] D[B] ˙ ˙ B[v] = B[v] for all v ∈ D[B].

for all

u ∈ D[B],

The operator B˙ admits a unique non-negative self-adjoint extension if and only if ˙ v) (Bv, = 0, ˙ |(v, ϕ−a )|2 v∈Dom(B) inf

(9.95)

˙ where a > 0. for all non-zero vectors ϕ−a from the defect subspace N−a of B, Proof. Let B be a non-negative self-adjoint extension of B˙ and let a > 0. Then A(a) = (aI − B)(aI + B)−1 = −I + 2a(aI + B)−1 is an sc-extension of symmetric ˙ ˙ −1 . It follows that contraction A˙ (a) = (aI − B)(aI + B) (a)

(a) A(a) ≤ AM ⇐⇒ (BF + aI)−1 ≤ (B + aI)−1 ≤ (BK + aI)−1 . μ ≤A

Suppose B is a non-negative self-adjoint operator in H such that (BF + aI)−1 ≤ (B + aI)−1 ≤ (BK + aI)−1 , for some a > 0. Since (BF + aI)−1 f = (BK + aI)−1 f = (B˙ + aI)−1 f, ˙ the latter inequalities yield that B is an extension of for all f ∈ (B˙ + aI)Dom(B), ˙ B. By Theorem 9.2.13 the operator A˙ (a) admits a unique sc-extension if and only if , ,2 , ˙ (a) , ,(A ϕ, h), sup = ∞ for all h ∈ H Dom(A˙ (a) ), h = 0. 2 ϕ∈Dom(A˙ (a) ) DA˙ (a) ϕ ˙ Because H Dom(A˙ (a) ) = N−a (the defect subspace of B), ˙ v), DA˙ (a) ϕ2 = (Bv,

˙ ϕ = (aI + B)v,

˙ and (A˙ (a) ϕ, ϕ−a ) = ((aI − B)v, ϕ−a ) = 2a(v, ϕ−a ) for all ϕ−a ∈ N−a , we get ˙ that B admits a unique non-negative self-adjoint extension if and only if condition (9.95) is satisﬁed.

296 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Let B be a non-negative self-adjoint extension of B˙ and let Aμ = (I − BF )(I + BF )−1 , A = (I − B)(I + B)−1 .

AM = (I − BK )(I + BK )−1 ,

Then I + Aμ ≤ I + A ≤ I + AM . It follows from Theorem 2.1.2 that Ran((I + Aμ )1/2 ) ⊆ Ran((I + A)1/2 ) ⊆ Ran((I + AM )1/2 ), and

(I + AM )−1/2 f 2 ≤ (I + A)−1/2 f 2 , f ∈ Ran((I + A)1/2 ), (I + A)−1/2 g2 ≤ (I + Aμ )−1/2 g2 , g ∈ Ran((I + Aμ )1/2 ).

˙ = D[BF ] ⊆ D[B] ⊆ D[BK ], and By Proposition 9.5.1 we get D[B] B[u] ≥ BK [u] for all u ∈ D[B], ˙ ˙ B[v] ≤ B[v] for all v ∈ D[B]. ˙ In addition, using the ﬁrst equality from (9.49), we obtain B[v] = B[v] for all ˙ v ∈ D[B]. Remark 9.5.5. Let C1 and C2 be two non-negative self-adjoint operators. It is well known (see [163]) that the following conditions are equivalent (i) C1 ≤ C2 in the sense of quadratic forms; (ii) (C1 + aI)−1 ≥ (C2 + aI)−1 for some (then for all) positive number a. Remark 9.5.6. It follows from (9.3) and the deﬁnition of the Friedrichs extension that ˙ v) (Bv, 1/2 inf = 0 ⇐⇒ ϕ−a ∈ / Ran(BF ). 2 ˙ |(v, ϕ )| v∈Dom(B) −a Therefore, B˙ admits a unique non-negative self-adjoint extension if and only if 1/2 Ran(BF ) ∩ N−a = {0} at least for one (then for all) a > 0. From Proposition 9.5.1 and relations (9.89), (9.90) we obtain (a)

D[BF ] = Ran((I + Aμ )1/2 ), (a) (a) (a) D[BK ] = Ran((I + Aμ )1/2 ) Ran((AM − Aμ )1/2 ), a > 0.

(9.96)

(a)

Moreover, since Ran((A(a) − Aμ )1/2 ) ⊆ N−a , the direct decomposition ˙ (N−a ∩ D[B]) , D[B] = D[BF ]+

(9.97)

˙ holds for an arbitrary non-negative self-adjoint extension B of B. Using Proposition 9.5.1 and Theorem 9.5.3 we obtain the following theorem by T. Ando and K. Nishio [16].

9.5. The restricted Phillips-Kato extension problem

297

Theorem 9.5.7. The following relations describing D[BK ] and BK [u] hold: ˙ u)|2 |(Bf, D[BK ] = u ∈ H : sup <∞ , ˙ f) (Bf, ˙ f ∈Dom(B) (9.98) ˙ u)|2 |(Bf, BK [u] = sup , u ∈ D[BK ]. ˙ f) (Bf, ˙ f ∈Dom(B) 1/2 ˙ one has Proof. Let u ∈ D[BK ] = Dom(BK ). Then for all f ∈ Dom(B) 1/2

1/2

˙ u)|2 |(Bf, |(BK f, u)|2 |(BK f, BK u)|2 = = . 1/2 1/2 ˙ f) (Bf, BK f 2 BK f 2 1/2 ˙ is a dense set in Ran(B 1/2 ). From Theorem 9.5.3 we obtain that BK Dom(B) K Hence, 1/2 1/2 |(BK f, BK u)|2 1/2 sup = BK u2 = BK [u]. 1/2 ˙ BK f 2 f ∈Dom(B)

We have proved that D[BK ] ⊂ BK [u]. In order to prove the inverse inclusion we use the Cayley transform ˙ ˙ −1 , A˙ = (I − B)(I + B) Let u ∈ H and let

AM = (I − BK )(I + BK )−1 .

˙ u)|2 |(Bf, < ∞. ˙ f) (Bf, ˙ f ∈Dom(B) sup

This is equivalent to ˙ |((I − A)ϕ, u)|2 < ∞, DA˙ ϕ2 ˙ ϕ∈Dom(A) sup

˙ ˙ where c > 0. Since i.e., |((I − A)ϕ, u)|2 ≤ cDA˙ ϕ2 for all ϕ ∈ Dom(A), ˙ = (I − AM )1/2 (I − AM )1/2 ϕ, (I − A)ϕ DA˙ ϕ2 = DAM ϕ2 = (I + AM )1/2 (I − AM )1/2 ϕ2 , ˙ is dense in Ran((I − AM )1/2 ) (see Proposition 9.2.5), we and (I − AM )1/2 Dom(A) have |((I − AM )h, u)|2 ≤ cDAM h2 , for all h ∈ H. It follows that for h ∈ H, |((I − AM )1/2 h, (I − AM )1/2 u)|2 ≤ c(I + AM )1/2 (I − AM )1/2 h2 ⇐⇒ |(g, (I − AM )1/2 u)|2 ≤ c(I + AM )1/2 g2 , g ∈ H.

298 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Now from (9.3) we obtain (I − AM )1/2 u ∈ Ran((I + AM )1/2 ). Since Ran((I − AM )1/2 ∩ Ran((I + AM )1/2 ) = Ran(DAM ), we get u ∈ Ran((I + AM )1/2 ). By Proposition 9.5.1, D[BK ] = Ran((I + AM )1/2 ).

This completes the proof.

˙ v], u, v ∈ D[B] ˙ and the Second RepreRemark 9.5.8. The equality B[u, v] = B[u, sentation Theorem yield 1/2

||B 1/2 u|| = ||BF u||,

1/2

˙ u ∈ Dom(BF ) = D[B].

˙ and let f ∈ D[B] ∩ Dom(B˙ ∗ ). Then Let u˙ ∈ D[B] (B 1/2 u, B 1/2 f ) = B[u, ˙ f ] = (B˙ u, ˙ f ) = (u, ˙ B˙ ∗ f ). ˙ there exists a sequence {u˙ n } such that On the other hand for each u ∈ D[B] 1/2

1/2

lim u˙ n = u, lim BF u˙ n = BF u.

n→∞

n→∞

1/2

Since B 1/2 is closed and Dom(B 1/2 ) ⊃ Dom(BF ), we obtain lim B 1/2 u˙ n = B 1/2 u.

n→∞

Therefore for any non-negative self-adjoint extension B of B˙ one has ˙ f ∈ D[B] ∩ Dom(B˙ ∗ ). B[u, f ] = (u, B˙ ∗ f ), u ∈ D[B],

(9.99)

Example. Let H = L2 (R2 ). Consider a self-adjoint and non-negative operator ˆ ˆ = fˆ(p) ∈ L2 (R2 ) : |p|2 fˆ(p) ∈ L2 (R2 ) , (Aˆf)(p) = |p|2 fˆ(p), Dom(A) + Here and below p = (p1 , p1 ), dp = dp1 dp2 , |p| = p21 + p22 . Clearly, (Aˆ1/2 fˆ)(p) = |p|fˆ(p), fˆ ∈ Dom(Aˆ1/2 ) = fˆ(p) ∈ L2 (R2 ) : |p|fˆ(p) ∈ L2 (R2 ) . Let a symmetric operator Aˆ0 be given by (Aˆ0 fˆ)(p) = |p|2 fˆ(p),

ˆ : Dom(Aˆ0 ) = {fˆ ∈ Dom(A)

The deﬁciency indices of Aˆ0 are (1, 1) and

4 c Nλ = , c ∈ C , |p|2 − λ

fˆ(p)dp = 0}. R2

λ ∈ C \ R+

9.5. The restricted Phillips-Kato extension problem

299

is a defect subspace of Aˆ0 . Obviously, N−1 ∩ Dom(Aˆ1/2 ) = {0}. Hence, from (9.97) it follows that Aˆ is the Friedrichs extension of Aˆ0 . Moreover, since 1 ∈ / L2 (R2 ), |p|(|p|2 + 1) we get N−1 ∩ Ran(Aˆ1/2 ) = {0}. Remark 9.5.6 yields that Aˆ is the unique nonnegative self-adjoint extension of Aˆ0 . The application of the Fourier transform 1 ˆ Ff (x) := f (p) = f (x) exp(−ip · x) dx, 2π R2 to Aˆ0 and Aˆ leads to the operators A0 and A in L2 (R2 , dx), x = (x1 , x2 ): A0 = F −1 Aˆ0 F,

ˆ A = F −1 AF,

Dom(A0 ) = {f ∈ W22 (R2 ), f (0) = 0}, A0 f = −Δf, f ∈ Dom(A0 ), and

Dom(A) = W22 (R2 ),

Af = −Δf,

(9.100)

f ∈ Dom(A),

where W22 (R2 ) is the Sobolev space and Δ is the Laplacian. Since F is a unitary operator, the operator A0 has a unique non-negative self-adjoint extension A. Theorem 9.5.9. The Friedrichs and Kre˘ın-von Neumann extensions of B˙ are transversal if and only if Dom(B˙ ∗ ) ⊆ D[BK ]. (9.101) ˙ Assume condition (9.101). Then Nλ ⊂ Proof. Let Nλ be the defect subspace of B. D[BK ] for all λ ∈ C \ [0, +∞). In particular N−1 ⊂ D[BK ]. It follows from (9.96) that Ran((AM − Aμ )1/2 ) = N−1 . Hence, also Ran(AM − Aμ ) = N−1 . Because 2(BF + I)−1 = I + Aμ , 2(BK + I)−1 = I + AM , we have

1 (BK + I)−1 − (BF + I)−1 = (AM − Aμ ). 2 −1 −1 Therefore Ran (BK + I) − (BF + I) = N−1 . Now Proposition 3.4.8 yields that BF and BK are transversal. Conversely, suppose that BF and BK are transversal, i.e., that Dom(BF ) + Dom(BK ) = Dom(B˙ ∗ ). Since ˙ ⊆ D[BK ] Dom(BF ) ⊂ D[BF ] = D[B] and Dom(BK ) ⊂ D[BK ], we obtain Dom(B˙ ∗ ) ⊆ D[BK ].

300 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Now we consider the case of bounded non-densely-deﬁned non-negative sym˙ metric operator B. Theorem 9.5.10. Let B˙ be a bounded, non-densely-deﬁned, non-negative, and sym˙ = H0 . Let B˙ ∗ ∈ [H, H0 ] be the admetric operator in a Hilbert space H, Dom(B) ˙ Put B˙ 0 = PH0 B, ˙ L = H H0 , where PH0 is an orthogonal projection joint of B. in H onto H0 . Then the following statements are equivalent: (i) B˙ admits bounded non-negative self-adjoint extensions in H; (ii) sup

f ∈H0

˙ ||2 ||Bf < ∞; ˙ f) (Bf,

1/2 (iii) B˙ ∗ L ⊆ Ran(B˙ 0 ). 1/2

˙ f ) = ||B˙ f ||2 , f ∈ H0 , and Proof. Since (Bf, 0 B˙ ∗ = B˙ 0 PH0 + B˙ ∗ PL , conditions (i) and (ii) are equivalent due to Theorem 2.1.2. Suppose B˙ admits a bounded non-negative self-adjoint extension B. Then for f ∈ H0 one has ˙ ||2 = ||Bf ||2 = ||B 1/2 B 1/2 f ||2 ≤ ||B 1/2 ||2 ||B 1/2 f ||2 ||Bf ˙ f ) = ||B 1/2 ||2 ||B˙ 1/2 f ||2 . = ||B 1/2 ||2 (Bf, f ) = ||B 1/2 ||2 (Bf, 0 It follows that statement (ii) holds true. [−1/2] ˙ ∗ Now suppose that (iii) is fulﬁlled. Then the operator L0 := B˙ 0 B L is [−1/2] 1/2 ∗ ˙ ˙ bounded, where B0 is the Moore-Penrose inverse to B0 . Let L0 ∈ [H0 , L] be the adjoint to L0 . Set ˙ H0 + (B˙ ∗ + L∗0 L0 )PL . B0 = BP

(9.102)

Then B0 is a bounded extension of B˙ in H. Let PL be the orthogonal projection operator in H onto L. For h ∈ H we have ˙ H h + (B˙ ∗ + L∗ L0 )PL h, PH h + PL h) (B0 h, h) = (BP 0 0 0 1/2 = ||B˙ 0 PH0 h||2 + ||L0 PL h||2 + 2Re (PH0 h, B˙ ∗ PL h) 1/2 1/2 [−1/2] ˙ ∗ = ||B˙ 0 PH0 h||2 + ||L0 PL h||2 + 2Re (B˙ 0 PH0 h, B˙ 0 B PL h) 1/2 2 = ||B˙ 0 PH0 h + L0 PL h|| . ˙ Therefore (i) is Thus, B0 is a non-negative bounded self-adjoint extension of B. equivalent to (iii). Remark 9.5.11. 1) It is easy to see that the conditions 1. sup

f ∈H0

˙ ||2 ||Bf < ∞, ˙ f) (Bf,

9.5. The restricted Phillips-Kato extension problem

301

2. there exists c > 0 such that ˙ g)|2 ≤ c(Bf, ˙ f )||g||2 , f ∈ H0 , g ∈ H, |(Bf, 3. there exists c > 0 such that ˙ g)|2 ≤ c(Bf, ˙ f )||g||2 , f ∈ H0 , g ∈ L |(Bf, are equivalent. 2) If B0 is given by (9.102) and W ∈ [L, L], W ≥ 0, then the operator B = B0 + W PL , ˙ is also a non-negative bounded self-adjoint extension of B. A linear operator T on a Hilbert space H is called accretive if Re (T f, f ) ≥ 0,

f ∈ Dom(T ).

An accretive operator is called maximal accretive or m-accretive if it does not admit accretive extensions in H. The following conditions for accretive operator T are equivalent: • the operator T is m-accretive; • the resolvent set ρ(T ) contains a point from the open left half-plane; • the operator T is densely deﬁned and closed, and T ∗ is an accretive operator. The resolvent set ρ(T ) of m-accretive operators contains the open left half-plane and 1 (T − λI)−1 ≤ , Re λ < 0. |Re λ| It is well known [163] that if T is an m-accretive operator, then the one-parameter semigroup U (t) = exp(−tT ), t ≥ 0, is contractive. Conversely, if the family {U (t)}t≥0 is a strongly continuous semigroup of bounded operators in a Hilbert space H, with U (0) = I (C0 -semigroup) and U (t) is a contraction for each t, then the generator T of U (t): T u := lim

t→+0

(I − U (t))u , t

u ∈ Dom(T ),

where domain Dom(T ) is deﬁned by condition: (I − U (t))u Dom(T ) = u ∈ H : lim t→+0 t

exists

,

302 Chapter 9. Canonical L-systems with Contractive and Accretive Operators is an m-accretive operator in H. Then the approximation by the Euler formula −n t U (t) = s − lim I + T , t ≥ 0, n→∞ n (see [163]) holds in the strong operator topology. The formulas S = (I − T )(I + T )−1 ,

T = (I − S)(I + S)−1 ,

establish a bijective correspondence between all m-accretive operators T and all contractions S such that ker(I + S) = {0}. Indeed, if f = (I + T )−1 φ, then Sφ = f − T f, φ = f + T f, and ||φ||2 − ||Sφ||2 = 4Re (T f, f ).

(9.103)

We call an accretive operator T α-sectorial if there exists a value of α ∈ (0, π/2) such that |Im (T f, f )| ≤ tan α Re (T f, f ),

f ∈ Dom(T ).

(9.104)

It is known [163] that if T is an m-accretive and α-sectorial operator, then the operator (−T ) is a generator of a semi-group of contractions holomorphic in the sector | arg ζ| < π/2 − α, i.e., the Cauchy problem ⎧ ⎨ dx + T x = 0, (9.105) dt ⎩ x(0) = 0, x0 ∈ Dom(T ), generates a contractive semi-group U (t) = exp(−tT ), t ≥ 0 that can be analytically extended as a semi-group of contractions holomorphic in the sector | arg ζ| < π/2 − α. For the remainder of this chapter we will mostly consider m-accretive and both m-accretive and α-sectorial operators (m-α-sectorial). Lemma 9.5.12. Let T be a densely-deﬁned m-accretive operator on a Hilbert space H. Then the operator S = (I − T )(I + T )−1 , belongs to the class CH (α) (is an α-co-sectorial contraction) if and only if T is an α-sectorial operator. Proof. Let f = (I + T )−1 φ, then Sφ = f − T f,

φ = f + T f.

Hence, f=

1 (I + S)φ, 2

Tf =

1 (I − S)φ. 2

9.5. The restricted Phillips-Kato extension problem

303

Moreover, (T f, f ) =

1 1 1 ∗ (I − S)φ, (I + S)φ = (I + S )(I − S)φ, φ = (F φ, φ), 2 2 4

where F = 14 (I + S ∗ )(I − S). Let us ﬁnd Re F and Im F . Using the deﬁnition of operator S we have 5 6 1 1 1 1 ∗ ∗ Re F = (I + S )(I − S) + (I − S )(I + S) = (I − S ∗ S), 2 4 4 4 and

5 6 1 1 1 1 S − S∗ ∗ ∗ Im F = (I + S )(I − S) − (I − S )(I + S) = . 2i 4 4 2 2i

Now let cot α |Im (T f, f )| ≤ Re (T f, f ),

f ∈ Dom(T ),

which would imply that cot α |2(Im (Sφ, φ)| ≤ ((I − S ∗ S)φ, φ), which proves the lemma.

Let B˙ be a closed, densely-deﬁned non-negative symmetric operator in Hilbert space H. The Phillips-Kato extension problem in the restricted sense consists of existence and description of m-accretive and α-sectorial extensions T of B˙ such that B˙ ⊂ T ⊂ B˙ ∗ . (9.106) Obviously, the operators T satisfying (9.106) are quasi-self-adjoint extensions of ˙ One can easily show that the fractional-linear transformasymmetric operator B. tion B = (I − A)(I + A)−1 , A = (I − B)(I + B)−1 , establishes a one-to-one correspondence between all quasi-self-adjoint m-accretive ˙ ˙ −1 . extensions B of B˙ and all qsc-extensions of A˙ = (I − B)(I + B) Theorem 9.5.13. Let B˙ be a densely-deﬁned, closed, and non-negative operator. The following conditions are equivalent: (i) BF = BK , (ii) B˙ admits a non-self-adjoint quasi-self-adjoint m-accretive extension, (iii) admits a non-self-adjoint quasi-self-adjoint m-α-sectorial extension for any α ∈ (0, π/2). Proof. It follows from (9.91) and (9.92) that BF = BK , and hence AF = AK . The proof of the theorem then follows from Corollaries 9.2.14 and 9.2.15.

304 Chapter 9. Canonical L-systems with Contractive and Accretive Operators The next theorem gives characterizations of quasi-self-adjoint m-accretive extensions. Theorem 9.5.14. Let B˙ be a non-negative densely-deﬁned symmetric operator and ˙ The following conditions are equivalet B be a maximal accretive extension of B. lent: (1) B ⊂ B˙ ∗ ; (2) Dom(B) ⊂ D[BK ] and Re (Bf, f ) ≥ BK [f ] for all f ∈ Dom(B); ˙ f )|2 ≤ (Bg, ˙ g) Re (Bf, f ) for all f ∈ Dom(B), g ∈ Dom(B). ˙ (3) |(Bg, The extension B is quasi-self-adjoint and m-α-sectorial if and only if 1/2

Dom(B) ⊂ Dom(BK ) and the sesquilinear form ω[f, h] := (Bf, h) − BK [f, h], f, h ∈ Dom(B),

(9.107)

is sectorial with the same semi-angle α and the vertex at the origin. ˙ then for all g ∈ Dom(B), ˙ for all f ∈ Proof. If B is an accretive extension of B, Dom(B), and for all t ∈ R, it follows that 0 ≤ Re (B(tg + f ), tf + g) ˙ g) + t Re (Bg, ˙ f ) + Re (Bf, g) + Re (Bf, f ). = t2 (Bg, ˙ f ) = (g, Bf ). Hence If in addition B ⊂ B˙ ∗ , then Dom(B) ⊂ Dom(B˙ ∗ ) and (Bg, ˙ g) + 2tRe (Bg, ˙ f ) + Re (Bf, f ) ≥ 0, t2 (Bg, ˙ f )|2 ≤ (Bg, ˙ g)Re (Bf, f ), and therefore for all t ∈ R. Now we get |Re (Bg, ˙ f )|2 ≤ (Bg, ˙ g)Re (Bf, f ), |(Bg, ˙ and all f ∈ Dom(B), i.e., (1) ⇒ (3). The equivalence (3) ⇐⇒ for all g ∈ Dom(B) (2) follows from (9.98). ˙ ˙ −1 and A = (I−B)(I+ Let us show that (3) implies (1). Let A˙ = (I−B)(I+ B) −1 ˙ = (I + B)Dom( ˙ ˙ B) . Then A˙ is a symmetric contraction deﬁned on Dom(A) B) and A is a contractive extension of A˙ deﬁned on H. Then the inequality in (3) can be rewritten as ˙ ˙ ˙ |((I − A)ϕ, (I + A)h)|2 ≤ ((I − A)ϕ, (I + A)ϕ)Re ((I − A)h, (I + A)h), ˙ and all h ∈ H. Using the relation Aϕ ˙ = Aϕ and simplifying for all ϕ ∈ Dom(A) we obtain 2 |(DA ϕ − 2iIm Aϕ, h)|2 ≤ DA ϕ2 DA h2

˙ for all ϕ ∈ Dom(A),

9.5. The restricted Phillips-Kato extension problem

305

2 and all h ∈ H. From (9.3) we obtain that DA ϕ − 2iIm Aϕ ∈ Ran(DA ) for all ˙ ϕ ∈ Dom(A) and [−1] 2 ˙ DA (DA ϕ − 2iIm Aϕ)2 ≤ DA ϕ2 , ϕ ∈ Dom(A), [−1]

where DA is the the Moore-Penrose inverse of DA described in Lemma 9.1.1. 2 Since DA ϕ ∈ Ran(DA ), we get Im Aϕ ∈ R(A) and [−1]

DA ϕ − 2iDA Im Aϕ2 ≤ DA ϕ2 , Hence

˙ ϕ ∈ Dom(A).

[−1]

DA ϕ2 + 4DA Im Aϕ2 ≤ DA ϕ2 .

˙ This means that A∗ ⊃ A, ˙ i.e., A is It follows that Im Aϕ = 0 for all ϕ ∈ Dom(A). ˙ Therefore, B is a quasi-self-adjoint extension of B. ˙ a qsc-extension of A. Suppose that the sesquilinear form ω given by (9.107) is α-sectorial with vertex at the origin. Then Re (Bf, f ) ≥ BK [f ] for all f ∈ Dom(B). Therefore B ˙ On the other hand is a maximal accretive and quasi-self-adjoint extension of B. for all f ∈ Dom(B) we have tan αRe (Bf, f ) ± Im (Bf, f ) = tan αRe ω[f ] ± Im ω[f ] + BK [f ] ≥ 0. ˙ Hence B is an m-α-sectorial extension of B. ˙ Conversely, let B be quasi-self-adjoint and an m-α-sectorial extension of B. ˙ ⊂ Dom(BK ) and all f ∈ Hence Dom(B) ⊂ D[BK ]. Since for each ϕ ∈ Dom(B) ˙ ˙ f ), we get D[BK ] one has BK [f, ϕ] = (f, Bϕ) and BK [ϕ, f ] = (Bϕ, ω[f − ϕ] = ω[f ], ˙ Because for all f ∈ Dom(B) and all ϕ ∈ Dom(B). inf

˙ ϕ∈Dom(B)

BK [f − ϕ] = 0,

for all f ∈ D[BK ], for given f ∈ D[BK ] and for every ε > 0 one can ﬁnd ϕ0 ∈ ˙ such that Dom(B) BK [f − ϕ0 ] < ε. It follows that (tan α)Re ω[f ] ± Im ω[f ] = (tan α)Re ω[f − ϕ0 ] ± Im ω[f − ϕ0 ] = (tan α)Re (B(f − ϕ0 , f − ϕ0 ) ± Im (B(f − ϕ0 , f − ϕ0 ) − BK [f − ϕ0 ] > −ε. Since ε is an arbitrary positive number, the form ω is with the semi-angle α and vertex at the origin.

306 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Deﬁnition 9.5.15. Let B˙ be a closed densely-deﬁned non-negative operator. A quasi-self-adjoint m-accretive extension B of B˙ is called extremal if inf

˙ ϕ∈Dom(B)

Re (B(f − ϕ), f − ϕ) = 0 for all f ∈ Dom(B).

From Proposition 9.2.9, Remark 9.2.10, and relations (9.44), (9.103) we immediately obtain the following statement. Proposition 9.5.16. A quasi-self-adjoint m-accretive extension B of a denselydeﬁned symmetric operator B˙ is extremal if and only if its Cayley transform T = (I − B)(I + B)−1 has the property (DT2 )N = 0, where N = N−1 is the ˙ Thus, if B is an extremal extension of B, ˙ then T is an defect subspace of B. ˙ ˙ −1 (see Deﬁnition 9.2.8). extremal qsc-extension of A˙ = (I − B)(I + B) If the deﬁciency number of B˙ is ﬁnite, then the following statements are equivalent: ˙ (i) B is a quasi-self-adjoint m-accretive extremal extension of B, ˙ (ii) B ∗ is a quasi-self-adjoint m-accretive extremal extension of B, ˙ (iii) T is an extremal qsc-extension of A, ˙ (iv) T ∗ is an extremal qsc-extension of A. Remark 9.5.17. If T is a quasi-self-adjoint and extremal m-accretive extension such that T = T ∗ , then T is not sectorial. Now we establish limiting properties of the Kre˘ın-Langer Q-functions corresponding to the Friedrichs and Kre˘ın non-negative self-adjoint extensions BF and ˙ We will use PropoBK of a non-negative densely-deﬁned symmetric operator B. sition 9.3.6. Let B be a non-negative self-adjoint extension of B˙ and let A be its fractional-linear transformation A = (I − B)(I + B)−1 . Then the resolvents of B and A are given by (B − λI)−1 = −

−1 1 1−λ (I + A) A − I , 1+λ 1+λ

λ ∈ ρ(B).

It follows that I + (λ + 1)(B − λI)−1 = −

2 1+λ

−1 1−λ A− I . 1+λ

Observe that A is an sc-extension of a non-densely-deﬁned symmetric contraction ˙ ˙ −1 . A˙ = (I − B)(I + B)

9.5. The restricted Phillips-Kato extension problem

307

Furthermore, suppose Γ(λ) ∈ [N , H] is the γ-ﬁeld corresponding to B and Q(λ) is the Kre˘ın-Langer Q-function of B˙ corresponding to the γ-ﬁeld (see Deﬁnition 7.4.2 and Deﬁnition 7.5.1). Let z0 = −1. Then from (7.54) we obtain −1 1−λ Q(λ) = C − 2X0∗ A − I X0 , 1+λ ˙ Note that where X0 = Γ(−1) ∈ [N , N−1 ], N−1 is the deﬁciency subspace of B. ˙ N−1 = H Dom(A). Hence 1−λ ∗ Q(λ) = C − 2X0 QA X0 , (9.108) 1+λ where QA is deﬁned by (9.70). Theorem 9.5.18. Let B be a non-negative self-adjoint extension of a non-negative ˙ Then: densely-deﬁned symmetric operator B. 1) The operator B coincides with the Friedrichs extension of B˙ if and only if lim (Q(x)f, f )N = −∞,

x↓−∞

∀f ∈ N .

(9.109)

2) The operator B coincides with the Kre˘ın extension of B˙ if and only if lim (Q(x)f, f )N = +∞,

x↑−0

∀f ∈ N .

(9.110)

Proof. Let B = BF . Then A = Aμ . Since Ran((I + Aμ )1/2 ) ∩ N−1 = {0}, from Proposition 9.2.5, equalities (9.9), (9.10), and (9.4), we get lim ((Aμ − zI)−1 h, h) = +∞

z↑−1−0

for all h ∈ N−1 . Clearly x → −∞ ⇐⇒ z =

1−x → −1 − 0. 1+x

It follows from (9.108) that lim (Q(x)f, f )N = −∞,

x↓−∞

∀f ∈ N .

If B = BK , then A = AM . As before using Ran((I − Aμ )1/2 ) ∩ N−1 = {0} one has lim ((AM − zI)−1 h, h) = −∞,

z↑+1+0

for all h ∈ N−1 . Since x → −0 ⇐⇒ z =

1−x → +1 + 0, 1+x

we get lim (Q(x)f, f )N = +∞ for all f ∈ N . The converse statements follow from x↑−0

Proposition 9.3.6.

308 Chapter 9. Canonical L-systems with Contractive and Accretive Operators

9.6 Bi-extensions of non-negative symmetric operators We consider semi-bounded (in particular non-negative) symmetric densely-deﬁned ˙ operators A, ˙ x) ≥ m(x, x), ˙ (Ax, x ∈ Dom(A). According to the classical von Neumann theorem there exists a self-adjoint extension A of A˙ with an arbitrary close to m lower bound. It was shown later by Friedreichs that operator A˙ actually admits a self-adjoint extension with the same lower bound. In this section we are going to show that for the case of a selfadjoint bi-extension of A˙ the analogue of von Neumann’s theorem is true while the analogue of the Friedrichs theorem, generally speaking, does not hold. Theorem 9.6.1. Let A˙ be a semi-bounded operator with a lower bound m and Aˆ be its symmetric extension with the same lower bound. Then A˙ admits a self-adjoint ˆ if and only bi-extension A with the same lower bound and containing Aˆ (A ⊃ A) if there exists a number k > 0 such that , ,2 , ˆ , (9.111) ,((A − mI)f, h), ≤ k((Aˆ − mI)f, f ) h2+, ˆ h ∈ H+ . for all f ∈ Dom(A), Proof. Let H+ ⊆ H ⊆ H− be the rigged triplet generated by A˙ and R be a Riesz-Berezansky operator corresponding to this triplet. In the Hilbert space H+ consider the operator B˙ := R(Aˆ − mI),

˙ = Dom(A). ˆ Dom(B)

˙ f )+ = ((Af ˆ − mI)f, f ) ≥ 0 for all f ∈ Dom(B). ˙ Observe that A is a Then (Bf, ˙ ˆ self-adjoint bi-extension of A containing A if and only if the operator B := RA is a (+)-bounded and (+)-self-adjoint extension of the operator B˙ in H+ . It follows from Theorem 9.5.10 and Remark 9.5.11 that the operator B˙ admits (+)-nonnegative bounded self-adjoint extension in H+ if and only if there exists k > 0 such that ˙ h)+ |2 ≤ k(Bf, ˙ f )+ ||h||2 , f ∈ Dom(B), ˙ h ∈ H+ . |(Bf, + This is equivalent to (9.111)

Remark 9.5.11 yields that if A˙ has at least one self-adjoint bi-extension A containing Aˆ with the same lower bound, then it has inﬁnitely many such biextensions. Corollary 9.6.2. Inequalities (9.111) hold if and only if there exists a constant C > 0 such that |((Aˆ − mI)f, ϕa )|2 ≤ C((Aˆ − mI)f, f ) ϕa 2+ , ˆ and all ϕa such that (A˙ ∗ − (m − a)I)ϕa = 0, (a > 0). for all f ∈ Dom(A)

(9.112)

9.6. Bi-extensions of non-negative symmetric operators

309

Proof. Suppose (9.111). Then for h = ϕa ∈ ker(A˙ ∗ − (m − a)I)) we have (9.112). Now let us show that (9.111) follows from (9.112). It is known that there exists ˆ with a self-adjoint extension A of Aˆ (for instance, the Friedrichs extension of A) the lower bound m. If λ is a regular point for A, then H+ = Dom(A) Nλ .

(9.113)

Indeed, if f ∈ H+ = Dom(A˙ ∗ ), then there exists an element g ∈ Dom(A) such that (A˙ ∗ − λI)f = (A − λI)g. This implies (A˙ ∗ − λI)(f − g) = 0 and hence (f − g) ∈ Nλ for any f ∈ H+ and g ∈ Dom(A), which conﬁrms (9.113). Further, applying the Cauchy-Schwartz inequality we obtain |((Aˆ − mI)f, g)|2 ≤ ((Aˆ − mI)f, f )((A − mI)g, g) ≤ C((Aˆ − mI)f, f )g2 , C > 0,

(9.114)

+

ˆ and g ∈ Dom(A). Clearly, all the points of the form (m − a), for f ∈ Dom(A) ˙ Thus (a > 0) are regular points for A and the points of a regular type for A. (9.113) implies H+ = Dom(A) Nm−a . (9.115) Let h ∈ H+ be an arbitrary vector. Applying (9.115) we get h = g + ψa , where g ∈ Dom(A) and ψa ∈ Nm−a . Adding up inequalities (9.112) and (9.114) and taking into account that the norms · and · + are equivalent on Nm−a we get (9.111). The following theorem is the analogue of the classical von Neumann’s result. Theorem 9.6.3. Let ε be an arbitrary small positive number and A˙ be a semibounded operator with the lower bound m. Then there exist inﬁnitely many semibounded self-adjoint bi-extensions with the lower bound (m − ε). Proof. First we show that the inequality |((A˙ − (m − ε)I)f, g)|2 ≤ k((A˙ − (m − ε)I)f, f )g2+, ˙ g ∈ M, and k > 0. Indeed, holds for all f ∈ Dom(A), |((A˙ − (m − ε)I)f, g)| = |(f, (A˙ ∗ − (m − εI)g)| ≤ |(f, A˙ ∗ g)| + |m − ε| · |(f, g)| ≤ f · A∗ g + |m − ε| · f · g 1 |m − ε| ≤ √ ((A˙ − (m − ε)I)f, f )1/2 g+ + √ ((A˙ − (m − ε)I)f, f )1/2 g+ ε ε 1 + |m − ε| ˙ √ = ((A − (m − ε)I)f, f )g+ . ε The statement of the theorem follows from Theorem 9.6.1 and Remark 9.5.11.

310 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Theorem 9.6.4. A non-negative densely-deﬁned operator A˙ admits a non-negative self-adjoint bi-extension if and only if the Friedrichs and Kre˘ın-von Neumann extensions of A˙ are transversal. Proof. Suppose that the Friedrichs extension AF and the Kre˘ın-von Neumann extension AK of the operator A˙ are transversal. Then due to Theorem 9.5.9 the 1/2 inclusion Dom(A˙ ∗ ) ⊂ D[AK ] holds. This means that H+ ⊆ Dom(AK ). Since 1/2 ||h||+ ≥ ||h|| for all h ∈ H+ , and AK is closed in H, the closed graph theorem 1/2 yields now that AK ∈ [H+ , H], i.e., there exists a number c > 0 such that 1/2

||AK u||2 = AK [u] ≤ c||u||2+ . 1/2

1/2

It follows that the sesquilinear form AK [u, v] = (AK u, AK v), u, v ∈ H+ is bounded on H+ . Therefore, by the Riesz theorem, there exists an operator AK ∈ [H+ , H− ] such that (AK u, v) = AK [u, v],

u, v ∈ H+ , u ∈ H+ .

Due to AK [u] ≥ 0 for all u ∈ D[AK ], the operator AK is non-negative. Since (AK u, v) = AK [u, v] for all u ∈ Dom(AK ) and all v ∈ D[AK ], we get (AK u, v) = (AK u, v),

u ∈ Dom(AK ), v ∈ H+ .

Hence AK ⊃ AK , i.e., AK is t-self-adjoint bi-extension of A˙ with quasi-kernel AK . Conversely, let A˙ admit a non-negative self-adjoint bi-extension. Then from Theorem 9.6.1 we get the equality ˙ h)|2 ≤ k(Af, ˙ f )||h||2 , |(Af, + ˙ and all h ∈ H+ = Dom(A˙ ∗ ), and some k > 0. Applying for all f ∈ Dom(A) Theorem 9.5.7 we get that H+ ⊆ D[AK ]. Now Theorem 9.5.9 yields that AF and AK are transversal. Corollary 9.6.5. If a non-negative densely-deﬁned symmetric operator A˙ admits a non-negative self-adjoint bi-extension, then it also admits a non-negative t-selfadjoint bi-extension A containing AK as a quasi-kernel. It follows from Theorem 9.6.4 that if AK = AF , then the operator A˙ does not admit non-negative self-adjoint bi-extensions. Consequently, in this case the analogue of the Friedrichs theorem is not true. The following theorem provides a criterion on when the analogue of the Friedrichs theorem does hold. Theorem 9.6.6. A non-negative densely-deﬁned symmetric operator A˙ admits a non-negative self-adjoint bi-extensions if and only if ∞ t d(E(t)h, h) < ∞ for all h ∈ N−a , a > 0, (9.116) 0

˙ where E(t) is a spectral function of the Kre˘ın-von Neumann extension AK of A.

9.7. Accretive bi-extensions

311

Proof. The inequality (9.116) is equivalent to the inclusion 1/2

N−a ⊂ Dom(AK ) = D[AK ]. Since −a is a regular point of AK , the direct decomposition ˙ −a , Dom(A˙ ∗ ) = Dom(AK )+N holds. So, from Theorem 9.5.9 we get that (9.116) is equivalent to transversality of AF and AK . The latter is equivalent to existence of a non-negative self-adjoint bi-extension of A˙ (see Theorem 9.6.4). Observe, that since N−a is a subspace in H, AK is closed in H, condition (9.116) is equivalent to the following: there exists a positive number k > 0, depending on a, such that ∞ t d(E(t)h, h) < k||h||2 , ∀h ∈ N−a , a > 0. (9.117) 0

On the other hand, (9.116) is equivalent (see proof of Theorem 9.6.4) to the existence of k > 0 such that ∞ t d(E(t)f, f ) < k||f ||2+ , ∀f ∈ Dom(A˙ ∗ ). 0

9.7 Accretive bi-extensions We recall (see page 301) that a closed, densely-deﬁned, linear operator T in a Hilbert space H is maximal accretive if it is accretive and T has a regular point in the left half-plane. A bi-extension A ∈ [H+ , H− ] is called an accretive bi-extension if Re (Af, f ) ≥ 0 for all f ∈ H+ . In this section we will study the existence of ˙ accretive (∗)-extensions of a given accretive operator T ∈ Ω(A). ˙ generated Lemma 9.7.1. Let A be a (∗)-extension of an accretive operator T ∈ Ω(A) via (4.60) by a self-adjoint operator A. Then A is accretive if and only if the form Re (T h, h) + (Ag, g) + 2Re (T h, g),

(9.118)

is non-negative for all h ∈ Dom(T ) and g ∈ Dom(A). ˙ Proof. Let A be an accretive (∗)-extension of an accretive operator T ∈ Ω(A). Then for h ∈ Dom(T ) and g ∈ Dom(A), Re (Af, f ) ≥ 0, f = g + h. It follows from Theorem 4.4.7 that the form (9.118) is non-negative. On the other hand, if the form (9.118) is non-negative, then for all h ∈ Dom(T ) and g ∈ Dom(A) the transversality of Dom(T ) and Dom(A) and Theorem 4.4.7 imply that A is accretive.

312 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Let AF and AK be Friedrich’s and Kre˘ın-von Neumann’s extensions of op˙ we denote the set of all erator A˙ of the form (9.91) and (9.92). By class Ξ(A) ˙ maximal accretive extensions of operator A. In particular, both AF and AK be˙ It follows from Lemma 9.7.1 that if T ∈ Ξ(A) ˙ and if (∗)-extension long to Ξ(A). ˙ ˙ we deﬁne A of T generated by A is accretive, then A ∈ Ξ(A). On the class Ξ(A) Cayley transform described in the Section 1.4 and given by the formula K(T ) = (I − T )(I + T )−1 ,

˙ T ∈ Ξ(A).

(9.119)

˙ This Cayley transform sets a one-to-one correspondence between the class Ξ(A) ∗ and a set of contractions Q ∈ [H, H] such that both Q and Q are extensions of a symmetric contraction ˙ ˙ −1 , S˙ = (I − A)(I + A) ˙ = (I + A)Dom( ˙ ˙ Clearly, Q and Q∗ are both deﬁned on a subspace Dom(S) A). ˙ qsc-extensions of S. Put ˙ N = H Dom(S). (9.120) ∗ ˙ ˙ Notice that N = N−1 = ker(A + I) (the deﬁciency subspace of A). Let Sμ = K(AF ) and SM = K(AK ). Using Theorem 9.2.7 and formula (9.42) we get that Q ∈ [H, H] is a qsc-extension of a symmetric contraction S˙ if and only if it can be represented in the form Q=

1 1 (SM + Sμ ) + (SM − Sμ )1/2 Z(SM − Sμ )1/2 , 2 2

(9.121)

where Z is a contraction in the subspace Ran(SM − Sμ ) ⊆ N. Clearly, if Z is a self-adjoint contraction, then (9.121) provides a description of all sc-extensions of a symmetric contraction S. ˙ contains mutually transversal operators if and Lemma 9.7.2. 1) The class Ξ(A) only if AF and AK are mutually transversal. ˙ Then T1 and T2 are mutually transversal if 2) Let T1 and T2 belong to Ξ(A). and only if (K(T1 ) − K(T2 ))N = N. Proof. It follows from (9.121) that K(T1 ) − K(T2 ) =

1 (SM − Sμ )1/2 (Z1 − Z2 )(SM − Sμ )1/2 , 2

where Zl , (l = 1, 2) are the corresponding to Tl contractions in Ran(SM − Sμ ). Relation (9.119) yields K(T1 ) − K(T2 ) = 2 (I + T1 )−1 − (I + T2 )−1 . Thus (I + T1 )−1 − (I + T2 )−1 =

1 (SM − Sμ )1/2 (Z1 − Z2 )(SM − Sμ )1/2 . 4

9.7. Accretive bi-extensions

313

Furthermore, using Theorem 4.4.2 we get that −1 −1 (I + T ) − (I + T ) N−1 = N−1 1 2

Ran(SM − Sμ ) = Ran(SM − Sμ ) = N = N−1 , ⇐⇒ Ran(Z1 − Z2 )N = N.

In what follows we assume that AK and AF are mutually transversal. Let A1 ˙ Consider a form deﬁned and A2 be two mutually transversal operators from Ξ(A). on Dom(A1 ) × Dom(A2 ) as B(f1 , f2 ) = (A1 f1 , f1 ) + (A2 f2 , f2 ) + 2Re (A1 f1 , f2 ),

(9.122)

where fl ∈ Dom(Al ), (l = 1, 2). Let φl =

1 (I + Al )fl , 2

Sl φl =

1 (I − Al )fl , 2

be the Cayley transform of Al for l = 1, 2. Then fl = (I + Sl )φl ,

Al fl = (I − Sl )φl ,

(l = 1, 2).

(9.123)

Substituting (9.123) into (9.122) we obtain a form deﬁned on H × H, ˜ 1 , φ2 ) = φ1 + φ2 2 − S1 φ1 + S2 φ2 2 − 2Re ((S1 − S2 )φ1 , φ2 ) . B(φ Let us set F =

1 (S1 − S2 ), 2

G=

1 (S1 + S2 ), 2

u=

1 (φ1 + φ2 ), 2

v=

1 (φ1 − φ2 ). (9.124) 2

˜ 1 , φ2 ) = 4H(u, v) where Then B(φ H(u, v) = u2 + (F v, v) − (F u, u) − F v + Gu2 .

(9.125)

Moreover, F ± G are contractive operators. From the above reasoning we conclude that non-negativity of the form B(f1 , f2 ) on Dom(A1 ) × Dom(A2 ) is equivalent to non-negativity of the form H(u, v) on H × H. Lemma 9.7.3. The form H(u, v) in (9.125) is non-negative for all u, v ∈ H if and only if operator F deﬁned in (9.124) is non-negative. Proof. If H(u, v) ≥ 0 for all u, v ∈ H then H(0, v) ≥ 0 for all v ∈ H. Hence (F v, v) ≥ F v2 ≥ 0, i.e., F ≥ 0. Conversely, let F ≥ 0. Since both operators F ± G are self-adjoint contractions, then −I ≤ F + G ≤ I and −I ≤ F − G ≤ I. This implies −(I − F ) ≤ G ≤ I − F, and thus G = (I − F )1/2 Z(I − F )1/2 , (9.126)

314 Chapter 9. Canonical L-systems with Contractive and Accretive Operators where Z is a self-adjoint contraction. Then (9.126) yields that for all u, v ∈ H, F v + Gu = F v2 + Gu2 + 2Re (F v, Gu) ≤ F v2 + (I − F )Z(I − F )1/2 u, Z(I − F )1/2 u , , , , + 2 , F (I − F )1/2 Z(I − F )1/2 u, v , = F v2 + Z(I − F )1/2 u2 − (F Z(I − F )1/2 u, Z(I − F )1/2 u) , , , , + 2 , F Z(I − F )1/2 u, (I − F )1/2 v , ≤ F v2 + Z(I − F )1/2 u2 − (F Z(I − F )1/2 u, Z(I − F )1/2 u) + (F Z(I − F )1/2 u, Z(I − F )1/2 u) + (F (I − F )1/2 v, (I − F )1/2 v) = F v2 + Z(I − F )1/2 u2 + (F v, v) − F v2 ≤ (F v, v) + u2 − (F u, u). Therefore, for all u, v ∈ H H(u, v) = u2 − (F u, u) + (F v, v) − F v + Gu2 ≥ 0.

The lemma is proved.

In what follows we will use Theorem 3.4.9, Theorem 4.4.3, and formulas (6.13). Theorem 9.7.4. Let A = A˙ ∗ − R−1 A˙ ∗ (I − PAA ˆ ) be a self-adjoint (∗)-extension of a ˙ with a self-adjoint quasi-kernel Aˆ ∈ Ξ(A), ˙ and non-negative symmetric operator A, generated (via (4.60)) by a self-adjoint extension A. Then the following statements are equivalent: (i) A is non-negative, ˆ − K(A) N, (see (9.119)–(9.120)) is positively deﬁned, (ii) K(A) (iii) (Aˆ + I)−1 ≥ (A + I)−1 , and Aˆ is transversal to A, (iv) Aˆ ≤ A and Aˆ is transversal to A. Proof. Let (Af, f ) ≥ 0 for all f ∈ H+ . Then by Theorem 4.4.7 we have that the form ˆ h) + (Ah, g) + 2Re (Ag, ˆ h), B(g, h) = (Ag,

ˆ h ∈ Dom(A)), (g ∈ Dom(A),

ˆ × Dom(A). Consequently, the form H(u, v) given by is non-negative on Dom(A) (9.125) is non-negative for all u, v ∈ H where F =

1 ˆ − K(A) and G = 1 K(A) ˆ + K(A) . K(A) 2 2

9.7. Accretive bi-extensions

315

Using Lemma 9.7.3 we conclude that F N ≥ 0 and applying Lemma 9.7.2 yields F N = N. This proves that (i) ⇒ (ii). The implication (ii) ⇒ (i) can be shown by reversing the argument. Since ˆ = −I + 2(Aˆ + I)−1 , K(A) = −I + 2(A + I)−1 , K(A) we get that (ii) ⇐⇒ (iii). Remark 9.5.5 yields (iii) ⇐⇒ (iv).

˙ admits non-negative (∗)-extenTheorem 9.7.5. A self-adjoint operator Aˆ ∈ Ξ(A) ˆ sions if and only if A is transversal to AF . ˆ − K(AF ) N is positively deﬁned. Proof. If Aˆ is transversal to AF , then K(A) Applying Theorem 9.7.4 we obtain that A = A˙ ∗ − R−1 A˙ ∗ (I − PAA ˆ F ), is a non-negative (∗)-extension. ˆ Conversely, if A = A˙ ∗− R−1 A˙ ∗ (I −PAA ˆ ) is a (∗)-extension of A, then via ˆ − K(A) N is positively deﬁned. But then due Theorem 9.7.4 we get that K(A) to the chain of inequalities ˆ ≥ K(A) ≥ K(AF ), K(A) ˆ − K(AF ) N is positively deﬁned as well. According to the operator K(A) Lemma 9.7.2, Aˆ is transversal AF . ˙ then all self-adjoint (∗)We note that if Aˆ is a self-adjoint extension of A, ˆ extensions of A coincide with t-self-adjoint bi-extensions of A˙ with the quasiˆ Consequently, Theorem 9.7.5 gives the criterion of the existence of a kernel A. non-negative t-self-adjoint bi-extension of A˙ and hence provides the conditions when Friedreichs theorem for t-self-adjoint bi-extensions is true. Now we focus on the study of non-self-adjoint accretive (∗)-extensions of ˙ operator T ∈ Ξ(A). ˙ generated by an Lemma 9.7.6. Let A be a (∗)-extension of operator T ∈ Ξ(A) ˙ ˆ operator A ∈ Ξ(A). Then the quasi-kernel A of the operator Re A is deﬁned by the formula 1 f = (Q + I)g + (S + I)(Q∗ − S)−1 (Q − Q∗ )g, 2 (9.127) 1 ˆ Af = (I − Q)g + (I − S)(Q∗ − S)−1 (Q − Q∗ )g, 2 where g ∈ H, Q = K(T ), Q∗ = K(T ∗ ), and S = K(A). Proof. Let A = A˙ ∗ − R−1 A˙ ∗ (I − PT A ) (of the form (4.60)) be a (∗)-extensions of operator T generated by a self-adjoint extension A. Let ˙ ⊕ (U + I)Ni , Dom(A) = Dom(A)

316 Chapter 9. Canonical L-systems with Contractive and Accretive Operators ˆ where Aˆ is a where U ∈ [Ni , N−i ] is a unitary mapping. Suppose f ∈ Dom(A), ∗ quasi-kernel of Re A. Due to the transversality of T and A and T and A we have f = u + (U + I)ϕ,

f = v + (U + I)ψ,

where u ∈ Dom(T ), v ∈ Dom(T ∗ ), and ϕ, ψ ∈ Ni . Also Af = T u + A˙ ∗ (U + I)ϕ − iR−1 (I − U)ϕ, A∗ f = T ∗ v + A˙ ∗ (U + I)ψ − iR−1 (I − U )ψ, and ˆ = 1 (Af + A∗ f ) Af 2 1 = T u + T ∗ v + A˙ ∗ (U + I)ϕ + A˙ ∗ (U + I)ψ − iR−1 (I − U )(ϕ + ψ) . 2 ˆ ∈ H, then ϕ = −ψ and hence any vector f ∈ Dom(A) ˆ is uniquely Since Af represented in the form f = u + φ,

u ∈ Dom(T ), φ ∈ (U + I)Ni ,

or in the form f = v − φ, v ∈ Dom(T ∗ ). By Corollary 4.4.4 Aˆ is transversal to A and Re A = A˙ ∗ − R−1 (I − PAA ˆ ). ˆ Dom(A). ˙ It follows from Thus MAˆ (U + I)Ni = M, where MAˆ = Dom(A) MT (U + I)Ni = M,

˙ where MT = Dom(T ) Dom(A),

that PAA ˆ MT = MA ˆ and hence for any u ∈ Dom(T ) there exists a φ ∈ (U + I)Ni ˆ and f ∈ Dom(A) such that f = u + φ. Similarly, for any v ∈ Dom(T ) there exists ˆ such that f = u − φ. Since a φ ∈ (U + I)Ni and f ∈ Dom(A) Dom(T ) = (I + Q)H, Dom(T ∗ ) = (I + Q∗ )H, Dom(A) = (I + S)H, and

˙ = Q∗ Dom(S) ˙ = S Dom(S), ˙ Q Dom(S) ˆ there are uniquely deﬁned g, g∗ ∈ H and we conclude that for any f ∈ Dom(A) h ∈ N such that f = (Q + I)g + (S + I)h,

f = (Q∗ + I)g∗ − (S + I)h.

(9.128)

Conversely, for every g ∈ H (respectively, g∗ ∈ H) there are g∗ ∈ H (respectively, ˆ Since A˙ ∗ (Q + I)g = g ∈ H) and h ∈ N, such that (9.128) holds with f ∈ Dom(A). ∗ ∗ ∗ ∗ (I − Q)g, A˙ (I + Q )g∗ = (I − Q )g∗ , and A˙ (I + S)h = (I − S)h, then ˆ = (I − Q)g + (I − S)h, Af

ˆ = (I − Q∗ )g∗ − (I − S)h. Af

(9.129)

9.7. Accretive bi-extensions

317

From (9.128) and (9.129) we have 2h = g∗ − g and 2Sh = Q∗ g∗ − g, which implies 2(Q∗ − S)h = (Q − Q∗ )g.

(9.130)

Since T ∗ and A are mutually transversal, according to Lemma 9.7.2 (Q∗ − S) N is an isomorphism on N. Then (9.130) implies h=

1 ∗ (Q − S)−1 (Q − Q∗ )g. 2

(9.131)

Substituting (9.131) into (9.128) and (9.129) we obtain (9.127).

˙ and A ∈ Ξ(A) ˙ be a transversal to T self-adjoint Lemma 9.7.7. Let T ∈ Ξ(A) operator. If the operator [K(T ) + K(T ∗ ) − 2K(A)] N, is an isomorphism of the space N (deﬁned in (9.120)), then the quasi-kernel Aˆ of the real part of the operator A = A˙ ∗ − R−1 A˙ ∗ (I − PT A ) is a Cayley transform of the operator Sˆ = S + (Q − S)(Re Q − S)−1 (Q∗ − S), where S = K(A) and Re Q = (1/2)[K(T ) + K(T ∗ )]. Proof. Let A = A˙ ∗ − R−1 A˙ ∗ (I − PT A ). Then by virtue of Lemma 9.7.6, formula (9.127) deﬁnes the quasi-kernel Aˆ of the operator Re A. It also follows from (9.127) that ˆ = 2g + (Q∗ − S)−1 (Q − Q∗ )g. f + Af Let PN and PS˙ denote the orthoprojection operators in H according to (9.120) ˙ respectively. Then onto N and Dom(S), 2g + (Q∗ − S)−1 (Q − Q∗ )g = 2PS˙ g + (Q∗ − S)−1 (2Q∗ − 2S + Q − Q∗ )PN g = 2PS˙ g + 2(Q∗ − S)−1 (Re Q − S)PN g, and

ˆ = 2P ˙ g + 2(Q∗ − S)−1 (Re Q − S)PN g. (I + A)f S

(9.132)

From the statement of the lemma we have that (Re Q − S) N is an isomorphism ˆ = H and the Cayley transform is well of the space N. Hence, (9.132) Ran(I + A) ˆ deﬁned for A. Let ˆ + A) ˆ −1 . Sˆ = (I − A)(I It follows from (9.127) that 1 (Sˆ + I)φ = (Q + I)g + (S + I)(Q∗ − S)−1 (Q − Q∗ )g, 2 1 ˆ (I − S)φ = (I − Q)g + (I − S)(Q∗ − S)−1 (Q − Q∗ )g. 2

318 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Therefore, 1 φ = g + (Q∗ − S)−1 (Q − Q∗ )g, 2 1 ˆ Sφ = Qg + S(Q∗ − S)−1 (Q − Q∗ )g 2 and hence

ˆ = Sφ + (Q − S)PN g, Sφ φ = PS˙ g + (Q∗ − S)−1 (Re Q − S)PN g.

(9.133)

Using the second half of (9.133) we have PN g = (Re Q − S)−1 (Q∗ − S)PN φ.

(9.134)

Substituting, (9.134) into the ﬁrst part of (9.133) we obtain ˆ = Sφ + (Q − S)(Re Q − S)−1 (Q∗ − S)φ, Sφ

which proves the lemma.

˙ By the class ΞAT we denote the set of all non-negative selfLet T ∈ Ξ(A). adjoint operators A ⊃ A˙ satisfying the following conditions: 1. [K(T ) + K(T ∗ ) − 2K(A)] N is a non-negative operator in N, where N is deﬁned in (9.120); 2. K(A)+2[K(T )−K(A)][K(T )+K(T ∗)−2K(A)]−1 [K(T ∗ )−K(A)] ≤ K(AK ).1 ˙ Theorem 9.7.8. A (∗)-extension of operator T ∈ Ξ(A), A = A˙ ∗ − R−1 A˙ ∗ (I − PT A ), ˙ is accretive if and only if A ∈ ΞAT . generated by a self-adjoint operator A ⊃ A, Proof. We prove the necessity part ﬁrst. Let A = A˙ ∗ − R−1 A˙ ∗ (I − PT A ) be an accretive (∗)-extension, then Re A is a non-negative (∗)-extension of the quasi˙ But according to Lemma 9.7.6 Aˆ is deﬁned by formulas (9.127). kernel Aˆ ∈ Ξ(A). ˆ then (9.132) implies that the operator Since (−1) is a regular point of operator A, (Re Q − S) N =

1 [K(T ) + K(T ∗ ) − 2K(A)] N, 2

is an isomorphism of the space N. According to Lemma 9.7.7 we have ˆ = K(A) + 2[K(T ) − K(A)][K(T ) + K(T ∗ ) − 2K(A)]−1 [K(T ∗ ) − K(A)]. K(A) ˆ is a self-adjoint contractive extension of S, ˙ then K(A) ˆ ≤ K(AK ). Also, Since K(A) since Re A is generated by A and Re A ≥ 0, then by Theorem 9.7.4 the operator 1 When

K(T ∗ )

we write [K(T ) + K(T ∗ ) − 2K(A)]−1 we mean the operator inverse to [K(T ) + − 2K(A)] N.

9.7. Accretive bi-extensions

319

ˆ − K(A)] N is non-negative. Consequently, the operator [K(T ) + K(T ∗ ) − [K(A) 2K(A) N is non-negative as well and we conclude that A ∈ ΞAT . Now we prove suﬃciency. Let A ∈ ΞAT , then by Lemma 9.7.7, Aˆ is a Cayley ˙ Since transform of a self-adjoint extension Sˆ of the operator S. ˆ − K(A)] N, [Sˆ − S] N = [K(A) is a non-negative operator, then due to Theorem 9.7.4 the operator Re A is a ˆ That is why A is an accretive (∗)-extension of non-negative (∗)-extension of A. operator T . ˙ admits accretive (∗)-extensions if and only Theorem 9.7.9. An operator T ∈ Ξ(A) if T is transversal to AF . Proof. If T admits accretive (∗)-extensions, then the class ΞAT is non-empty, i.e., ˙ such that the operator there exists a self-adjoint operator A ∈ Ξ(A) [(K(T ) + K(T ∗) − 2K(A)] N, is non-negative. But then [(K(T ) + K(T ∗ ) − 2K(AF )] N is non-negative as well. This yields that [K(T ) − K(AF )] N is an isomorphism of N. Then by Lemma 9.7.2 T and AF are mutually transversal. This proves the necessity. Now let us assume that T and AF are mutually transversal. We will show that in this case AF ∈ ΞAT . By Lemma 9.7.2, [K(T ) − K(AF )] N is an isomorphism of the space N. Then using formula (9.121) we have K(T ) = Q =

1 1 (SM + Sμ ) + (SM − Sμ )1/2 Z(SM − Sμ )1/2 , 2 2

where Z ∈ [N, N] is a contraction. Furthermore, (Q − Sμ ) N =

1 (SM − Sμ )1/2 (Z + I)(SM − Sμ )1/2 N. 2

Thus, Z + I is an isomorphism of the space N. Moreover, Re Z + I ≥ 0 and for every f ∈ N, ((Re Z + I)f, f ) =

1 f 2 − Zf 2 + (Z + I)f 2 . 2

(9.135)

But since (Z + I)f 2 ≥ af 2 , where a > 0, f ∈ N, we have ((Re Z + I)f, f ) ≥ bf 2 ,

(b > 0).

Hence, Re Z + I is a non-negative operator implying that [(1/2)(K(T ) + K(T ∗ ) − K(A)] N =

1 (SM − Sμ )1/2 (Re Z + I)(SM − Sμ )1/2 N, 2

320 Chapter 9. Canonical L-systems with Contractive and Accretive Operators is non-negative too. Also (9.135) implies Re Z + I ≥

1 ∗ (Z + I)(Z + I). 2

It is easy to see then that (Re Z + I)−1 ≤ 2(Z + I)−1 (Z ∗ + I)−1 . Therefore, 1 (Z + I)(Re Z + I)−1 (Z ∗ + I) ≤ I. 2

(9.136)

Now, since K(AF ) + 2[K(T ) − K(A)][K(T ) + K(T ∗ ) − 2K(A)]−1 [K(T ∗ ) − K(A)] 1 = Sμ + (SM − Sμ )1/2 (Z + I)(Re Z + I)−1 (Z ∗ + I)(SM − Sμ )1/2 , 2 then applying (9.136) we obtain K(AF ) + 2[K(T ) − K(A)][K(T ) + K(T ∗ ) − 2K(A)]−1 [K(T ∗ ) − K(A)] ≤ K(AK ). Thus, AF belongs to the class ΞAT and applying theorem (9.7.8) we conclude that A = A˙ ∗ − R−1 A˙ ∗ (I − PAT ) is an accretive (∗)-extension of T . ˙ be transversal to AF . Then an accretive (∗)Theorem 9.7.10. Let T ∈ Ξ(A) extension A of T generated by AF has a property that Re A ⊃ AK if and only if T and T ∗ are extremal extensions of A˙ (see Deﬁnition 9.5.15). Proof. Suppose Re A ⊃ AK and Re A = A˙ ∗ − R−1 A˙ ∗ (I − PT AF ). Then by Lemma 9.7.7 we have SM = Sμ + (Q − Sμ )(Re Q − Sμ )−1 (Q∗ − Sμ ). Thus,

(Z + I)(Re Z + I)−1 (Z ∗ + I) = 2I,

where Q = K(T ) =

(9.137)

1 1 (SM + Sμ ) + (SM − Sμ )1/2 Z(SM − Sμ )1/2 . 2 2

It is easy to see that (Z ∗ + I)(Re Z + I)−1 (Z + I) = (Z + I)(Re Z + I)−1 (Z ∗ + I).

(9.138)

Then it follows from (9.137) and (9.138) that Z ∗ Z = ZZ ∗ = I, i.e., Z is a unitary operator in N. From Propositions 9.5.16 and 9.2.9 we get that ˙ both operators T and T ∗ are extremal m-accretive extensions of A. The second part of the theorem is proved by reversing the argument.

9.8. Realization of Stieltjes functions

321

Let α ∈ [0, π/2). A bi-extension A of a non-negative symmetric operator A˙ is called α-sectorial if |(Im Af, f )| ≤ tan α(Re Af, f ),

f ∈ H+ .

Clearly, non-negative self-adjoint bi-extensions are 0-sectorial, and formally accretive bi-extensions one can consider as π/2-sectorial. Theorem 9.7.11. Let T be a quasi-self-adjoint and maximal α-sectorial extension ˙ If T is transversal to AF , then T admits α-sectorial (∗)-extensions. of A. Proof. Let A be generated via (4.60) by AF and is given by A = A˙ ∗ − R−1 A˙ ∗ (I − PT AF ). By Theorem 9.7.9 the operator A is an accretive (∗)-extension with quasi-kernel T . If f ∈ H+ is decomposed as f = φ + h, where φ ∈ Dom(AF ), h ∈ Dom(T ), then by Theorem 4.4.7 we have (Af, f ) = (AF φ, φ) + (T h, h) + 2Re (T h, φ). Let AK [u, v] be the closed form associated with the Kre˘ın-von Neumann extension AK of A˙ and let w[h] = (T h, h) − AK [h],

h ∈ Dom(T ).

According to (9.99) we have (T h, φ) = AK [h, φ]. Hence, using the equality (AF φ, φ) = AK [φ] (see Theorem 9.5.4), one obtains (Af, f ) = AF [φ, φ] + AK [h] + AK [h, φ] + AK [φ, h] + w[h] = AK [φ + h] + w[h]. From Theorem 9.5.14 we have, for all h ∈ Dom(T ), |Im w[h]| ≤ (tan α)Re w[h]. This yields |(Im Af, f )| = |Im w[h]| ≤ tan αRe w[h] ≤ tan α(AK [φ + h] + Re w[h]) = tan α(Re Af, f ). So A is α-sectorial (∗)-bi-extension of T .

9.8 Realization of Stieltjes functions The scalar versions of the following deﬁnition can be found in [159].

322 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Deﬁnition 9.8.1. An operator-valued Herglotz-Nevanlinna function V (z) in a ﬁnitedimensional Hilbert space E is called a Stieltjes function if V (z) is holomorphic in Ext[0, +∞) and Im[zV (z)] ≥ 0. (9.139) Im z Consequently, an operator-valued Herglotz-Nevanlinna function V (z) is Stieltjes if zV (z) is also a Herglotz-Nevanlinna function. Applying (6.7) for this case we get that n # zk V (zk ) − z¯l V (¯ zl ) hk , hl ≥ 0, (9.140) zk − z¯l E k,l=1

for an arbitrary sequence {zk } (k = 1, . . . , n) of (Im zk > 0) complex numbers and a sequence of vectors {hk } in E. Similar to (6.52), the formula holds true for the case of a Stieltjes function. Indeed, if V (z) is a Stieltjes operator-valued function, then ∞ dG(t) V (z) = γ + , (9.141) t−z 0

where γ ≥ 0 and G(t) is a non-decreasing on [0, +∞) operator-valued function such that ∞ (dG(t)h, h)E < ∞, h ∈ E. (9.142) 1+t 0

Theorem 9.8.2. Let Θ be an L-system of the form (6.31)–(6.36) with a densely˙ Then the impedance function VΘ (z) deﬁned non-negative symmetric operator A. deﬁned by (6.47) is a Stieltjes function if and only if the operator A of the L-system Θ is accretive. Proof. Let us assume ﬁrst that A is an accretive operator, i.e., (Re Af, f ) ≥ 0, for all f ∈ H+ . Let {zk } (k = 1, . . . , n) be a sequence of (Im zk > 0) complex numbers and hk be a sequence of vectors in E. Let us write Khk = δk ,

gk = (Re A − zk I)−1 δk ,

g=

n #

gk .

(9.143)

k=1

Since (Re Ag, g) ≥ 0, we have n #

(Re Agk , gl ) ≥ 0.

(9.144)

k,l=1

By formal calculations one can have (Re A)gk = δk + zk (Re A − zk I)−1 δk , and n # k,l=1

(Re Agk , gl ) =

n # (δk , (Re A − zl I)−1 δl ) k,l=1

+ (zk (Re A − zk I)−1 δk , (Re A − zk I)−1 δl ) .

9.8. Realization of Stieltjes functions

323

Using obvious equalities (Re A − zk I)−1 Khk , Khl = VΘ (zk )hk , hl E , and

(Re A − z¯l I)−1 (Re A − zk I)−1 Khk , Khl =

VΘ (zk ) − VΘ (¯ zl ) h k , hl zk − z¯l

, E

we obtain n # k,l=1

n # zk VΘ (zk ) − z¯l VΘ (¯ zl ) ((Re A)gk , gl ) = h k , hl ≥ 0, zk − z¯l E

(9.145)

k,l=1

which implies that VΘ (z) is a Stieltjes function. Now we prove necessity. First we assume that A˙ is a prime operator. Then the equivalence of (9.145) and (9.144) implies that (Re Ag, g) ≥ 0 for any g from c.l.s.{Nz }. According to Lemma 6.6.4 and (6.94), a symmetric operator A˙ with the equal deﬁciency indices is prime if and only if c.l.s. Nz = H. z=z¯

Thus (Re Ag, g) ≥ 0 for any g ∈ H+ and therefore A is an accretive operator. Now let us assume that A˙ is not a prime operator. Then there exists a subspace H1 ⊂ H on which A˙ generates a self-adjoint operator A1 . Let us denote by A˙ 0 an operator induced by A˙ on H0 = H H1 . As we have shown in the proof of Theorem 6.6.1 the decomposition 0 1 H+ = H+ ⊕ H+ ,

0 1 H+ = Dom(A˙ ∗0 ), H+ = Dom(A1 ),

(9.146)

is (+)-orthogonal. Since A˙ is a non-negative operator, then ˙ g) ≥ 0, (Re Ag, g) = (A1 g, g) = (Ag,

1 ∀g ∈ H+ = Dom(A1 ).

On the other hand, operator A˙ 0 is prime in H0 and hence (by Lemma 6.6.4) c.l.s. N0z = H0 , where N0z are the deﬁciency subspaces of the symmetric operz=z¯

ator A˙ 0 in H0 . Then the equivalence of (9.145) and (9.144) again implies that 0 (Re Ag, g) ≥ 0 for any g ∈ H+ . Taking into account decomposition (9.146) we conclude that Re (Ag, g) ≥ 0 holds for all g ∈ H+ and hence A is accretive. Let α ∈ (0, π2 ). Now we introduce sectorial subclasses S α of operator-valued Stieltjes functions. An operator-valued Stieltjes function V (z) belongs to S α if Kα =

6 n 5 # zk V (zk ) − z¯l V (¯ zl ) − (cot α) V ∗ (zl )V (zk ) hk , hl ≥ 0, (9.147) zk − z¯l E

k,l=1

324 Chapter 9. Canonical L-systems with Contractive and Accretive Operators for an arbitrary sequence {zk } (k = 1, . . . , n) of (Im zk > 0) complex numbers and a sequence of vectors {hk } in E. For 0 < α1 < α2 < π2 , we have S α1 ⊂ S α2 ⊂ S, where S denotes the class of all Stieltjes functions (which corresponds to the case α = π2 ), as follows from the inequality Kα1 ≤ Kα2 ≤ K π2 . The following theorem reﬁnes the result of Theorem 9.8.2 as applied to the class S α . Theorem 9.8.3. Let Θ be a scattering L-system of the form (6.31)–(6.36) with a ˙ Then the impedance function densely-deﬁned non-negative symmetric operator A. α VΘ (z) deﬁned by (6.47) belongs to the class S if and only if the operator A of the L-system Θ is α-sectorial. Proof. The outline of the proof can be replicated from the proof of Theorem 9.8.2. Then all we need is to show that (9.147) is equivalent to relation (9.29) of Deﬁnition 9.2.1 of sectoriality. Suppose that A is α-sectorial, then (9.29) holds for all g ∈ H+ and hence cot α · |(Im A g, g)| ≤ (Re A g, g), g ∈ H+ . (9.148) Consequently, it follows from (9.143) that n #

(Re A gk , g ) ≥ (cot α)

k,=1

n #

(Im A gk , g ).

k,=1

Taking into account that Im A = KK ∗ , we get n #

(Re A(Re A − zk I)−1 Khk , (Re A − z I)−1 Kh )

k,=1

≥ cot α ·

n #

(KK ∗ (Re A − zk I)−1 Khk , (Re A − z I)−1 Kh ),

k,=1

which leads to n #

(K ∗ (Re A − z¯ I)−1 Re A(Re A − zk I)−1 Khk , h )E

k,=1

≥ cot α ·

n # k,=1

(K ∗ (Re A − z¯ I)−1 KK ∗ (Re A − zk I)−1 Khk , h )E .

9.8. Realization of Stieltjes functions

325

Using resolvent identity K ∗ (Re A−¯ z I)−1 Re A(Re A − zk I)−1 K = K∗

z¯ (Re A − z¯ I)−1 − zk (Re A − zk I)−1 K, z¯ − zk

(9.149)

we obtain (9.147). The converse statement is proved by reversing the argument. Thus we have shown that (9.147) is equivalent to (9.148) and this completes the proof. At this point we would like to introduce a special subclass of scalar Stieltjes functions. Let π 0 ≤ α1 ≤ α2 ≤ . 2 We say that a scalar Stieltjes function V (z) belongs to the class S α1 ,α2 if tan α1 = lim V (x), x→−∞

tan α2 = lim V (x). x→−0

Theorem 9.8.4. Let Θ be a scattering L-system of the form A K 1 Θ= , H+ ⊂ H ⊂ H− C

(9.150)

(9.151)

˙ Let also A be an αwith a densely-deﬁned non-negative symmetric operator A. ˙ sectorial (∗)-extension of T ∈ Λ(A). Then the impedance function VΘ (z) deﬁned by (6.47) belongs to the class S α1 ,α2 , tan α2 ≤ tan α, and T is (α2 − α1 )-sectorial with the exact angle of sectoriality (α2 − α1 ).2 Proof. Since A is an α-sectorial (∗)-extension of T , then (9.148) holds and we can apply Theorem 9.8.3. Then using (9.149) for zk = z = z we obtain Im (zV (z)) = (Im z)K ∗ (Re A − zI)−1 Re A(Re A − z¯I)−1 K.

(9.152)

Applying Theorem 9.8.3 again we have Im (zVΘ (z)) ≥ (cot α)VΘ∗ (z)VΘ (z). Im z

(9.153)

It follows from (9.152) that Im (zVΘ (z)) = K ∗ (Re A − xI)−1 Re A(Re A − xI)−1 K. z→x Im z lim

2 We

(9.154)

say that the angle of sectoriality α is exact for an α-sectorial operator T if tan α = |Im (T f,f )| supf ∈Dom(T ) |Re (T f,f )| .

326 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Here we used the fact that if A is α-sectorial, then A is accretive and x < 0 is a regular point for the quasi-kernel of Re A. Thus, (9.154) yields VΘ (x)− lim

z→x

Im (zVΘ (z)) = K ∗ (Re A − xI)−1 K Im z − K ∗ (Re A − xI)−1 Re A(Re A − xI)−1 K = K ∗ (Re A − xI)−1 [I − Re A(Re A − xI)−1 ]K = K ∗ (Re A − xI)−1 [Re A − xI − Re A](Re A − xI)−1 K = −xK ∗ (Re A − xI)−1 (Re A − xI)−1 K ≥ 0.

Therefore, Im (zVΘ (z)) , (x < 0). Im z Since VΘ (x) > 0 for x < 0, then applying (9.153) and (9.155) yields VΘ (x) ≥ lim

z→x

(9.155)

VΘ (x) ≥ (cot α)VΘ2 (x), and therefore VΘ (x) ≤ tan α,

(x < 0).

(9.156)

It follows from Theorem 9.8.2 that the impedance function VΘ (z) of an L-system with accretive operator A has the integral representation (9.141), i.e., ∞ VΘ (z) = γ + 0

dG(t) . t−z

(9.157)

Then (9.156) and (9.157) yield VΘ (x) = γ + and thus

∞ 0

0

∞

dG(t) ≤ tan α. t−x

dG(t) < ∞ and t

∞

γ+ 0

(x < 0).

dG(t) ≤ tan α. t

Let us write tan α1 = γ,

tan α2 = γ +

0

∞

dG(t) . t

(9.158)

Using (9.158) we obtain that VΘ (z) ∈ S α1 ,α2 and tan α2 ≤ tan α. According to Theorem 8.2.4 for the system Θ of the form (9.151) there is a system Θ of the form (8.29) with the main operator S = (I − T )(I + T )−1 and such that (8.30) holds, i.e., 1−z WΘ (z) = WΘ (−1)WΘ , z ∈ ρ(T ), z = −1. (9.159) 1+z

9.8. Realization of Stieltjes functions

327

We also know from Lemma 9.5.12 that T is α-sectorial if and only if S is α-cosectorial contraction. It follows from Theorems 9.4.1 and 9.4.2 that the exact angle of co-sectoriality of S can be calculated by cot β =

1 + VΘ (1)VΘ (−1) . |VΘ (−1) − VΘ (1)|

(9.160)

Let us compute VΘ (1) and VΘ (−1) using (9.159) and (6.48). We get VΘ (1) = −i(I + WΘ−1 (−1)WΘ (0))−1 (WΘ−1 (−1)WΘ (0) − I), VΘ (−1) = −i(I + WΘ−1 (−1)WΘ (−∞))−1 (WΘ−1 (−1)WΘ (−∞) − I), and WΘ (0) =

1 − iVΘ (0) 1 − iVΘ (−∞) 1 + iVΘ (−1) , WΘ (−∞) = , WΘ−1 (−1) = . 1 + iVΘ (0) 1 + iVΘ (−∞) 1 − iVΘ (−1)

This yields VΘ (1) =

VΘ (−1) − VΘ (−∞) , 1 + VΘ (−1)VΘ (−∞)

VΘ (−1) =

VΘ (−1) − VΘ (0) . 1 + VΘ (−1)VΘ (0)

Taking into account (9.160) we get cot β =

1 + VΘ (0)VΘ (−∞) 1 + tan α2 · tan α1 = = cot(α2 − α1 ). VΘ (0) − VΘ (−∞) tan α2 − tan α1

Corollary 9.8.5. Let Θ of the form (9.151) be an L-system as in the statement of Theorem 9.8.4 and let α be the exact angle of sectoriality of the operator T of the system Θ. Then VΘ (z) ∈ S 0,α . Proof. According to Theorem 9.8.4 the exact angle of sectoriality is given by α2 − α1 , where tan α1 = lim VΘ (x), tan α2 = lim VΘ (x). x→−∞

x→−0

It was also shown that tan α ≥ tan α2 . On the other hand, since in the statement of the current corollary α is the exact angle of sectoriality of T , then α = α2 − α1 and hence tan(α2 − α1 ) ≥ tan α2 . Therefore, α1 = 0. Remark 9.8.6. It follows that under assumptions of Corollary 9.8.5, the impedance function VΘ (z) has the form ∞ dG(t) VΘ (z) = . t−z 0 Theorem 9.8.7. Let Θ be an L-system of the form (9.151), where A is a (∗)˙ and A˙ is a closed densely-deﬁned non-negative symmetric extension of T ∈ Λ(A) operator with deﬁciency numbers (1, 1). If the impedance function VΘ (z) belongs to the class S α1 ,α2 , then A is α-sectorial, where + tan α = tan α2 + 2 tan α1 (tan α2 − tan α1 ).

328 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Proof. Since VΘ (z) ∈ S α1 ,α2 , we have VΘ (z) = tan α1 +

∞

0

dG(t) t−z

and

tan α2 = tan α1 +

∞

0

dG(t) . t

(9.161)

Let {zk }, k = 1, . . . , n be arbitrary numbers in C+ and {ξk }, k = 1, . . . , n be arbitrary complex numbers. By direct substitution one gets , ,2 ∞ , √ ,2 n n n ,# , ,# # zk VΘ (zk ) − z¯l VΘ (¯ zl ) ¯ ξl t ,, , , , ξl , + , ξk ξl = (tan α1 ) , , dG(t). , , , zk − z¯l t − zl ,

k,l=1

l=1

0

l=1

(9.162) Furthermore, n #

VΘ (zl )VΘ (¯ zk )ξl ξ¯k =

k,l=1

n # tan α1 +

0

k,l=1

× tan α1 +

∞

0

dG(t) t − z¯k

∞

dG(t) t − zl

, ,2 ∞# n n ,# , ξ dG(t) , , l ξl ξ¯k = , (tan α1 )ξl + , . , t − z l , 0 l=1

(9.163)

l=1

It follows from (9.163) that , , , , , , ∞# n n n n ,# ,# , , ∞# ξl dG(t) ,, ξl dG(t) ,, , , , , tan α1 ξl + ξl , + , , , ≤ tan α1 , , , , , , 0 t − zk , t − zl , 0 l=1 l=1 l=1 l=1 , , , , √ n n ,# , , ∞ 1 # ξl tdG(t) ,, , , , √ = tan α1 , ξl , + , , , , , 0 t l=1 t − zl , l=1 ⎛ ⎞1/2 , n , √ ,,2 1/2 ∞ ,,# n ,# , ∞ dG(t) ξ t , , , , l ⎝ ≤ tan α1 , ξl , + , , dG(t)⎠ , , , t t − zl , 0 0 l=1

l=1

⎡

⎤1/2 , n ,2 , n √ ,,2 ,# , ∞ ,# ξ t , , , , l ≤ (tan α1 )1/2 ⎣tan α1 , ξl , + , , dG(t)⎦ , , , t − zl , 0 l=1

+ 0

∞

l=1

⎤1/2 , n ,2 , n √ ,,2 1/2 , , ∞ ,# # dG(t) ξl t , , , ⎣tan α1 ,, ξl , + , , dG(t)⎦ , , , t t − zl , 0

= = tan1/2 α1 +

⎡

l=1

0

∞

dG(t) t

1/2

>⎛

l=1

⎞1/2 n # z V (z ) − z ¯ V (¯ z ) k Θ k l Θ l ¯ ⎝ ξk ξl ⎠ . zk − z¯l k,l=1

9.8. Realization of Stieltjes functions

329

Using (9.162), (9.163) we obtain =

n #

1/2

VΘ (zl )VΘ (¯ zk )ξl ξ¯k ≤ tan

α1 +

k,l=1

×

n # k,l=1

∞ 0

dG(t) t

1/2 >2

zk VΘ (zk ) − z¯l VΘ (¯ zl ) ¯ ξk ξl . zk − z¯l

Applying Theorem 9.8.3 we get that A is α-sectorial and using (9.161) yields = tan α = tan1/2 α1 +

∞

0

dG(t) t

1/2 >2

√ √ = tan α1 + (tan α2 − tan α1 ) + 2 tan α1 · tan α2 − tan α1 √ √ = tan α2 + 2 tan α1 · tan α2 − tan α1 .

The next statement immediately follows from Theorems 9.8.4 and 9.8.7. Theorem 9.8.8. Let Θ be an L-system of the form (9.151) with a densely-deﬁned ˙ Then A is α-sectorial (∗)-extension of an non-negative symmetric operator A. ˙ with the exact angle α ∈ (0, π/2) if and only if α-sectorial operator T ∈ Λ(A) ∞ dG(t) VΘ (z) = ∈ S 0,α . t − z 0 Moreover, the angle α can be found via the formula ∞ dG(t) tan α = . t 0

(9.164)

Now we deﬁne a class of realizable Stieltjes functions. At this point we need to note that since Stieltjes functions form a subset of Herglotz-Nevanlinna functions, then according to Deﬁnition 6.4.1 and Theorems 6.4.3 and 6.5.2, we have that the class of all realizable Stieltjes functions is a subclass of N (R). To see the speciﬁcations of this class we recall that aside of integral representation (9.141), any Stieltjes function admits a representation (6.52). It was shown in Chapter 7 that a Herglotz-Nevanlinna operator-function can be realized as the impedance function of an L-system if and only if in the representation (6.52) X = 0 and

+∞

Qh = −∞

for all h ∈ E such that

∞ −∞

t dG(t)h, 1 + t2

(dG(t)h, h)E < ∞.

(9.165)

(9.166)

330 Chapter 9. Canonical L-systems with Contractive and Accretive Operators holds. Considering this we obtain Q=

1 [V (−i) + V ∗ (−i)] = γ + 2

+∞

0

t dG(t). 1 + t2

(9.167)

Combining (9.165) and (9.167) we conclude that γh = 0 for all h ∈ E such that (9.166) holds. Deﬁnition 9.8.9. An operator-valued Stieltjes function V (z) in a ﬁnite-dimensional Hilbert space E belongs to the class S(R) if in the representation (9.141) γh = 0 for all h ∈ E such that

∞ 0

(dG(t)h, h)E < ∞.

(9.168)

We are going to focus though on the subclass S0 (R) of S(R) whose deﬁnition is the following. Deﬁnition 9.8.10. An operator-valued Stieltjes function V (z) ∈ S(R) belongs to the class S0 (R) if in the representation (9.141) we have ∞ (dG(t)h, h)E = ∞, (9.169) 0

for all non-zero h ∈ E. An L-system Θ of the form (6.31)-(6.36) is called an accretive L-system if its operator A is accretive. The following theorem gives the analogue of the Theorem 7.1.4 for the functions of the class S0 (R). Theorem 9.8.11. Let Θ be an accretive L-system of the form (6.31)–(6.36) with an ˙ Then invertible channel operator K and a densely-deﬁned symmetric operator A. its impedance function VΘ (z) of the form (6.47) belongs to the class S0 (R). Proof. Since our L-system Θ is accretive, then by Theorem 9.8.2, VΘ (z) is a Stieltjes function. Now let us show that VΘ (z) belongs to S0 (R). It follows from the ˙ and Theorem 6.2.10 that E1 = K −1 L, where L = H Dom(A)

4 +∞ E1 = h ∈ E : (dG(t)h, h)E < ∞ . 0

˙ = H and consequently L = {0}. Next, E1 = {0}, But Dom(A) ∞ (dG(t)h, h)E = ∞, 0

for all non-zero h ∈ E, and therefore VΘ (z) ∈ S0 (R).

9.8. Realization of Stieltjes functions

331

Inverse realization theorem analogous to the Theorem 7.1.5 can be stated and proved for the classes S0 (R) as well. Theorem 9.8.12. Let an operator-valued function V (z) belong to the class S0 (R). Then V (z) can be realized as an impedance function of a minimal accretive Lsystem Θ of the form (6.31)–(6.36) with an invertible channel operator K, a ˙ Dom(T ) = Dom(T ∗ ), and densely-deﬁned non-negative symmetric operator A, a preassigned direction operator J for which I + iV (−i)J is invertible.3 Proof. We have already noted that the class of Stieltjes function lies inside the wider class of all Herglotz-Nevanlinna functions. Thus all we actually have to show is that S0 (R) ⊂ N0 (R) and that the realizing L-system in the proof of the Theorem 7.1.5 appears to be an accretive L-system. The former is rather obvious and follows directly from the deﬁnition of the class S0 (R). To see that the realizing L-system is accretive we need to recall that the model L-system Θ was constructed in the proof of the Theorem 7.1.5 and then it was shown that VΘ (z) = V (z). But V (z) is a Stieltjes function and hence so is VΘ (z). Applying Theorem 9.8.2 yields the desired result. Now it directly follows from Theorems 9.8.8 and 9.8.12 that if a function V (z) ∈ S0 (R) is also a member of the class S α1 ,α2 , then it can be realized by an accretive L-system Θ whose operator A is an α-sectorial (∗)-extension of an α-sectorial operator T with the exact angle α ∈ (0, π/2) found by (9.164). Let us deﬁne a subclass of the class S0 (R). Deﬁnition 9.8.13. An operator-valued Stieltjes function V (z) of the class S0 (R) is said to be a member of the class S0K (R) if ∞ (dG(t)h, h)E = ∞, (9.170) t 0 for all non-zero h ∈ E. Below we state and prove a direct and inverse realization theorem for this subclass. Theorem 9.8.14. Let Θ be an accretive L-system of the form (6.31)–(6.36) with an ˙ If the invertible channel operator K and a densely-deﬁned symmetric operator A. Kre˘ın-von Neumann extension AK is a quasi-kernel for Re A, then the impedance function VΘ (z) of the form (6.47) belongs to the class S0K (R). Conversely, if V (z) ∈ S0K (R), then it can be realized as the impedance function of an accretive L-system Θ of the form (6.31)-( 6.36) with Re A containing AK as a quasi-kernel and a preassigned direction operator J for which I +iV (−i)J is invertible. 3 We

cally.

have already mentioned that if J = I this invertibility condition is satisﬁed automati-

332 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Proof. We begin with the proof of the second part. First we use realization Theorems 6.5.1, 7.1.5, and 9.8.12 to construct a minimal model L-system Θ whose impedance function is V (z). Then we will show that (9.170) is equivalent to the fact that the self-adjoint operator A deﬁned in (6.69), that we constructed in these theorems to be a quasi-kernel for Re A, coincides with AK , that is the Kre˘ın-von Neumann extension of the symmetric operator A˙ of the form (6.76). Let L2G (E) be a model space constructed in the proof or theorem (6.5.1). Let also E(s) be the orthoprojection operator in L2G (E) deﬁned by f (t), 0 ≤ t ≤ s, E(s)f (t) = (9.171) 0, t > s, where f (t) ∈ C00 (E, [0, +∞)). Here C00 (E, [0, +∞)) is the set of continuous compactly supported functions f (t), ([0 < t < +∞)) with values in E. Then for the operator A, that is the operator of multiplication by independent variable deﬁned in (6.69) in the proof of Theorem 6.5.1, we have ∞ A= s dE(s), (9.172) 0

and E(s) is the resolution of identity of the operator A. By construction provided in the proof of Theorem 6.5.1, the operator A is the quasi-kernel of Re A, where A is an accretive (∗)-extension of the model system. Let us calculate (E(s)f (t), f (t)) and (Af (t), f (t)) (here we use L2G (E) scalar product). ∞ s (E(s)f (t), f (t)) = (dG(t)E(s)f (t), f (t))E = (dG(t)f (t), f (t))E , 0

⎧ s ⎨

∞ (Af (t), f (t)) =

sd 0

⎩

0

⎫ ⎬ (dG(t)f (t), f (t))E

0

⎭

(9.173)

∞ =

s d(G(s)x(s), x(s))E . (9.174) 0

The equality A = AK holds (see Proposition 9.5.2) if for all ϕ ∈ N−a , ϕ = 0 ∞ (dE(t)ϕ, ϕ) = ∞, (9.175) t 0 where N−a is the deﬁciency subspace of the operator A˙ corresponding to the point (−a), (a > 0). But according to Theorem 6.5.1 (see also (6.74)) we have

4 h 2 Nz = ∈ LG (E) | h ∈ E , t−z

and hence N−a =

4 h ∈ L2G (E) | h ∈ E . t+a

(9.176)

9.8. Realization of Stieltjes functions

333

Taking into account (9.170) we have for all h ∈ E, ∞ 0

(dE(s)ϕ, ϕ)L2G (E) s

∞ =

h h (dE(s) t+a , t+a )L2G (E)

0

s

∞ = 0

(dG(s)h, h)E . s(s + a)2

Hence the operator A = AK iﬀ ∞ 0

(dG(t)h, h)E = ∞, t(t + a)2

∀h ∈ E, h = 0.

Let us transform (9.170) ∞ ∞ (dG(t)h, h)E (t + a)2 h h = dG(t) , t t t+a t+a E 0 0 ∞ ∞ h h h h = t dG(t) , + 2a dG(t) , t+a t+a E t+a t+a E 0 0 ∞ (dG(t)h, h)E + a2 . t(t + a)2 0

(9.177)

(9.178)

Since Re A is a non-negative self-adjoint bi-extension of A˙ in the model system, then we can apply Theorem 9.6.6 to get (9.117). Then ﬁrst two integrals in (9.178) converge for a ﬁxed a because of (9.117) and equality ∞ ∞ h h dG(t) , = d (E(t)ϕ, ϕ) , ϕ ∈ N−a . t+a t+a E 0 0 Therefore the divergence of integral in (9.170) completely depends on divergence of the last integral in (9.178). Now we can prove the ﬁrst part of the theorem. Let Θ be our L-system with AK that is a quasi-kernel for Re A, and the impedance function VΘ (z). Without loss of generality we can consider Θ as a minimal system, otherwise we would take the principal part of Θ that is minimal and has the same impedance function (see Theorem 6.6.1 and Remark 6.6.3). Furthermore, VΘ (z) can be realized as an impedance function of the model L-system Θ1 constructed in the proof of Theorem 7.1.5. Some of the elements of Θ1 were already described above during the proof of the second part of the theorem. If the L-system Θ1 is not minimal, we consider its principal part Θ1,0 that is described by (6.93) and has the same impedance function as Θ1 . Since both Θ and Θ1,0 share the same impedance function VΘ (z) they also have the same transfer function WΘ (z) and thus we can apply Theorem 6.6.10 on bi-unitary equivalence. According to this theorem the quasi-kernel operator A0 of Θ1,0 is unitarily equivalent to the quasi-kernel AK in Θ. Consequently, property (9.175) of AK gets transferred by the unitary equivalence mapping to the corresponding property of A0 making it, by Proposition 9.5.2, the Kre˘ın-von

334 Chapter 9. Canonical L-systems with Contractive and Accretive Operators Neumann self-adjoint extension of the corresponding symmetric operator A˙ 0 of Θ1,0 . But this (see Theorem 6.6.1) implies that the quasi-kernel operator A of Θ1 (deﬁned by (6.69) and (9.172)) is also the Kre˘ın-von Neumann self-adjoint extension and hence has property (9.175) that causes (9.177). Using (9.177) in conjunction with (9.178) we obtain (9.170). That proves the theorem.

9.9 Realization of inverse Stieltjes functions A scalar version of the following deﬁnition can be found in [159]. Deﬁnition 9.9.1. We will call an operator-valued Herglotz-Nevanlinna function V (z) in a ﬁnite-dimensional Hilbert space E an inverse Stieltjes if V (z) is holomorphic in Ext[0, +∞) and Im[V (z)/z] ≥ 0. (9.179) Im z Combining (9.179) with (6.7) we obtain n # V (zk )/zk − V (¯ zl )/¯ zl h k , hl ≥ 0, zk − z¯l E k,l=1

for an arbitrary sequence {zk } (k = 1, . . . , n) of (Im zk > 0) complex numbers and a sequence of vectors {hk } in E. It can be shown (see [159]) that every inverse Stieltjes function V (z) in a ﬁnite-dimensional Hilbert space E admits the integral representation ∞ 1 1 V (z) = α + zβ + − dG(t), (9.180) t−z t 0 where α ≤ 0, β ≥ 0, and G(t) is a non-decreasing on [0, +∞) operator-valued function such that ∞ (dG(t)h, h) < ∞, ∀h ∈ E. t + t2 0 The following deﬁnition provides the description of all realizable inverse Stieltjes operator-valued functions. Deﬁnition 9.9.2. An operator-valued inverse Stieltjes function V (z) in a ﬁnitedimensional Hilbert space E is a member of the class S −1 (R) if in the representation (9.180) we have i) β = 0, ii) for all h ∈ E with

∞ 0

αh = 0,

(dG(t)h, h)E < ∞.

In what follows we will, however, be mostly interested in the following subclass of S −1 (R).

9.9. Realization of inverse Stieltjes functions

335

Deﬁnition 9.9.3. An inverse Stieltjes function V (z) ∈ S −1 (R) is a member of the class S0−1 (R) if ∞ (dG(t)h, h)E = ∞, 0

for all h ∈ E, h = 0. It is not hard to see that S0−1 (R) is the analogue of the class N0 (R) introduced in Section 7.1 and of the class S0 (R) discussed in Section 9.8. ˙ is called accumulative if A (∗)-extensions A of an operator T ∈ Λ(A) (Re Af, f ) ≤ (A˙ ∗ f, f ) + (f, A˙ ∗ f ),

f ∈ H+ .

(9.181)

An L-system Θ of the form (6.31)-(6.36) is called accumulative if its operator A is accumulative, i.e., satisﬁes (9.181). It is easy to see that if an L-system is accumulative, then (9.181) implies that the operator A˙ of the system is nonnegative and both operators T and T ∗ are accretive. The following statement is the direct realization theorem for the functions of the class S0−1 (R). Theorem 9.9.4. Let Θ be an accumulative L-system of the form (6.31)–(6.36) with ˙ = H. Then its impedance function an invertible channel operator K and Dom(A) VΘ (z) of the form (6.47) belongs to the class S0−1 (R). Proof. First we will show that VΘ (z) is an inverse Stieltjes function. Let {zk } (k = 1, . . . , n) be a sequence of non-real (zk = z¯k ) complex numbers and ϕk ˙ (zk = z¯k ) is a sequence of elements of Nzk , the defect subspace of the operator A. Then for every k there exists hk ∈ E such that ϕk = zk (Re A − zk I)−1 Khk ,

(k = 1, . . . , n).

Taking into account that A˙ ∗ ϕk = zk ϕk , formula (9.182), and letting ϕ = we get

(9.182) )n k=1

(A˙ ∗ ϕ, ϕ) + (ϕ, A˙ ∗ ϕ) − (Re Aϕ, ϕ) n # = (A˙ ∗ ϕk , ϕl ) + (ϕk , A˙ ∗ ϕl ) − (Re Aϕk , ϕl ) =

=

k,l=1 n #

([−Re A + zk + z¯l ]ϕk , ϕl )

k,l=1 n #

(Re A − z¯l I)−1 (¯ zl (Re A − z¯l I) − zk (Re A − zk I))(Re A − zk I)−1 zk z¯l (zk − z¯l ) k,l=1 × Khk , Khl

ϕk

336 Chapter 9. Canonical L-systems with Contractive and Accretive Operators n # z¯l K ∗ (Re A − zk I)−1 K − zk K ∗ (Re A − zl I)−1 K = h k , hl zk z¯l (zk − z¯l ) k,l=1 n # z¯l VΘ (zk ) − zk VΘ (¯ zl ) = hk , hl ≥ 0. zk zl (zk − z¯l ) k,l=1

The last line can be re-written as n # VΘ (zk )/zk − VΘ (¯ zl )/¯ zl hk , hl ≥ 0. zk − z¯l

(9.183)

k,l=1

Letting in (9.183) n = 1, z1 = z, and h1 = h we get

VΘ (z)/z − VΘ (¯ z )/¯ z h, h z − z¯

≥ 0,

(9.184)

which means Im (VΘ (z)/z) ≥ 0, Im z and therefore VΘ (z)/z is a Herglotz-Nevanlinna function. In Theorem 6.4.3 we have shown that VΘ (z) ∈ N (R). Applying (9.179) we conclude that VΘ (z) is an inverse Stieltjes function. Now we will show that VΘ (z) belongs to S −1 (R). As any inverse Stieltjes function VΘ (z) has its integral representation (9.180) where α ≤ 0, β ≥ 0, and

∞ 0

(dG(t)h, h) < ∞, t + t2

∀h ∈ E.

In a neighborhood of zero the expression (t + t2 ) is equivalent to the (t + t3 ) and in a neighborhood of the point at inﬁnity 1 1 < . t + t3 t + t2 Hence,

∞ 0

(dG(t)h, h) < ∞, t + t3

∀h ∈ E.

Furthermore, ∞ 1 t t 1 VΘ (z) = α + zβ + − + − dG(t) t−z 1 + t2 1 + t2 t 0 ∞ ∞ dG(t) 1 t = α− + zβ + − dG(t). t + t3 t−z 1 + t2 0 0

9.9. Realization of inverse Stieltjes functions

337

On the other hand, as it was shown in Section 6.4, a Herglotz-Nevanlinna function can be realized if and only if it belongs to the class N (R) and hence in representation (6.52) F = 0 and +∞ t Qh = dG(t)h, 2 −∞ 1 + t for all h ∈ E such that (6.54) holds. Considering this and the uniqueness of the function G(t) we obtain ∞ +∞ dG(t) t α− f= dG(t)f, (9.185) 3 t+t 1 + t2 0 0 +∞ for all f ∈ E such that −∞ (dG(t)f, f )E < ∞. Solving (9.185) for α we get

∞

αf = 0

1 dG(t)f, t

(9.186)

for the same selection of f . The left-hand side of (9.186) is non-positive but the right-hand side is non-negative. This means that α = 0 and VΘ (z) ∈ S −1 (R). The proof of the fact that VΘ (z) ∈ S0−1 (R) is similar to the proof of Theorem 7.1.4 and follows from Theorem 6.2.10. The inverse realization theorem can be stated and proved for the class S0−1 (R) as follows. Theorem 9.9.5. Let an operator-valued function V (z) belong to the class S0−1 (R). Then V (z) can be realized as an impedance function of an accumulative minimal L-system Θ of the form (6.31)–(6.36) with an invertible channel operator K, a non-negative densely-deﬁned symmetric operator A˙ and J = I. Proof. The class S0−1 (R) is a subclass of N0 (R) and hence it is realizable by a minimal L-system Θ with a densely-deﬁned symmetric operator A˙ and J = I. Thus all we have to show is that the L-system Θ we have constructed in the proof of Theorem 7.1.5 is an accumulative L-system, i.e., satisfying the condition (9.181), that is (Re Af, f ) ≤ (A˙ ∗ f, f ) + (f, A˙ ∗ f ), f ∈ H+ . Since the L-system Θ is minimal then the operator A˙ is prime. Applying Lemma 6.6.4 provides us with (6.94) which implies c.l.s. Nz = H, z=z

z = z¯.

(9.187)

In the proof of Theorem 9.9.4 we have shown that (Re Aϕ, ϕ) ≤ (A˙ ∗ ϕ, ϕ) + (ϕ, A˙ ∗ ϕ),

ϕ=

n # k=1

ϕk ,

ϕk ∈ Nzk ,

(9.188)

338 Chapter 9. Canonical L-systems with Contractive and Accretive Operators is equivalent to (9.183), where zk are deﬁned by (9.182). Combining (9.187) and (9.188) we get property (9.181) and conclude that Θ is an accumulative L-system. Now we deﬁne a subclass of the class S0−1 (R). Deﬁnition 9.9.6. An operator-valued Stieltjes function V (z) of the class −1 S0−1 (R) is said to be a member of the class S0,F (R) if ∞ t (dG(t)h, h)E = ∞, 2+1 t 0

(9.189)

for all non-zero h ∈ E. Theorem 9.9.7. Let Θ be an accumulative L-system of the form (6.31)–(6.36) with ˙ If an invertible channel operator K and a symmetric densely-deﬁned operator A. Friedreichs extension AF is a quasi-kernel for Re A, then the impedance VΘ (z) of −1 the form (6.47) belongs to the class S0,F (R). −1 Conversely, if V (z) ∈ S0,F (R), then it can be realized as an impedance of an accumulative L-system Θ of the form (6.31)–(6.36) with Re A containing AF as a quasi-kernel and a preassigned direction operator J for which I + iV (−i)J is invertible. Proof. Following the framework of the proof of Theorem 9.8.14, we begin with the proof of the second part. First we use realization Theorems 6.5.1, 7.1.5, and 9.9.5 to construct a minimal model L-system Θ whose impedance function is V (z). Then we will show that (9.170) is equivalent to the fact that self-adjoint operator A deﬁned in (6.69), that we constructed in these theorems to be a quasi-kernel for Re A, coincides with AF , that is the Friedreichs extension of the symmetric operator A˙ of the form (6.76). Let L2G (E) be a model space constructed in the proof or theorem (6.5.1). Let also E(s) be the orthoprojection operator in L2G (E) deﬁned by (9.171). Then for the operator A deﬁned in 6.69 in the proof of Theorem 6.5.1 we have ∞ A= t dE(t), 0

and E(t) is the spectral function of operator A. As we have shown in the proof of Theorem 9.8.14 the relations (9.173) and (9.174) hold. The equality A = AF holds (see Proposition 9.5.2) if for all ϕ ∈ N−a ∞ t (dE(t)ϕ, ϕ)E = ∞, (9.190) 0

where N−a is the deﬁciency subspace of the operator A˙ corresponding to the point (−a), (a > 0). But according to Theorem 6.5.1 we have N−a described by (9.176). Taking into account (9.189) we have, for all h ∈ E, ∞ ∞ ∞ h h s (dG(s)h, h)E s(dE(s)ϕ, ϕ)L2G (E) = sd E(s) , = . t + a t + a L2 (E) (s + a)2 0

0

G

0

9.9. Realization of inverse Stieltjes functions

339

Hence the operator A = AF iﬀ ∞ 0

t (dG(t)h, h)E = ∞, (t + a)2

∀h ∈ E, h = 0.

Let us transform (9.189) ∞ ∞ t t(t + a)2 h h (dG(t)h, h) = dG(t) , E t2 + 1 t2 + 1 t+a t+a E 0 0 ∞ 2 t t h h = · dG(t) , 2 t2 + 1 (t + a) t + a t + a E 0 ∞ 2 t h h + 2a dG(t) , (t + a)2 (t2 + 1) t+a t+a E 0 ∞ 1 t (dG(t)h, h)E + a2 · . 2+1 t (t + a)2 0

(9.191)

(9.192)

Consider the obvious inequality t2 1 t2 − (t + a)2 (2t + a)(−a) − = = < 0. 2 2 2 2 2 (t + a) (t + 1) t + 1 (t + a) (t + 1) (t + a)2 (t2 + 1) Taking into account this inequality and the fact that the integral ∞ (dG(t)h, h)E t2 + 1 0 converges for all h ∈ E, we conclude that the second integral in (9.192) is convergent. Let us denote this integral as Q. Then using (9.192) and obvious estimates we obtain ∞ ∞ t t h h (dG(t)h, h) ≤ dG(t) , E t2 + 1 (t + a)2 t+a t+a E 0 0 ∞ t (dG(t)h, h)E + 2aQ + a2 , 2 (t + a) 0 or

∞ 0

t (dG(t)h, h)E ≤ (a2 + 1) t2 + 1

0

∞

t (t + a)2

h h dG(t) , + 2aQ. t+a t+a E

Since V (z) ∈ S0−1 (R), then (9.186) holds and the integral on the left diverges causing the integral on the right-hand to side diverge as well. Thus A = AF . Now we can prove the ﬁrst part of the theorem. Let Θ be our L-system with AF that is a quasi-kernel for Re A, and the impedance function VΘ (z). Then VΘ (z) can be realized as an impedance function of the model L-system Θ1 constructed

340 Chapter 9. Canonical L-systems with Contractive and Accretive Operators in the proof of Theorem 7.1.5. Repeating the argument of the second part of the proof of Theorem 9.8.14 with AK replaced by AF we conclude that the quasi-kernel operator A of Θ1 (deﬁned by (6.69)) is the Friedreichs self-adjoint extension and hence has property (9.190) that in turn causes (9.191) for any a > 0. Let a = 1, then by (9.191), ∞ ∞= 0

t (dG(t)h, h)E ≤ (t + 1)2

∞ 0

t (dG(t)h, h)E , t2 + 1

∀h ∈ E, h = 0,

and hence the integral on the right diverges and (9.189) holds. This completes the proof.

Chapter 10

L-systems with Schr¨odinger operator In this chapter we apply some of the previous results to the problems related to the Schr¨odinger operator. First we give a complete characterization of all (∗)extensions of ordinary diﬀerential operators. Then we use it to provide a thorough description of all L-systems with a Schr¨odinger operator Th . Moreover, we describe the class of scalar Stieltjes/(inverse Stieltjes)-like functions that can be realized as impedance functions of L-systems with a Schr¨odinger operator Th . The formulas that restore an L-system uniquely from a given Stieltjes/(inverse Stieltjes)-like function as the impedance function of this L-system are derived. These formulas allow us to solve the inverse problem and ﬁnd the exact value of the parameter h in the deﬁnition of Th as well as a real parameter μ that appears in the construction of the elements of the L-system being realized. A detailed study of these formulas shows the dynamics of the restored parameters h and μ in terms of the changing free term in the integral representation of a realizable function. We also provide a full description of accretive, sectorial, and extremal boundary value problems for a Schr¨odinger operator Th on the half-line in terms of the boundary parameter h.

10.1 (∗)-extensions of ordinary diﬀerential operators In this section we will give a complete description of all (∗)-extensions of ordinary diﬀerential operators. First we note that for the case of a densely-deﬁned symmetric operator A˙ the set of formulas (4.60) can be re-written (see p. 110). Let U be an isometric operator from the defect subspace Ni of the symmetric operator A˙ onto the defect subspace N−i . Then the formulas below establish a one-to one correspondence between all (∗)-extensions A of an operator T and all isometries U, Af = A˙ ∗ f + iR−1 (U − I)ϕ,

A∗ f = A˙ ∗ f + iR−1 (U − I)ψ,

f ∈ H+ , (10.1)

Y. Arlinskii et al., Conservative Realizations of Herglotz–Nevanlinna Functions, Operator Theory: Advances and Applications, DOI 10.1007/978-3-7643-9996-2_10, © Springer Basel AG 2011

341

342

Chapter 10. L-systems with Schr¨ odinger operator

where ϕ, ψ ∈ Ni are uniquely determined from the conditions f − (U + I)ϕ ∈ Dom(T ),

f − (U + I)ψ ∈ Dom(T ∗ ),

and R ∈ [H− , H+ ] is the Riesz-Berezansky operator of the triplet H+ ⊂ H ⊂ H− . If the symmetric operator A˙ has deﬁciency indices (n, n), then formulas (10.1) can be rewritten in the form Af = A˙ ∗ f +

n #

Δk (f )χk ,

A∗ f = A˙ ∗ f +

k=1

n #

δk (f )χk ,

(10.2)

k=1

where {χj }n1 ∈ H− is a basis in the subspace R−1 (U − I)Ni , and {Δk }n1 , {δk }n1 , are bounded linear functionals on H+ with the properties Δk (f ) = 0,

∀f ∈ Dom(T ),

δk (f ) = 0,

∀f ∈ Dom(T ∗ ).

(10.3)

We will give a description of all (∗)-extensions of a non-symmetric diﬀerential operator on a half-line. Consider the following self-adjoint quasi-diﬀerential expression on L2 [a, +∞], l(y) = (−1)n (p0 y (n) ) + (−1)n−1 (p1 y (n−1) ) + . . . + pn y, where p01(x) , p1 (x), . . . , pn (x) are locally summable functions on [a, +∞). Denote by D ∗ the set of functions y ∈ L2 [a, +∞) for which the quasi-derivatives [207] y [k] = y [n+k]

dk y dn y [n] , k = 0, 1 . . . , n − 1, y = p , 0 dxk n−k dxn d y d [n+k−1] = pk n−k − y , k = 1, 2 . . . , n, dx dx

are locally absolutely continuous and y [2k] = l(y) belongs to L2 [a, +∞). Consider the symmetric operator

˙ = l(y), Ay y ∈ D∗ , k = 1, 2, . . . , 2n, y [k−1] (a) = 0, and suppose that this operator has deﬁciency indices (n, n). Consider the operators ⎧ ⎧ ∗ T y = l(y), T y = l(y), ⎪ ⎪ ⎪ ⎪ ⎨ y ∈ D∗ , ⎨ y ∈ D∗ , (j = 1, 2, . . . , n). 2n 2n ) ) ⎪ ⎪ ⎪ ⎪ υjk y [k−1] (a) = 0, υ∗jk y [k−1] (a) = 0, ⎩ ⎩ k=1

k=1

(10.4) ˙ = Dom(T ) ∩ Dom(T ∗ ) and ρ(T ) = ∅. It is well known Suppose also that Dom(A) that Dom(A˙ ∗ ) = D∗ , A˙ ∗ y = l(y), y ∈ D∗ .

10.1. (∗)-extensions of ordinary diﬀerential operators

343

Consider H+ = Dom(A˙ ∗ ) = D∗ with the inner product (y, z)+ = (y, z)L2 [a,+∞) + (l(y), l(y))L2 [a,+∞) , n

and construct the rigged Hilbert space H+ ⊂ L2 [a, +∞) ⊂ H− . Let {υj }i and n {υ∗j }i be the set of elements in H− generating the functionals (f, υj ) =

2n #

υjk f

[k−1]

(a),

(f, υ∗j ) =

k=1

2n #

υ∗jk f [k−1] (a),

(j = 1, . . . , n).

k=1

n

n

Let {χj }i ∈ H− be the set of elements linearly independent with {υj }i and generating the functionals (f, χj ) =

2n #

χjk f [k−1] (a),

(j = 1, 2, . . . , n).

k=1

We also suppose that the operator

: = l(y), Ay y ∈ H+ , (y, χj ) = 0,

(j = 1, 2, . . . , n),

is self-adjoint in L2 [a, +∞). Theorem 10.1.1. Let T be a diﬀerential operator of the form (10.4).Then the formulas Ay = l(y) +

n #

(y, υj )ckj χk ,

A∗ y = l(y) +

k,j=1

n #

(y, υ∗j )dkj χk ,

(10.5)

k,j=1

establish a one-to-one correspondence between the set of (∗)-extensions A of the operator T and the set of matrices C = cjk and D = dkj uniquely determined by the matrices U = υjk , U∗ = υ∗jk , V∗ = χjk by the relation 5 6 5 6 0 I 0...± 1 ∗ ∗ ∗ U∗ D V − V CU = , ±I := . (10.6) −I 0 ±1 . . . 0 Proof. Let A be a (∗)-extension of an operator T of the form (10.4). Then, as we mentioned above, A has the form (10.2) with the property (10.3). Therefore Δk (f ) =

n #

(f, υj )ckj ,

δk (f ) =

j=1

n #

(f, υ∗j )dkj ,

j=1

and the matrices C = ckj and D = dkj are invertible. It follows from (10.2) that (A˙ ∗ f, g) − (f, A˙ ∗ g) =

n # k,j=1

(υ∗j , g)dkj (f, χk ) −

n # k,j=1

(f, υj )ckj (χk , g).

(10.7)

344

Chapter 10. L-systems with Schr¨ odinger operator

Applying the Lagrange formula for diﬀerential operators and taking into account that the deﬁciency indices of the operator A˙ are (n, n), we get, using (10.7) with g = f , that # − [f [k−1] (a)[2n−k] (a) − f [2n−k] (a)f [k−1] (a)] k

=

## ( dkj υ∗jl χkm )f [l−1] (a)f [m−1] (a) l,m

k,j

(10.8)

## − ( ckj υjl χkm )f [l−1] (a)f [m−1] (a). l,m k,j

Hence, we obtain (10.6). Going back to (10.6) we get (10.8), and thus (10.5) determines a (∗)-extension of an operator T of the form (10.4).

10.2 Canonical L-systems with Schr¨ odinger operator We begin this section by applying Theorem 10.1.1 in the context of a non-selfadjoint Schr¨odinger operator. Let H = L2 [a, +∞) and l(y) = −y + q(x)y, where q is a real locally summable function. Suppose that the symmetric operator

˙ = −y + q(x)y, Ay (10.9) y(a) = y (a) = 0, has deﬁciency indices (1,1). Let D∗ be the set of functions locally absolutely continuous together with their ﬁrst derivatives such that l(y) ∈ L2 [a, +∞). Consider H+ = Dom(A˙ ∗ ) = D ∗ with the scalar product ∞ (y, z)+ = y(x)z(x) + l(y)l(z) dx, y, z ∈ D∗ . a

Let H+ ⊂ L2 [a, +∞) ⊂ H− be the corresponding triplet of Hilbert spaces. Consider elements υ, υ∗ , χ ∈ H− , generating functionals 1

(Im h) 2 (f, χ) = [μf (a) − f (a)], |μ − h| (f, υ) = hf (a) − f (a), and operators

T y = l(y) = −y + q(x)y, (y, υ) = hy(a) − y (a) = 0,

Im h > 0, Im μ = 0,

(f, υ∗ ) = hf (a) − f (a),

T ∗ y = l(y) = −y + q(x)y, (y, υ∗ ) = hy(a) − y (a) = 0,

: = l(y) = −y + q(x)y, Ay , Im μ = 0. (y, χ) = μy(a) − y (a) = 0,

(10.10)

10.2. Canonical L-systems with Schr¨odinger operator

345

:=A ?∗ . By Theorem 10.1.1, the (∗) - extension A of an It is well known [3] that A operator T of the form (10.10) can be represented as Ay = −y + q(x)y + c(y, υ)χ,

A∗ y = −y + q(x)y + d(y, υ∗ )χ,

where the numbers c and d satisfy the relation (10.6). According to (10.4) and (10.10), the matrices U, U∗ and V have the form U = h,

−1 ,

U∗ = h, −1 ,

V=

(Im h)1/2 μ, |μ−h|

−(Im h)1/2 |μ−h|

.

Therefore, d U∗∗ V − c V ∗ U = d

5

6

h −1

1/2

(Im h) |μ−h|

1/2

h) μ − (Im |μ−h|

5

6 0 1 = . −1 0

⎡

−

⎤

(Im h)1/2 μ ⎦ c ⎣ |μ−h| (Im h)1/2 − |μ−h|

h, −1

Solving this matrix equation with respect to c and d we get c=

|μ − h| (μ − h)(Im h)

1 2

,

d=

|μ − h| 1

(μ − h)(Im h) 2

.

Consider now the elements μδ(x − a) + δ (x − a) and hδ(x − a) + δ (x − a) in H− , and generating functionals (f, μδ(x − a) + δ (x − a)) = μf (a) − f (a), (f, hδ(x − a) + δ (x − a)) = hf (a) − f (a), where δ(x − a) and δ (x − a) are the delta-function and the derivative of the delta-function at the point a. Thus, we have proved the following. Theorem 10.2.1. The set of all (∗)-extensions A of a non-self-adjoint Schr¨ odinger operator Th of the form (10.10) can be represented in the form 1 [y (a) − hy(a)] [μδ(x − a) + δ (x − a)], μ−h 1 A∗ y = −y + q(x)y − [y (a) − hy(a)] [μδ(x − a) + δ (x − a)]. μ−h Ay = −y + q(x)y −

(10.11)

Moreover, the formulas (10.11) establish a one-to-one correspondence between the set of all (∗)-extensions of a Schr¨ odinger operator Th of the form (10.10) and all real numbers μ ∈ [−∞, +∞].

346

Chapter 10. L-systems with Schr¨ odinger operator

Consider the symmetric operator A˙ of the form (10.9) with defect indices (1,1), generated by the diﬀerential operation l(y) = −y + q(x)y. Let ϕk (x, λ), (k = 1, 2) be the solutions of the following Cauchy problems: ⎧ ⎧ ⎨ l(ϕ1 ) = λϕ1 ⎨ l(ϕ2 ) = λϕ2 ϕ1 (a, λ) = 0 , ϕ2 (a, λ) = −1 . (10.12) ⎩ ⎩ ϕ1 (a, λ) = 1 ϕ2 (a, λ) = 0 It is well known [3] that there exists a function m∞ (λ) (note that −m∞ (λ) is a Herglotz-Nevanlinna function called the Weyl-Titchmarsh function) for which ϕ(x, λ) = ϕ2 (x, λ) + m∞ (λ)ϕ1 (x, λ)

(10.13)

belongs to L2 [a, +∞). Consider an L-system with a (∗)-extension of a non-self-adjoint Schr¨odinger operator described in Section 10.1 as a state-space operator. One can easily check that the (∗)-extension Ay = −y + q(x)y −

1 [y (a) − hy(a)] [μδ(x − a) + δ (x − a)], μ−h

Im h > 0,

of the non-self-adjoint Schr¨odinger operator Th of the form (10.10) satisﬁes the condition A − A∗ Im A = = (., g)g, (10.14) 2i where 1 (Im h) 2 g= [μδ(x − a) + δ (x − a)], (10.15) |μ − h| and δ(x−a), δ (x) are the delta-function and its derivative at the point a. Moreover, 1

(y, g) =

(Im h) 2 [μy(a) − y (a)], |μ − h|

(10.16)

where y ∈ H+ , g ∈ H− , and H+ ⊂ L2 (a, +∞) ⊂ H− is the triplet of Hilbert spaces as discussed in Theorem 10.2.1. Let E = C, then we deﬁne Kc = cg,

K ∗ y = (y, g),

y ∈ H+

and obtain Im A = KK ∗ . Applying (6.36) the array A K Θ= H+ ⊂ L2 [a, +∞) ⊂ H−

c ∈ C,

1 C

(10.17)

(10.18)

is a canonical L-system with state-space operator A of the form (10.11), the direction operator J = 1, and the channel operator K of the form (10.17). Indeed, the following theorem applies.

10.2. Canonical L-systems with Schr¨odinger operator Theorem 10.2.2. The system Θ of the form (Th − zI)x = β[μδ(x − a) + δ (x − a)]ϕ− , ϕ+ = ϕ− − 2iβ[μx(a) − x (a)],

347

ϕ± ∈ C,

(10.19)

with β > 0 and Th (Im h > 0) deﬁned by (10.10), is a scattering L-system if and 1

only if β = formula

(Im h) 2 |μ−h|

. The transfer function WΘ (z) of this system is given by the μ − h m∞ (z) + h . μ − h m∞ (z) + h

(10.20)

(m∞ (z) + μ) Im h , (μ − Re h) m∞ (z) + μRe h − |h|2

(10.21)

WΘ (z) = Moreover, the function VΘ (z) =

is the transfer function of the impedance system (Aξ − zI)x = β[μδ(x − a) + δ (x − a)]ψ− , ψ+ = β[μx(a) − x (a)]), where Aξ has the form

ψ± ∈ C,

Aξ y = −y + q(x)y , ξy(a) = y (a)

and ξ=

μRe h − |h|2 . μ − Re h

(10.22)

Proof. Let Θ of the form (10.19) be an L-system with J = I. Then by Deﬁnition 6.3.4 there is a (∗)-extension A of Th such that Im A = Kβ Kβ∗ , where Kβ c = cβ[μδ(x − a) + δ (x − a)],

Kβ∗ y = β[μy(a) − y (a)],

y ∈ H+ , c ∈ C. (10.23) All the (∗)-extensions of Th , however, are described by (10.11). Comparing formula (10.23) with formulas (10.14)–(10.17), applying uniqueness Theorem 4.3.9, and performing straightforward calculations, we obtain |β|2 =

(Im h) . |μ − h|2 1 2

h) Since β is real and positive we have β = (Im . Formulas (10.20) and (10.21) are |μ−h| obtained by direct computations. The remaining statement of the theorem follows from the fact that Aξ with ξ described by (10.22) is the quasi-kernel of Re A that corresponds to system (10.19) and the deﬁnition of the impedance system.

348

Chapter 10. L-systems with Schr¨ odinger operator 1 2

h) Now let β = (Im . We need to show that (10.19) is an L-system. As we |μ−h| already mentioned, all the (∗)-extensions of Th are described by (10.11). Hence, (10.14) and (10.15) imply that Im A = KK ∗ only if β has the above value. Consequently, (10.19) is an L-system only for that β.

One can also see that (10.21) implies Im VΘ (i) = −Im h

|μ − h|2 Im m∞ (i) , ,2 . 2 (μ − Re h) , μRe h−|h|2 , ,m∞ (i) + μ−Re h ,

(10.24)

Now we can give a good illustration to Theorems 8.2.1 and 8.2.2 of Section 8.2 for the case when a J-unitary factor B = −1. Since Theorem 10.2.1 establishes a one-to-one correspondence between all (∗)-extensions A of the form (10.11) and all real numbers μ ∈ [−∞, +∞], we can parameterize all A’s via μ’s and label them accordingly. We set then 1 [y (a) − hy(a)] [μδ(x − a) + δ (x − a)], μ−h 1 A∗μ y = −y + qy − [y (a) − hy(a)] [μδ(x − a) + δ (x − a)], μ−h

(10.25)

1 [y (a) − hy(a)] [ξδ(x − a) + δ (x − a)], ξ−h 1 A∗ξ y = −y + qy − [y (a) − hy(a)] [ξδ(x − a) + δ (x − a)], ξ−h

(10.26)

Aμ y = −y + qy −

and Aξ y = −y + qy −

with ξ deﬁned by (10.22). Let Aμ Θμ = H+ ⊂ L2 [a, +∞) ⊂ H−

and Θξ =

Aξ H+ ⊂ L2 [a, +∞) ⊂ H−

Kμ

1 , C

(10.27)

Kξ

1 , C

(10.28)

be the corresponding L-systems, where Kμ and Kξ are parameterized and labeled according to (10.16)-(10.17). From relation (10.20) it follows that WΘμ (z) =

μ − h m∞ (z) + h , μ − h m∞ (z) + h

WΘξ (z) =

ξ=

μRe h − |h|2 , μ − Re h

Because

ξ − h m∞ (z) + h . ξ − h m∞ (z) + h

(10.29)

10.3. Accretive and sectorial boundary problems for a Schr¨odinger operator 349 one obtains ξ−h = ξ−h

μRe h−|h|2 μ−Re h μRe h−|h|2 μ−Re h

−h −h

=

μRe h − |h|2 − μh + Re hh μRe h − |h|2 − μh + Re hh

μ(Re h − h) − h(h − Re h) −iμIm h + ihIm h = μ(Re h − h) − h(h − Re h) iμIm h − ihIm h −(μ − h)Im h μ−h = =− . (μ − h)Im h μ−h =

Therefore, from (10.29) we get WΘμ (z) = −WΘξ (z).

(10.30)

The relations (10.30) and (6.48) imply VΘμ (z) = Kμ∗ (Re Aμ − zI)−1 Kμ = i[WΘμ (z) − I] [WΘμ (z) + I]−1 = i[−WΘξ (z) − I][−WΘξ (z) + I]−1 = −VΘ−1 (z). ξ Finally,

VΘμ (z) = −VΘ−1 (z). ξ

(10.31)

10.3 Accretive and sectorial boundary problems for a Schr¨ odinger operator Suppose that the symmetric operator A˙ of the form (10.9) with deﬁciency indices (1,1) is non-negative. Theorem 10.3.1. A Schr¨ odinger operator Th , (Im h > 0) of the form (10.10) is accretive if and only if the function ¯ ¯ 1 − [(m∞ (z) + h)/(m ∞ (z) + h)][(m∞ (−1) + h)/(m∞ (−1) + h)] ¯ ¯ 1 + [(m∞ (z) + h)/(m ∞ (z) + h)][(m∞ (−1) + h)/(m∞ (−1) + h)] (10.32) is holomorphic in Ext[0, +∞), Vh (−0) = 0, Vh (−∞) = 0, and Vh (z) = −i

1 + Vh (−0) Vh (−∞) ≥ 0.

(10.33)

Moreover, an accretive operator Th of the form (10.10) is α-sectorial if and only if inequality (10.33) is strict. In this case the exact value of the angle α is α = arccot

1 + Vh (−0)Vh (−∞) . |Vh (−∞) − Vh (−0)|

(10.34)

350

Chapter 10. L-systems with Schr¨ odinger operator

Proof. Taking into account that (−1) is a point of regular type for the operator A˙ of the form (10.9), we can show that (−1) is a regular point for the operator Th of the form (10.10). Indeed, assume that (−1) is an eigenvalue for Th . Then for some non-zero f ∈ Dom(Th ) we have Af = −f , or Re Af + i(f, g)g = −f, where g is deﬁned by (10.15). This implies (Re Af, f ) + i|(f, g)|2 = −(f, f ). The right-hand side of the above equation is real and hence (f, g) = 0. Using (10.15) and (10.10) yields the system

hf (a) − f (a) = 0, μf (a) − f (a) = 0. Taking into account that Im h = 0 and μ is real, the system has only the solution ˙ But this contradicts the fact f (a) = f (a) = 0, which means that f ∈ Dom(A). ˙ Consequently, (−1) is also a that (−1) is a point of regular type for the operator A. point of regular type for the operator Th . Indeed, (I +Th )−1 can not be unbounded ˙ −1 is bounded and Ran(I + Th) is diﬀerent from Ran(I + A) ˙ no more since (I + A) than by one dimension. Thus, all we need to show is that Ran(I + Th ) = H. ˙ Now if we take a Assuming the contrary, we get Ran(I + Th ) = Ran(I + A). ˙ then the vector (Th + I)f = h ∈ vector f ∈ Dom(Th ) such that f ∈ / Dom(A), ˙ can be represented as Ran(I + Th ) = Ran(I + A) ˙ 0 = (I + Th )f, h = (I + A)f

˙ f0 ∈ Dom(A).

Hence (I + Th )(f − f0 ) = 0, which contradicts the existence of (I + Th )−1 . This proves that (−1) is a regular point for the operator Th . Consider the linear-fractional transformation Sh = (I − Th )(I + Th )−1 , of the operator Th . According to Lemma 9.5.12, the operator Sh is an α-co-sectorial contraction if and only if Th is an α-sectorial operator. Without loss of generality, we can assume that operator Th (and consequently Sh ) is prime. If Th is not prime then, since (−1) is its regular point, we can apply Lemma 6.6.9 and conclude that the non-negative operator A˙ of the form (10.9) is not prime as well. Then there is an invariant subspace H2 of H such that H = H1 ⊕ H2 and ˙ H2 , A˙ 2 = A

A˙ = A˙ 1 ⊕ A˙ 2 ,

where A˙ 2 is a non-negative self-adjoint operator in H2 . Similarly we can represent Th = Th,1 ⊕ A˙ 2 . Hence, the operator Sh will also split into Sh = Sh,1 ⊕ Sh,2 , where Sh,2 is a self-adjoint contraction. Applying (9.83), (9.84), (8.30) together with Lemma 9.5.12 and Theorem 9.4.2, we have a proof of the statement of the theorem.

10.3. Accretive and sectorial boundary problems for a Schr¨odinger operator 351 Consider a non-negative Schr¨odinger operator A˙ of the form (10.9) and Th deﬁned by (10.10). According to Theorem 10.2.1 the set of all (∗)-extensions A of Th ⊃ A˙ is given by (10.11). We can use (10.11) to describe all self-adjoint ¯ For a real ξ we have (∗)-extensions A of Th when h = h. 1 [y (a) − ξy(a)] [μδ(x − a) + δ (x − a)] μ−ξ ξ 1 = −y + q(x)y − [ y (a) − y(a)] [μδ(x − a) + δ (x − a)] μ−ξ ξ 1 1 = −y + q(x)y − [ y (a) − y(a)] [μδ(x − a) + δ (x − a)]. μ/ξ − 1 ξ

Ay = −y + q(x)y −

Setting h = 1/ξ we obtain the formula for self-adjoint (∗)-extensions of Th ⊃ A˙ A y = −y + q(x)y −

1 [hy (a) − y(a)] [μδ(x − a) + δ (x − a)]. hμ − 1

(10.35)

Let ϕk (x, λ), (k = 1, 2) be the solutions of Cauchy problems deﬁned in (10.12). As we have already mentioned in (10.13) ϕ(x, λ) = ϕ2 (x, λ) + m∞ (λ)ϕ1 (x, λ) belongs to L2 [a, +∞). Consider an equation (A − λI)ψ(x, λ) = χ,

(10.36)

where χ = μδ(x − a) + δ (x − a) and A is deﬁned by (10.35). It is known from Chapter 4 that its solution ψ(x, λ) belongs to Nλ , the deﬁciency subspace of the ˙ Thus we have operator A. ψ(x, λ) = g(λ)ϕ(x, λ), where ϕ(x, λ) is deﬁned by (10.13). It is easy to see that l(ψ) = λψ. Furthermore, (ψ, υ) = ψ(a, λ) − hψ (a, λ) = g(λ)[ϕ(a, λ) − hϕ (a, λ)] = g(λ)[−1 − h m∞ (λ)], where υ = δ(x − a) + hδ (x − a). Using (10.11) and (10.36) we have 1 g(λ)[−1 − h m∞ (λ)] = 1. hμ − 1 This implies g(λ) = and hence ((A − λI)−1 χ, χ) =

1 − hμ , 1 + h m∞ (λ) 1 − hμ [−μ − m∞ (λ)]. 1 + h m∞ (λ)

(10.37)

352

Chapter 10. L-systems with Schr¨ odinger operator

Let us set in (10.37) h = 0 and μ = −m∞ (−0) (assuming that m∞ (−0) < ∞). Then we get QF (λ) = ((A − λI)−1 χ, χ) = m∞ (−0) − m∞ (λ).

(10.38)

It is easy to see that the function QF of the form (10.38) satisﬁes all the conditions to be a Q-function that corresponds to the Friedrichs extension AF y = −y +q(x)y, ˙ Thus we can apply Theorem 9.5.18 to QF and obtain that QF (x) → y(a) = 0 of A. −∞ when x → −∞ on the real axis. Then (10.38) implies that for a real x, m∞ (x) → +∞ when x → −∞.

(10.39)

Theorem 10.3.2. A non-negative Schr¨ odinger operator A˙ of the form (10.9) admits non-self-adjoint accretive (α-sectorial, 0 < α < π/2) extensions Th of the form (10.10) if and only if m∞ (−0) < ∞. Proof. Let m∞ (−0) < 0. We write c = m∞ (−1) and b = m∞ (−0). First, we ﬁnd the values of h (Im h > 0) for which the inequality (10.33) holds. Using (10.32) and the above notations we get Vh (−0) =

(b − c)Im h ¯. cb + (c + b)Re h + h h

(10.40)

Taking into account the properties of the QF -function and (10.32) we have Vh (−∞) =

Im h . c + Re h

(10.41)

Furthermore, 1 + Vh (−0)Vh (−∞) = 1 +

(Im h)2 (b − c) ¯ + Re h) . [cb + (c + b)Re h + h h](c

(10.42)

Setting h = x + iy and using (10.42) we obtain 1 + Vh (−0)Vh (−∞) =

[cb + (c + b)x + x2 + y 2 ](c + x) + (b − c)y 2 . [cb + (c + b)x + x2 + y 2 ](c + x)

(10.43)

We introduce functions f (x, y) and g(x, y) to denote numerator and denominator of (10.43), respectively f (x, y) = x2 (b + 2c + x) + y 2 (b + x) + (2bc + c2 )x + bc2 , g(x, y) = x2 (x + b + 2c) + y 2 (x + c) + (2bc + c2 )x + bc2 . Let us ﬁnd out when f (x, y) ≥ 0. If x = −b, then f (−b, y) = 2b2 c − 2b2 c − bc2 + bc2 = 0,

(10.44)

10.3. Accretive and sectorial boundary problems for a Schr¨odinger operator 353

y

-c

-b

0

x

Figure 10.1: f (x, y)/g(x, y) ≥ 0 domain for y > 0. Clearly, for x > −b we have that y 2 (b + x) > 0 while for x < −b that y 2 (b + x) < 0. Consider the function f (x) = x3 + (b + 2c)x2 + (2bc + c2 )x + bc2 . Using the derivative to ﬁnd the intervals where f (x) is monotone we get that it is increasing on (−∞, −c) ∪ (− 2b+c 3 , +∞). Besides, since 2b + c 2b + c f (−c) = 0, f − < 0, −c < − < −b, 3 3 f (x, y) = f (x) + y 2 (b + x), then for any x < −b, f (x, y) < 0 for all y > 0 and for any x > −b, f (x, y) > 0 for all y > 0. Similar study can be applied to the function g(x, y). As the result we obtain that for any x > −b, g(x, y) > 0 for all y > 0 and for any x < −c, g(x, y) < 0 for all y > 0, since g(x, y) = f (x) + y 2 (x + c). Now let us describe the behavior of g(x, y) for −c < x < −b. Evidently, the domain where g(x, y) > 0 for −c < x < −b follows from the inequality y2 > −x2 − (b + c)x − bc. The roots of the quadratic expression on the right are x1 = −c and x2 = −b. Hence the domain satisfying the inequality f (x, y)/g(x, y) ≥ 0 has the form presented in Figure 10.1. We are going to check whether Vh (z) is holomorphic on Ext[0, +∞) if our parameter h belongs to this domain. First we will show that Vh (z) is not holomorphic on Ext[0, +∞) if h belongs to the domain shown in Figure 10.2 below. In order to do that it is enough to prove that there is a z ∈ Ext[0, +∞) such that 1+

¯ c+h m∞ (z) + h · ¯ = 0, m∞ (z) + h c + h

(10.45)

354

Chapter 10. L-systems with Schr¨odinger operator

y

-c

-b

x

0

Figure 10.2: Domain for h = x + iy

for h in the domain in Figure 10.2. Since h = x + iy, then equation (10.45) is equivalent to m∞ (z) = (−x2 − y 2 − cx)/(c + x) and hence QF (z) = m∞ (−0) − m∞ (z) = b −

−x2 − y 2 − cx c+x

x2 + (c + b)x + cb + y 2 = . c+x

(10.46)

By a direct check one conﬁrms that the fraction x2 + (c + b)x + cb + y 2 , c+x considered in the shaded region in Figure 10.1, is negative if and only if a point (x, y) belongs to that shaded region in Figure 10.2. Using Theorem 9.5.18, (10.38), and (10.39) we get that the QF -function is negative on the left real semi-axis, and hence equation (10.46) is solvable for z. Therefore, the function Vh (z) is holomorphic on Ext[0, +∞) only if h is in the domain (shaded region y > 0, x > −b = −m∞ (−0)) depicted in Figure 10.3, and inequality (10.33) holds for h from that domain as well. Thus, according to Theorem 10.3.1, any h from the ˙ Using domain in Figure 10.3 generates an accretive extension Th of the operator A. (10.40) and (10.41) we obtain 1 + Vh (−0)Vh (−∞) [cb + (b + c)x + x2 + y 2 ](c + x) + (b − c)y 2 = Vh (−∞) − Vh (−0) y([cb + (b + c)x + x2 + y 2 ] − (b − c)(c + x)) b+x m∞ (−0) + x = = . y y Therefore, we have shown that if for any given h, (Im h > 0) an operator Th of the form (10.10) is α-sectorial, then the exact value of the angle α is shown

10.3. Accretive and sectorial boundary problems for a Schr¨odinger operator 355

y h

a - m¥ (0)

x

0

Figure 10.3: Value for the angle α

in Figure 10.3. Conversely, let for some h, (Im h > 0) the operator Th of the form (10.10) be α-sectorial for some angle α, (0 < α < π/2). We will show that in this case m∞ (−0) < ∞. Assume the contrary, i.e., m∞ (−0) = ∞. Then by Theorem 10.3.1 the value of sectorial angle α is found via (10.34). Consequently, our assumption and (10.32) imply that Vh (−0) = Vh (−∞). But then (10.34) yields that α = arccot(∞) = 0 which is impossible. The following theorem immediately follows from the reasoning in the proof above. Theorem 10.3.3. If an accretive Schr¨ odinger operator Th , (Im h > 0) of the form (10.10) is α-sectorial, then α = arccot

Re h + m∞ (−0) Im h

.

(10.47)

Conversely, if h, (Im h > 0) is such that Re h + m∞ (−0) > 0, then operator Th of the form (10.10) is α-sectorial and α is determined by (10.47). It is easy to see that Th∗ = Th¯ and hence all the accretive and α-sectorial non-self-adjoint operators Th are described by only such values of parameter h for which Re h ≥ −m∞ (−0). Also, since a numeric range of an α-sectorial operator Th lies in the sector | arg ζ| ≤ α, then the spectrum of Th also belongs to this sector. Therefore, formula (10.47) allows one to “control” the spectrum with the help of a boundary parameter h. Now let us ﬁnd the value of the parameter h for which operator Th of the form (10.10) generates a Kre˘in-von Neumann extension (see Section 9.5) of operator A˙ ¯ belong to the line Re h = −m∞ (−0) (see Figure of the form (10.9). Let h = h 10.3). Then the operator (I − Th )(I + Th )−1 is a contraction according to Lemma

356

Chapter 10. L-systems with Schr¨odinger operator

9.5.12. Taking into account (9.84) we get 1 1 (SM + Sμ ) + ζ(h)(SM − Sμ ), (10.48) 2 2 ˙ ˙ −1 given + A) where SM and Sμ are the extreme sc-extensions of S˙ = (I − A)(I by (9.37) and (9.38), and |ζ(h)| ≤ 1 for all h’s in the domain in Figure 10.3. Since the left-hand side of (10.48) (for non-real h that belongs to the line Re h = −m∞ (−0)) is a contraction but not a α-co-sectorial contraction for any α, (0 < α < π/2), then (see Theorem 9.2.7) |ζ(h)| = 1, for all h ∈ (Re h = −m∞ (−0)). Besides, when h moves along the above line, ζ(h) traces a unit circle with points ±1 removed. By virtue of Theorem 8.2.4, the function V (z) of the form (9.83) constructed for the operator (I − Th)(I + Th )−1 possesses a property that V (1+) = Vh (−0), V (−1−) = Vh (−∞), where Vh (−0) and Vh (−∞) are deﬁned by (10.40) and (10.41). Taking into account (10.40) and (10.41), for h → −m∞ (−0) along the line Re h = −m∞ (−0) we have that ζ(h) → 1. Since the right-hand side of (10.48) coincides with soft extension SM , then for h = −m∞ (−0) the operator ˙ Therefore the set of all Th generates a Kre˘in-von Neumann extension AK of A. ˙ self-adjoint extensions of a non-negative operator A of the form (10.9) is deﬁned by operators Th of the form (10.10) under the condition that h is real and h ≥ −m∞ (−0). Consequently, the set of all accretive and α-sectorial boundary value problems for a Schr¨odinger operator is deﬁned by a parameter h from the domain depicted in Figure 10.3. Moreover, if: (I − Th )(I + Th )−1 =

1. h belongs to the real line, then we obtain all accretive self-adjoint boundary value problems; ¯ then we obtain all 2. h does not belong to a line Re h = −m∞ (−0) and h = h, α-sectorial boundary value problems and the exact value of the angle α is given by (10.47); ¯ lies on the line Re h = −m∞ (−0), then we obtain all accretive bound3. h = h ary value problems that are not α-sectorial for any α, (0 < α < π/2). Example. Consider a non-negative symmetric Schr¨odinger (Bessel type) operator with deﬁciency indices (1, 1): 2 ˙ = −y + ν − (1/4) y, Ay x2 y (1) = y(1) = 0,

ν=

k+1 , k = 0, 1, 2, . . . , 2

in the Hilbert space H = L2 (1, +∞). It was shown in [53] that

The Friedrichs extension AF

1 m∞,ν (−0) = ν − . 2 ˙ of A in this case has the form

AF y = −y +

ν 2 − (1/4) y, x2

y(1) = 0.

10.4. Functional model for symmetric operator with deﬁciency indices (1,1) 357 The Kre˘ın-von Neumann extension is the boundary value problem AK y = −y +

ν 2 − (1/4) y, x2

y (1) + (ν − (1/2)) y(1) = 0.

When ν = 12 , the Kre˘ın-von Neumann extension is simply the Neuman boundary value problem AK y = −y , y (1) = 0. This example shows that the Kre˘ın-von Neumann extension may be not only the Neuman but also a mixed boundary value problem.

10.4 Functional model for symmetric operator with deﬁciency indices (1,1) In this section we present material of auxiliary nature dealing with a functional model for a symmetric operator with deﬁciency indices (1,1). Let H be a Hilbert space and A˙ be a closed densely-deﬁned symmetric operator with deﬁciency indices (1,1). According to von Neumann formulas (1.13) every self-adjoint extension Aα of A˙ can be represented in the form ˙ + c(ϕ − e2iα ψ), ϕ ∈ Ni , ψ ∈ N−i , Dom(Aα ) = Dom(A) ˙ + c(iϕ + ie2iα ψ), h ∈ Dom(A), ˙ c ∈ C, Aα = Ah

(10.49)

where ϕ = ψ = 1 and α ∈ [0, π). As usual we set H+ = Dom(A˙ ∗ ) and construct a rigged Hilbert triplet H+ ⊂ H ⊂ H− . Let us now consider a Hilbert space H+,α = Dom(Aα ) equipped with the scalar product (f, g)+,α = (f, g) + (Aα f, Aα g),

(f, g ∈ H),

where Aα is some self-adjoint extension of A˙ in H of the form (10.49). Following the steps of Sections 2.1 and 6.5 we construct another rigged Hilbert space H+,α ⊂ H ⊂ H−,α with the Riesz-Berezansky operator R ∈ [H−,α , H+,α ] such that (f, g) = (f, Rg)+,α ,

g−,α = Rg+,α,

(∀f ∈ H+,α , g ∈ H),

(10.50)

and (f, g)−,α = (Rf, Rg)+,α ,

(∀f, g ∈ H−,α ).

Lemma 10.4.1. Let A˙ be a densely-deﬁned symmetric operator in Hilbert space H with deﬁciency indices (1, 1) and Aα be its self-adjoint extension in H. If an element g ∈ H−,α is such that g ∈ / H, then the operator Aˆα f = Aα f, is densely deﬁned in H.

Dom(Aˆα ) = {f ∈ Dom(Aα ) | (f, g) = 0},

(10.51)

358

Chapter 10. L-systems with Schr¨ odinger operator

Proof. Assuming that Dom(Aˆα ) = H, we have the existence of a vector φ ∈ H such that (f, φ) = 0 for all f ∈ Dom(Aˆα ). Taking in to account (10.49), (10.50), and (10.51) we get that, for all f ∈ Dom(Aˆα ), (f, Rg)+,α = 0 and (f, Rφ)+,α = 0. Therefore the vectors Rg and Rφ are proportional and hence g and φ are proportional as well. This contradicts the fact that g ∈ / H. Lemma 10.4.2. Let A˙ be a closed, densely-deﬁned, symmetric operator in Hilbert space H with deﬁciency indices (1, 1). Then H can be decomposed into the orthogonal sum H = H0 ⊕ H1 , (10.52) where both H0 and H1 are invariant under the resolvents Rzα of self-adjoint extensions Aα of A˙ for all α ∈ [0, π). Moreover, all the resolvents Rzα coincide on H1 . Proof. Let H0α be the subspace of H obtained by forming the closure of all ﬁnite linear combinations of vectors of the form Rzα ϕ, ϕ ∈ Ni for all non-real z. This subspace is evidently invariant under Rzα because of the resolvent identity Rzα − Rλα = (z − λ)Rzα Rλα . Since the adjoint of Rzα is Rzα¯ , then H0α ⊥ is invariant under Rzα¯ as well. Since the vector −zRzα ϕ, (ϕ ∈ Ni ) converges weakly to ϕ as z goes to inﬁnity along the imaginary axis, then ϕ belongs to H0α ⊥ . Let f ∈ H0α ⊥ . Then g = Rzα f ∈ H0α ⊥ , (Im z = 0) and (ϕ, Rzα f ) = 0, (10.53) for ϕ ∈ H0α and Rzα f ∈ H0α ⊥ . It is easy to see that the vector g = Rzα f does not belong to ker(A˙ ∗ ∓ iI). Let us assume the contrary, i.e., A˙ ∗ g = ±ig. Since g ∈ Dom(Aα ), then we get Aα g = ±ig, which is impossible for a self-adjoint operator Aα . It follows from the relation g = Rzα f that (Aα − zI)g = f, and Aα g = zg + f.

(10.54)

Consider the space H+ = Dom(A˙ ∗ ). Then (10.53) and (10.54) yield (ϕ, g)+ = 0. Indeed, (ϕ, g)+ = (ϕ, g) + (A˙ ∗ ϕ, A˙ ∗ g) = (ϕ, g) + i(ϕ, Aα g) = (ϕ, g) + i(ϕ, zg + f ) = 0. Since we have proved that g = Rzα f does not belong to ker(A˙ ∗ ∓ iI), then taking ˙ Therefore (10.54) implies into account (2.9) we get that g = Rzα f ∈ Dom(A). ˙ = zg + f, Aα g = Aβ g = Ag

(∀α, β ∈ [0, π)).

Thus all the family of resolvents Rzα coincide on H0α ⊥ and for any β ∈ [0, π) vectors Rzβ ϕ are orthogonal to H0α ⊥ . Since both α and β are arbitrary values from [0, π), then these spaces must coincide.

10.4. Functional model for symmetric operator with deﬁciency indices (1,1) 359 It was shown in Lemma 6.6.6 that if A˙ is a densely-deﬁned, prime, symmetric operator in a Hilbert space H with deﬁciency indices (1, 1), then c.l.s. Rzα ϕ = H,

(ϕ ∈ Ni ).

z=z¯

(10.55)

Let us now consider a self-adjoint operator in Lμ2 (R) deﬁned by Bx(t) = tx(t),

Dom(B) = {x(t) ∈ Lμ2 (R) | tx(t) ∈ Lμ2 (R)},

(10.56)

where the measure μ (non-decreasing, non-negative function μ(t) on R) satisﬁes the conditions 1 dμ(t) = ∞, dμ(t) = 1. (10.57) 1 + t2 R R Consider the operator μ ˙ ˙ Bx(t) = tx(t), Dom(B) = {x(t), tx(t) ∈ L2 (R) | x(t)μ(t) = 0}. (10.58) R

Utilizing Theorems 6.5.1 and 6.6.7 we conclude that the operator B˙ of the form (10.58) is closed, prime, and symmetric. Obviously, the deﬁciency indices of this operator is (1, 1). It was also shown in the proof of Theorem 6.5.1 that

4 c ∗ ˙ ker(B ∓ iI) = ,c∈C . t∓i Consider vectors ϕ=

1 , t−i

ψ=

Obviously, ϕ2Lμ (R) 2

=

ψ2Lμ(R) 2

= R

1 . t+i

1 dμ(t) = 1, 1 + t2

1 1 H+,0 − , (10.59) t−1 t+i i i 2cit 2ic Bx(t) = tu(t) + c + = tu(t) + = t u(t) + , (10.60) t−1 t+i 1 + t2 1 + t2 ˙ and x(t) ∈ Dom(B). It follows from (10.59)-(10.60) that where u(t) ∈ Dom(B) any vector x(t) ∈ H+,0 can be represented in the form and

˙ +c = Dom(B) = Dom(B)

x(t) = u(t) +

2ic , 1 + t2

(c ∈ C).

(10.61)

Taking into account (10.61) and R u(t)dμ(t) = 0, we have 2ic 2ic ˙ c ∈ C). (x, 1) = u(t) + dμ(t) = dμ(t), (z ∈ Dom(B), 2 2 1+t R R 1+t (10.62)

360

Chapter 10. L-systems with Schr¨ odinger operator

The relation (10.62) above shows that the functional (x, 1) = R x(t)dμ(t) is continuous on H+,0 and therefore x(t) ≡ 1 belongs to H−,0 . Hence we can apply Lemma 10.4.1 and relation (10.58) to conclude that symmetric operator B˙ is densely deﬁned. Theorem 10.4.3. Let A˙ be a closed densely-deﬁned, symmetric, prime operator with deﬁciency indices (1, 1) in a Hilbert space H, and let A be its self-adjoint extension. ˙ A) is unitary equivalent to the pair (B, ˙ B) of the form (10.58), Then the pair (A, μ (10.56) in some Hilbert space L2 (R), where the measure μ(t) satisﬁes (10.57). Proof. Let ϕ ∈ ker(A˙ ∗ − iI), ϕ = 1, and A have the resolution of identity E(t). Consider an operator U x(t) = x(t)(t − i)dE(t)ϕ. (10.63) R

We are going to show that U is an isometric map of Lμ2 (R) onto H where dμ(t) = (1 + t2 )d(E(t)ϕ, ϕ)H .

(10.64)

It follows from (10.64) that 2 (U x, U y)H = x(t)y(t)(t + 1)d(E(t)ϕ, ϕ)H = x(t)y(t)dμ(t) R R = x(t), y(t) Lμ (R) . 2

Let φ =

1 . t−i

Clearly, φ ∈ Lμ2 (R) and φLμ2 (R) = 1. It is also easy to see that −1

Rz (B)φ = (B − zI)

1 t−i

=

1 . (t − z)(t − i)

Combining (10.63) and (10.65) we obtain (t − i) dE(t) U Rz (B)φ = dE(t) ϕ = ϕ = Rz (A)ϕ. R (t − z)(t − i) R t−z

(10.65)

(10.66)

Since the operator A˙ is a closed densely-deﬁned, symmetric, prime operator with deﬁciency indices (1, 1), then we can apply Lemma 6.6.6 or Lemma 10.4.2 to obtain relation (10.55) for some α such that Aα = A and Rzα (Aα ) = Rz (A). Consequently, (10.66) implies that the closed linear span of all vectors of the form Rz (B)φ coincides with Lμ2 (R). Using the resolvent identity and relation (10.66) we have U Rz (B)Rλ (B)φ = Rz (A)Rλ (A)ϕ, and hence U Rz (B)U −1 Rλ (A)ϕ = Rz (A)Rλ (A)ϕ. Thus

U Rz (B)U −1 = Rz (A).

(10.67)

10.5. Accretive (∗)-extensions of a Schr¨ odinger operator Applying (10.63), we get 1 1 Uφ = U = (t − i)dE(t)ϕ = dE(t)ϕ = ϕ. t−i R t−i R

361

(10.68)

The latter means that U ker(B˙ ∗ − iI) = ker(A˙ ∗ − iI). Since 1 i 1 1 = − , (t + i)(t − i) 2 t+i t−i then i R−i (B)φ + φ = 2

i 1 ∈ ker(B˙ ∗ + iI). 2 t+i

On the other hand, applying (10.67) and (10.68) we obtain i 1 i i U = U R−i (B)φ + U φ = R−i (A)ϕ + ϕ. 2 t+i 2 2

(10.69)

By direct calculation one checks that the right-hand side of (10.69) belongs to ker(A˙ ∗ + iI). Therefore, operator U maps ker(B˙ ∗ + iI) onto ker(A˙ ∗ + iI). As a result we get ˙ −1 , A˙ = U BU A = U BU −1 .

This completes the proof.

A result analogous to Theorem 10.4.3 can be proved in a similar way for the case of deﬁciency indices (n, n).

10.5 Accretive (∗)-extensions of a Schr¨ odinger operator Suppose that the symmetric operator A˙ of the form

˙ = −y + q(x)y, Ay y(a) = y (a) = 0, (see also (10.9)) with deﬁciency indices (1,1) is non-negative, i.e., (Af, f ) ≥ 0 for ˙ all f ∈ Dom(A)). It was shown in Theorem 10.3.3 that the Schr¨odinger operator Th of the form

∗ Th y = −y + q(x)y Th y = −y + q(x)y , , (10.70) hy(a) − y (a) = 0 hy(a) − y (a) = 0 (see also (10.10)) is accretive if and only if Re h ≥ −m∞(−0).

(10.71)

362

Chapter 10. L-systems with Schr¨ odinger operator

For real h such that h ≥ −m∞ (−0) we get a description of all non-negative ˙ For h = −m∞ (−0) the corresponding self-adjoint extensions of an operator A. operator

AK y = −y + q(x)y, (10.72) y (a) + m∞ (−0)y(a) = 0, is the Kre˘ın-von Neumann extension of A˙ and for h = +∞ the corresponding operator

AF y = −y + q(x)y, (10.73) y(a) = 0, ˙ It follows from (10.71), (10.72) and (10.73) that is the Friedrichs extension of A. a non-negative operator A˙ of the form (10.10) admits non-self-adjoint accretive extensions if and only if m∞ (−0) < ∞. Theorem 10.5.1. Let Th (Im h > 0) be an accretive Schr¨ odinger operator of the form (10.70). Then for all real μ satisfying the inequality μ≥

(Im h)2 + Re h, m∞ (−0) + Re h

(10.74)

the operators 1 [hy(a) − y (a)] [μδ(x − a) + δ (x − a)], μ−h 1 A∗ y = −y + q(x)y + [hy(a) − y (a)] [μδ(x − a) + δ (x − a)] μ−h Ay = −y + q(x)y +

(10.75)

deﬁne the set of all accretive (∗)-extensions A of the operator Th . The operator Th has a unique accretive (∗)-extension A if and only if Re h = −m∞(−0). In this case this unique (∗)-extension has the form Ay = −y + q(x)y + [hy(a) − y (a)] δ(x − a),

(10.76)

A∗ y = −y + q(x)y + [hy(a) − y (a)] δ(x − a).

Proof. It follows from Theorem 10.2.1 that the set of all (∗)-extensions of the Schr¨odinger operator Th (Im h > 0) of the form (10.70) can be described by the formula (10.75). Suppose that A of the form (10.75) is an accretive (∗) - extension of the accretive Schr¨ odinger operator Th of the form (10.70). Then μ + m∞ (−0) ≥ 0. Let ξ < 0. By the direct computations using (10.20) one obtains Im WΘ (ξ) =

−2[μ + m∞ (ξ)Im h] |μ − h|2 |m∞ (ξ) + h|2 2

(10.77) 2

× [−(Re h) + (μ − m∞ (ξ))Re h + μm∞ (ξ) − (Im h) ].

10.5. Accretive (∗)-extensions of a Schr¨ odinger operator

363

Taking into account (see (10.39)) that m∞ (ξ) → +∞ monotonically as ξ → −∞, we get that μ + m∞ (ξ) > 0 for ξ < 0. Using (6.47) and (6.48) yields VΘ (z) = i[WΘ (z) + 1]−1 [WΘ (z) − 1] = i − 2i[WΘ (z) + 1]−1 . Therefore, Re VΘ (z) = −2[WΘ∗ (z) + 1]−1

WΘ (z) − WΘ∗ (z) [WΘ (z) + 1]−1 2i

and for ξ < 0, VΘ (ξ) = −2[WΘ∗ (ξ) + 1]−1 Im WΘ (ξ) [WΘ (ξ) + 1]−1 .

(10.78)

Comparing (10.78) and (6.47) we obtain Im WΘ (ξ) ≤ 0 for ξ < 0. It follows from (10.77) that −(Re h)2 − (μ − m∞ (ξ))Re h + μm∞ (ξ) − (Im h)2 ≥ 0 for all ξ < 0. Taking the limit ξ → −0 we get −(Re h)2 − (μ − m∞ (−0)) Re h + μm∞ (−0) − (Im h)2 ≥ 0, which is equivalent to (10.74). Suppose now that the inequality (10.74) holds. We will show that the operator A of the form (10.75) is accretive. Because m∞ (−0) < m∞ (ξ) for all ξ < 0, it is easy to verify that the inequality − (Re h)2 − (μ − m∞ (−0)) Re h + μm∞ (−0) − (Im h)2 ≤ −(Re h)2 − (μ − m∞ (ξ)) Re h + μm∞ (ξ) − (Im h)2

(10.79)

holds for μ ≥ Re h.

(10.80)

Using (10.74) one concludes that (10.80) and the inequality (10.74) are equivalent to the left-hand side of (10.79) being non-negative, i.e., Im WΘ (ξ) ≤ 0,

ξ < 0.

Therefore, applying (6.47) we get VΘ (ξ) ≥ 0 for ξ < 0. Since VΘ (z) is a HerglotzNevanlinna function, and it is non-negative on the negative axis, then VΘ (z) is a Stieltjes function. Therefore, it follows from Theorem 9.8.2 that A is accretive. When Re h = −m∞ (−0) then (10.74) implies that μ = ∞. Taking the limit μ → +∞ in (10.75) we get that Th has only one accretive (∗)-extension A if and only if Re h = −m∞ (−0). This unique (∗)-extension A has the form (10.76).

364

Chapter 10. L-systems with Schr¨ odinger operator

10.6 Stieltjes functions and L-systems with accretive Schr¨ odinger operator Next, we consider a scattering L-system of the form A K Θ= H+ ⊂ L2 [a, +∞) ⊂ H−

1 , C

where A is deﬁned by (10.11), and suppose that the operator

Th y = −y + q(x)y, y (a) = hy(a), is accretive and the symmetric operator

˙ = −y + q(x)y, Ay y (a) = y(a) = 0,

(10.81)

(10.82)

in L2 [a, +∞) has defect indices (1, 1). It was shown in Theorem 10.3.3 that the operator Th is accretive if and only if Re h ≥ −m∞ (−0). Assume that Re h = −m∞ (−0). Then, it follows from Theorem 10.5.1 that the (∗)-extension (10.11) of the operator Th (Re h = −m∞ (−0)) is accretive if and only if μ = ∞ (see (10.74)). From the relations (10.11) for μ = ∞ we get (10.76) and Re A = −y + q(x)y + [m∞ (−0)y(a) + y (a)]δ(x − a), Im A = (Im h)y(a)δ(x − a) = (y, g)g, Here

Im h > 0.

1

g = (Im h) 2 δ(x − a),

(10.83)

(10.84)

and the operator Re A of the form (10.83) is non-negative, i.e., (Re Ay, y) ≥ 0 for all y ∈ H+ . Moreover, according to Remark 9.8.6, ∗

VΘ (z)φ = K (Re A − zI)

−1

∞ Kφ = 0

dσ(t) φ, t−z

φ ∈ C.

So, we can associate VΘ (z) with the Stieltjes function ∞ VΘ (z) = 0

where

∞

∞ dσ(t) = ∞,

0

dσ(t) , t−z

0

dσ(t) < ∞. 1+t

(10.85)

(10.86)

10.6. Stieltjes functions and L-systems with accretive Schr¨odinger operator 365 Since Re A ⊃ AK (see (10.83)), where

AK y = −y + q(x)y, y (a) + m∞ (−0)y(a) = 0,

(10.87)

is the Kre˘ın-von Neumann extension of a non-negative operator A˙ of the form (10.82). Theorem 10.6.1. Let Θ be a minimal L-system of the form (10.18), where A is a (∗)-extension of the form (10.76) of the accretive Schr¨ odinger operator Th with Re h = −m∞ (−0). Then the spectral measure in the representation ∞ VΘ (z) = 0

dσ(t) t−z

(10.88)

satisﬁes the relation ⎛ ∞ 0

⎞2

⎜ dσ(t) ⎜ = Im h ⎜ sup 1 + t2 ⎝y∈Dom(AK ) ∞

⎟ ⎟ . 12 ⎟ ⎠ 2 2 (|y(x)| + |l(y)| ) dx |y(a)|

(10.89)

a

Lσ2 [0, +∞),

Proof. Consider the Hilbert space representation (10.85) and the operator

where σ(t) is the function from the

Λ˙ σ f = tf (t), Dom(Λ˙ σ ) = f ∈ Lσ2 [0, +∞) | tf (t) ∈ Lσ2 [0, +∞),

∞

f (t)dσ(t) = 0 .

0

According to Section 10.4, the operator Λ˙ σ is a prime, symmetric operator with deﬁciency indices (1, 1) and Λσ f = tf (t),

Dom(Λσ ) = {f ∈ Lσ2 [0, +∞) | tf (t) ∈ Lσ2 [0, +∞)} ,

is its self-adjoint extension. It was shown in Section 9.8 that Stieltjes function (10.88) can be realized as an impedance function VΘ (z) of some L-system of the form Λ Kσ 1 ΘΛ = , σ σ H+ ⊂ Lσ2 [0, +∞) ⊂ H− C where Λ = Re Λ + i(., 1)1,

σ (1 ∈ H− ),

(10.90)

and Re Λ ⊃ Λσ . Therefore, ∞ VΘ (z) = 0

dσ(t) = (K σ )∗ (Re Λ − zI)−1 K σ = (Re Λ − zI)−11, 1 t−z

= VΘΛ (z).

366

Chapter 10. L-systems with Schr¨ odinger operator

Notice that we may assume that the vector 1 is such that (Re Λ − zI)−11 = Indeed, since (Re Λ − zI)−11 = ∞ Im VΘ (i) = 0

ξ t−z ,

1 . t−z

(10.91)

ξ ∈ C, we get

∞ (2 dσ(t) ( dσ(t) ( −1 ( 2 = ((Re Λ − iI) 1( = |ξ| . t2 + 1 t2 + 1 0

So, |ξ|2 = 1. It follows from (6.44), (6.47) and (6.48) that −1 WΘ (z) = [I − iVΘ (z)] [I + iVΘ (z)]−1 = [I − iVΘ (z)] [I + iVΘΛ (z)]

= WΘΛ (z). By Theorem 6.6.10 there exists a triple of isometric operators (U+ , U, U− ) that σ maps the triplet H+ ⊂ L2 [a, +∞) ⊂ H− isometrically onto H+ ⊂ Lσ2 [0, +∞) ⊂ −1 σ H− with Λ = U− AU+ . Therefore, −1 Im Λ = U− Im AU+ . 1 It follows from (10.84) and (10.90) that U− g = 1, g = (Im h) 2 δ(x − a) and ( (2 ( ( 2 gH− = (1( σ . On the other hand, because Dom(AK ) is a subspace in H+

H−

and dense in L2 [a, +∞), we can construct a new triple of Hilbert spaces, writing K Dom(AK ) = H+ , K ⊂ H− K H+ ⊂ H+ ⊂ L2 [a, +∞) ⊂ H− . In the same manner, writing Dom(Λσ ) = H+ , we get ⊂ H− σ H+ ⊂ H+ ⊂ Lσ2 [0, +∞) σ . ⊂ H−

(10.92)

K K Since the operators AK and Λσ are unitarily equivalent and U+ H+ = U+ maps K K K H+ onto H+ , there exists a triple of operators (U+ , U, U− ) that maps the triplet K K of Hilbert spaces H+ ⊂ L2 [a, +∞) ⊂ H− onto the triplet of Hilbert spaces

H+ ⊂ Lσ2 [0, +∞) ⊂ H− .

(10.93)

It was shown in Theorem 6.5.2 that Dom(Λσ ) = Dom(Λ˙ σ ) +

c , 1 + t2

c ∈ C.

10.6. Stieltjes functions and L-systems with accretive Schr¨odinger operator 367 σ +. Consider the linear, continuous functional of the form (f, 1), (f ∈ H+ ) on H Due to the embedding (10.92), we can consider this functional on H+ and it can be represented in the form

(f, 1) = (f, : 1),

σ f ∈ H+ , : 1 ∈ H− , 1 ∈ H− .

(10.94)

Let R be the Riesz-Berezansky operator in the triple of Hilbert spaces (10.93) which maps H− isometrically onto H+ . As we showed before in the proof of Theorem 6.5.1, c Rc = , c ∈ C. 1 + t2 Therefore, R1 =

1 , 1 + t2

and 1 belongs to H− . We will now show that (f, 1) = (f, 1),

f ∈ H+ ,

1 ∈ H− ,

σ 1 ∈ H− .

(10.95)

Taking into account (10.91), (10.94), and properties of the resolvent of self-adjoint bi-extensions of a symmetric operator, we get

(Λσ − zI)−1 u, 1 = (Λσ − zI)−1 u, : 1 , : u, (Re Λ − zI)−11 = u, (Λσ − zI)−11 , 1 u, = u, (Λσ − zI)−1 c , t−z 1 1 u, = c u, , u ∈ Hσ = Lσ2 [0, +∞)), c ∈ C. t−z t−z : Therefore, c = 1 and 1 = 1. It follows from (10.95) that 1 |(f, 1)| |(f, 1)| 2 = sup = 1H − = (1H − ) 2 f H + f H + f ∈H + f ∈H + ⎛∞ ⎞ 12 12 1 1 1 dσ ⎠ . = ((1, 1)H − ) 2 = ((R1, 1)) 2 = ( , 1) =⎝ 1 + t2 1 + t2

sup

0

K On the other hand, for y ∈ H+ , g ∈ H− , K K K (y, g) = (U ∗ U y, g) = (U+ y, (U ∗ )∗ g) = (U+ y, U− g) = (U+ y, 1).

368

Chapter 10. L-systems with Schr¨ odinger operator

Therefore, sup y∈HK +

K |(U+ y, 1)| |(y, g)| = sup yHK yHK y∈HK + + +

⎛∞ ⎞ 12 K |(U+ y, 1)| |(f, 1)| ⎝ dσ ⎠ = sup = sup = . K y f 1 + t2 K U H f ∈H H + y∈H+ + + + 0

Taking into account (10.84) we get (10.89).

Theorem 10.6.2. Let Θ be an L-system of the form (10.18), where A is a (∗)extension of the form (10.11) of the accretive Schr¨ odinger operator Th of the form (10.81). Then its impedance function VΘ (z) is a Stieltjes function if and only if ⎧ ⎪ (Im h)2 ⎨μ ≥ + Re h, m∞ (−0) + Re h (10.96) ⎪ ⎩ Re h ≥ −m (−0). ∞ Proof. The proof of this theorem follows from Theorems 10.3.3 and 10.5.1 that Th is accretive if and only if Re h ≥ −m∞ (−0), as well as from Theorem 9.8.2 stating that VΘ (λ) is a Stieltjes function if and only if A is accretive. We can summarize this part of the section with the following general result. Theorem 10.6.3. An L-system Θ of the form ⎧ 1 ⎪ (Im h) 2 ⎪ ⎪ [μδ(x − a) + δ (x − a)]ϕ− , ⎨ (Th − zI)x = |μ − h| 1 ⎪ (Im h) 2 ⎪ ⎪ [μx(a) − x (a)], ⎩ ϕ+ = ϕ− − 2i |μ − h|

ϕ± ∈ C,

(10.97)

with an accretive Schr¨ odinger operator Th of the form (10.81) has the Stieltjes impedance function VΘ (z) if and only if relation (10.96) holds. Proof. The proof immediately follows from Theorems 10.2.2 and 10.6.2.

α1 ,α2

Now let us consider applications related to the subclass S of scalar Stieltjes functions. Let again Θ be an L-system of the form (10.18), where A is a (∗)extension (10.11) of the accretive Schr¨odinger operator Th (10.81). As we know from Theorem 10.3.3 and formula (10.47), the operator Th is α-sectorial if and only if Re h > −m∞(−0) and is accretive but not α-sectorial for any α ∈ (0, π/2) if and only if Re h = −m∞ (−0). It also follows from Theorem 10.5.1 that the operator A of Θ is accretive if and only if (10.74) holds. Using (10.21) we can write the impedance function VΘ (z) in the form VΘ (z) =

(m∞ (z) + μ) Im h . (μ − Re h) (m∞ (z) + Re h) − (Im h)2

(10.98)

10.6. Stieltjes functions and L-systems with accretive Schr¨odinger operator 369 Consider our system Θ with μ = +∞. Then in (10.98) we obtain VΘ (z) =

Im h . m∞ (z) + h

Then in this case lim VΘ (x) = lim

x→−∞

x→−∞

Im h = 0, m∞ (x) + h

(10.99)

since m∞ (x) → +∞ as x → −∞. Also, lim VΘ (x) =

x→−0

Im h . m∞ (−0) + h

Assuming that Th is α-sectorial and hence Re h > −m∞ (−0), we use (9.150) and obtain lim VΘ (x) = 0 = tan 0 = tan α1 ,

x→−∞

lim VΘ (x) =

x→−0

Im h = tan α2 . m∞ (−0) + h

On the other hand since Th is α-sectorial, then via Theorem 10.3.3 we have that tan α = tan α2 =

Im h , m∞ (−0) + h

and hence, by Corollary 9.8.5, VΘ (z) belongs to the class S 0,α . Let now μ = +∞ and satisﬁes inequality (10.74). Then (m∞ (x) + μ) Im h (μ − Re h) (m∞ (x) + Re h) − (Im h)2 Im h = = tan α1 , μ − Re h

(10.100)

(m∞ (−0) + μ) Im h = tan α2 . (μ − Re h) (m∞ (−0) + Re h) − (Im h)2

(10.101)

lim VΘ (x) = lim

x→−∞

x→−∞

and lim VΘ (x) =

x→−0

Therefore, in this case VΘ (z) ∈ S α1 ,α2 . Theorem 10.6.4. Let Θ be an L-system of the form (10.18), where A is a (∗)extension of an α-sectorial operator Th with the exact angle of sectoriality α ∈ (0, π/2). Then A is an α-sectorial (∗)-extension of Th (with the same angle of sectoriality) if and only if μ = +∞ in (10.11). Proof. It follows from (10.99)-(10.101) that in this case VΘ (z) ∈ S 0,α if and only if μ = +∞. Thus using Corollary 9.8.5 for the function VΘ (z) we obtain that A is α-sectorial (∗)-extension of Th .

370

Chapter 10. L-systems with Schr¨ odinger operator

We note that if Th is α-sectorial with the exact angle of sectoriality α, then it admits only one α-sectorial (∗)-extension A with the same angle of sectoriality α. Consequently, μ = +∞ and A has the form (10.76) as it was explained in the proof of Theorem 10.5.1. Theorem 10.6.5. Let Θ be an L-system of the form (10.18), where A is a (∗)extension of an α-sectorial operator Th with the exact angle of sectoriality α ∈ (0, π/2). Then A is accretive but not α-sectorial for any α ∈ (0, π/2) a (∗)extension of Th if and only if in (10.11) μ = μ0 =

(Im h)2 + Re h. m∞ (−0) + Re h

(10.102)

Proof. Let VΘ (z) be the impedance function of our system Θ. If in (10.100) we set μ = μ0 where μ0 is given by (10.102), then Im h m∞ (−0) + Re h 1 = = μ0 − Re h Im h tan α (10.103) π = tan − α = tan α1 , 2 where α1 = π2 − α. On the other hand, using (10.101) with μ = μ0 we obtain h)2 Im h m∞ (−0) + m∞(Im (−0)+Re h lim VΘ (x) = =∞ (Im h)2 x→−0 2 (10.104) m∞ (−0)+Re h (m∞ (−0) + Re h) − (Im h) π = tan = tan α2 . 2 lim VΘ (x) =

x→−∞

π

π

Hence, (10.103) and (10.104) yield that VΘ (z) ∈ S 2 −α, 2 . If we assume that A is a β-sectorial (∗)-extension, then by Theorem 9.8.7, + tan β ≤ tan α2 + 2 tan α1 (tan α2 − tan α1 ) = ∞. Therefore, A is accretive but not β-sectorial for any β ∈ (0, π/2).

Note that it follows from the above theorem that any α-sectorial operator Th with the exact angle of sectoriality α ∈ (0, π/2) admits only one accretive (∗)extension A. This extension takes form (10.11) with μ = μ0 where μ0 is given by (10.102).

10.7 Inverse Stieltjes functions and systems with Schr¨ odinger operator Consider an L-system of the form (10.18), with A˙ and A˙ ∗ given by (10.9) and μ = Re h = −m∞ (−0). Then we obtain the following scattering L-system: A K 1 Θ= , (10.105) H+ ⊂ L2 [a, +∞) ⊂ H− C

10.7. Inverse Stieltjes functions and systems with Schr¨ odinger operator

371

where i [y (a) − hy(a)] [δ (x − a) − m∞ (−0)δ(x − a)], Im h i A∗ y = −y + q(x)y + [y (a) − hy(a)] [δ (x − a) − m∞ (−0)δ(x − a)], Im h (10.106) Ay = −y + q(x)y −

and Re A = −y + q(x)y − y(a)[δ (x − a) − m∞ (−0)δ(x − a)], 1 Im A = [−y (a) − m∞ (−0)y(a)] [δ (x − a) − m∞ (−0)δ(x − a)] Im h 1 = (y, g)g, Im h where g=

1 1

(Im h) 2

(10.107)

[δ (x − a) − m∞ (−0)δ(x − a)].

Using relation (10.20) we get that the transfer function WΘ (z) of an L-system Θ of the form (10.105) has the representation WΘ (z) = I − 2iK ∗(A − zI)−1 K = −

m∞ (z) + h . m∞ (z) + h

(10.108)

Denote by ΘK an L-system of the form (10.18) with (10.76) and by ΘF an Lsystem of the form (10.105), (10.106). Then it follows from (10.108) that WΘF (z) = −WΘK (z). Moreover (see (6.48)), VΘF (z) = i[WΘF (z) − I][WΘF (z) + I]−1 = −i[WΘK (z) + I][−WΘK (z) + I]−1 = i[WΘK (z) + I][WΘK (z) − I]−1 = −{i[WΘK (z) − I][WΘK (z) + I]−1 }−1 = −VΘ−1 (z). K Since VΘK (z) is a Stieltjes function with representation ∞ dσ(t) VΘK (z) = , t−z 0 −VΘ−1 (z) is the inverse Stieltjes function and has the representation K ∞ 1 1 VΘF (z) = −VΘ−1 (z) = α + − dτ(t), K t−z t 0 where α ≤ 0, τ (t) is a nondecreasing function satisfying the relations ∞

∞ dτ(t) = ∞,

0

0

dτ (t) < ∞. t2 + t

372

Chapter 10. L-systems with Schr¨ odinger operator

It follows from (10.107) that Re A ⊃ AF , where

AF y = −y + q(x)y, y(a) = 0 is the Friedrichs extension of a non-negative operator A˙ of the form (6.44). Taking into account calculations similar to those in connection with Theorem 10.6.1, we get Theorem 10.7.1. Let Θ be a minimal L-system of the form (10.105), where A is a (∗)-extension given by (10.106) of the accretive Schr¨ odinger operator Th (μ = Re h = −m∞ (−0)). Then the spectral measure τ (t) in the representation ∞ VΘF (z) = VΘ (z) = α + 0

1 1 − t−z t

dτ(t),

(10.109)

satisﬁes the relation ⎛ ∞ 0

⎞2

dτ (t) 1 ⎜ ⎜ = ⎜ sup 2 1+t Im h ⎝y∈Dom(AF ) ∞

⎟ ⎟ . 12 ⎟ ⎠ 2 2 (|y(x)| + |l(y)| ) dx |y (a)|

(10.110)

a

As a consequence of Theorems 10.6.1 and 10.7.1, it follows that the spectral measures σ(t) and τ (t) in the representations (10.88) and (10.109), which correspond to the Kre˘ın-von Neumann extension AK and the Friedrichs extension AF , satisfy the relation ⎛ ⎞2 ∞ 0

dσ(t) 1 + t2

∞ 0

⎜ dτ (t) ⎜ = ⎜ sup 1 + t2 ⎝y∈Dom(AK ) ∞

⎟ ⎟ 12 ⎟ ⎠ (|y(x)|2 + |l(y)|2 ) dx |y(a)|

a

⎛

⎞2

⎜ ⎜ × ⎜ sup ⎝y∈Dom(AF ) ∞

⎟ ⎟ . 12 ⎟ ⎠ (|y(x)|2 + |l(y)|2 ) dx |y (a)|

a

Remark 10.7.2. It follows from relations (10.89) and (10.110) that ∞ (Im h) |y(a)|2 |y(x)|2 + |l(y)|2 dx ≥ ∞ dσ(t) , y (a) + m∞ (−0)y(a) = 0, a

a

∞

|y(x)|2 + |l(y)|

2

0

dx ≥

1+t2 2

|y (a)| ∞ dσ(t) ,

(Im h)

0

1+t2

y(a) = 0.

10.7. Inverse Stieltjes functions and systems with Schr¨ odinger operator

373

As we have already mentioned in Section 10.5, any Stieltjes function VΘ (z) has a parametric representation of the type (9.141) ∞ VΘ (z) = γ + 0

dσ(t) , t−z

(γ ≥ 0),

(10.111)

where σ(t) is a non-decreasing function for which ∞ 0

dσ(t) < ∞. t+1

(10.112)

In addition to (10.112), since the maximal symmetric part of Th has dense domain in L2 [a, +∞), one concludes that ∞ dσ(t) = ∞.

(10.113)

0

Thus, the function VΘ (z) has the parametric representation (10.111), (10.112), (10.113) if and only if the inequalities (10.96) hold. Theorem 10.7.3. Let Θ be an L-system of the form (10.105), where A is a (∗)extension of the form (10.11) of an accretive Schr¨ odinger operator Th of the form (10.81). Then its impedance function VΘ (z) is an inverse Stieltjes function if and only if −m∞ (−0) ≤ μ ≤ Re h. (10.114) Proof. Consider the (∗)-extensions Aμ and Aξ of the accretive Schr¨ odinger operator Th deﬁned by (10.25) and (10.26), respectively, with ξ= Let

Θμ =

Aμ Kμ 1 H+ ⊂L2 [a,+∞)⊂H− C

μRe h − |h|2 . μ − Re h

(10.115)

and Θξ =

Aξ Kξ 1 H+ ⊂L2 [a,+∞)⊂H− C

,

be the corresponding L-systems described in (10.27) and (10.28). Applying (10.29) and taking into account (10.115) we have WΘμ (z) = −WΘξ (z). The relations (10.116) and (6.48) imply VΘμ (z) = Kμ∗ (Re Aμ − zI)−1 Kμ = i[WΘμ (z) − I] [WΘμ (z) + I]−1 = i[−WΘξ (z) − I][−WΘξ (z) + I]−1 = −VΘ−1 (z), ξ

(10.116)

374

Chapter 10. L-systems with Schr¨ odinger operator

and VΘμ (z) = −VΘ−1 (z). As was shown in Theorem 10.6.2, the function VΘξ (z) is ξ a Stieltjes function if and only if ξ≥ Therefore, ξ=

(Im h)2 + Re h. m∞ (−0) + Re h

μRe h − |h|2 (Im h)2 ≥ + Re h. μ − Re h m∞ (−0) + Re h

(10.117)

Furthermore, μRe h − |h|2 μRe h − (Im h)2 − (Re h)2 (Im h)2 = =− + Re h. μ − Re h μ − Re h μ − Re h It follows from (10.117) that −

(Im h)2 (Im h)2 + Re h ≥ + Re h, μ − Re h m∞ (−0) + Re h

and

1 1 − ≥ 0. Re h − μ m∞ (−0) + Re h

The last inequality is equivalent to (10.114). When μ satisﬁes (10.114), then VΘξ (z) is a Stieltjes function and therefore VΘμ (z) is an inverse Stieltjes function. Theorem 10.7.4. Let Θ=

Aμ H+ ⊂ L2 [a, +∞) ⊂ H−

Kμ

1 , C

be a minimal scattering L-system, where Aμ is a (∗)-extension of the form (10.25) of the accretive Schr¨ odinger operator Th and μ≥

(Im h)2 + Re h. m∞ (−0) + Re h

Then the spectral measure σξ (t) in the representation ∞ VΘμ (z) = γ + 0

dσξ (t) , t−z

ξ=

μRe h − |h|2 μ − Re h

(10.118)

satisﬁes the relation ⎛ ∞ 0

dσξ (t) Im h = 2 1+t |μ − h|2

⎞2

⎜ ⎜ ⎜ sup ⎝y∈Dom(Aξ ) ∞ a

⎟ ⎟ , 12 ⎟ ⎠ (|y(x)|2 + |l(y)|2 ) dx |μy(a) − y (a)|

(10.119)

10.7. Inverse Stieltjes functions and systems with Schr¨ odinger operator where

:ξ y = −y + qy, A y (a) = ξy(a),

ξ=

μRe h − |h|2 , μ − Re h

375

(10.120)

is a non-negative self-adjoint extension of the non-negative operator A˙ of the form (10.82). h)2 Proof. It is easy to check that when μ ∈ m∞(Im (−0)+Re h + Re h, +∞ then ξ=

μRe h − |h|2 ∈ [−m∞ (−0), Re h). μ − Re h

:ξ of the form (10.120) is a non-negative self-adjoint extension of the Therefore, A non-negative minimal operator A˙ of the form (10.82). From (10.114) it follows that 5 6 μRe h − |h|2 μ − Re h Re Aμ = −y + qy + y(a) − y (a) |μ − h|2 |μ − h|2 × [μδ(x − a) + δ (x − a)], and that :ξ , Re Aμ ⊃ A

ξ=

μRe h − |h|2 . μ − Re h

Taking into account that Im Aμ = (., g)g, where g is deﬁned by (10.15), and following the proof of Theorem 10.6.1, one obtains relation (10.119). Theorem 10.7.5. Let Θξ =

Aξ H+ ⊂ L2 [a, +∞) ⊂ H−

Kξ

1 , C

be a minimal L-system, where Aξ is a (∗)-extension of the form (10.26) of the accretive Schr¨ odinger operator Th and −m∞ (−0) ≤ ξ ≤ Re h. Then the spectral measure τμ (t) in the representation ∞ VΘξ (z) = α + 0

1 1 − t−z t

dτμ (t),

(α ≤ 0),

satisﬁes the relation ⎛ ∞ 0

dτμ (t) Im h = 2 1+t |ξ − h|2

⎞2

⎜ ⎜ ⎜ sup ⎝y∈Dom(Aμ ) ∞ a

⎟ ⎟ , 12 ⎟ ⎠ (|y(x)|2 + |l(y)|2 ) dx |ξy(a) − y (a)|

(10.121)

376

Chapter 10. L-systems with Schr¨ odinger operator

where ξ= and

μRe h − |h|2 , μ − Re h

μ > Re h,

(10.122)

:μ y = −y + qy, A y (a) = μy(a),

is a non-negative self-adjoint extension of the non-negative operator A˙ of the form (10.82). Proof. It was established in the proof of Theorem 10.7.3 that the inequalities μ≥

(Im h)2 + Re h m∞ (−0) + Re h

(10.123)

and −m∞ (−0) ≤ ξ ≤ Re h, where ξ=

μ > Re h,

μRe h − |h|2 , μ − Re h

are equivalent. Consider two (∗)-extensions Aμ and Aξ of the accretive Schr¨ odinger operator Th . As μ satisﬁes (10.123), the operator-valued function VΘξ (z) is an inverse Stieltjes function and VΘμ (z) is a Stieltjes function by Theorem 10.7.3, and VΘμ (z) = −VΘ−1 (z). It follows from (10.26) that ξ Re Aξ = −y + qy 5 6 ξRe h − |h|2 ξ − Re h + y(a) − y (a) [ξδ(x − a) + δ (x − a)]. |ξ − h|2 |ξ − h|2 Using (10.122) we get μ=

ξRe h − |h|2 . ξ − Re h

:μ . Taking into account that Im Aξ = (., g)g, where Therefore, Re Aξ ⊃ A 1

(Im h) 2 g= [ξδ(x − a) + δ (x − a)], |ξ − h| and following the proof of Theorem 10.7.4, one will obtain relation (10.121). As a consequence of Theorems 10.7.4 and 10.7.5 we get the following: Let Aξ Kξ 1 Aμ Kμ 1 Θμ = H+ ⊂L2 [a,+∞)⊂H and Θ = ξ C H+ ⊂L2 [a,+∞)⊂H− C , −

10.7. Inverse Stieltjes functions and systems with Schr¨ odinger operator

377

where ξ is related to μ by (10.115), be L-systems described in (10.27) and (10.28), where Aμ and Aξ are (∗)-extensions of the form (10.25), (10.26) of the accretive Schr¨odinger operator Th and μ≥

(Im h)2 + Re h. m∞ (−0) + Re h

Then the spectral measures in the representations ∞ VΘμ (z) = γ + 0

dσξ (t) t−z

∞ and

VΘξ (z) = α + 0

1 1 − t−z t

dτμ (t),

satisfy the relation ⎛

, , μ , μ−h y(a) −

, , 1 ∞ ∞ ⎜ y (a) , μ−h dσξ (t) dτμ (t) ⎜ = ⎜ sup 12 2 2 1+t 1+t ⎝y∈Dom(Aξ ) ∞ 2 2 0 0 (|y(x)| + |l(y)| ) dx

⎞2 ⎟ ⎟ ⎟ ⎠

a

⎛

⎞2 (10.124) , , , μRe h−h2 , μ−Re h ⎜ , μ−h y(a) − μ−h y (a), ⎟ ⎜ ⎟ × ⎜ sup . 12 ⎟ ⎝y∈Dom(Aμ ) ∞ ⎠ 2 2 (|y(x)| + |l(y)| ) dx a

The proof of relation (10.124) follows from Theorems 10.7.4 and 10.7.5 by multiplying both parts of the relations (10.119) and (10.121), from the fact that ξ=

μRe h − |h|2 , μ−h

and μRe h − |h|2 μRe h − |h|2 − μh − hRe h −h= μ − Re h μ − Re h μ(Re h − h) − h(h − Re h) −iμIm h + ihIm h h−μ = = = iIm h . μ − Re h μ − Re h μ − Re h

ξ−h=

Remark 10.7.6. Using (10.24), we can rewrite relations (10.119) and (10.110) in the following way: sup y (a)=ξy(a)

sup ∞

y(a)=0

a

∞ a

|y(a)|2 Im m∞ (i) =− , |m∞ (i) + ξ|2 (|y(x)|2 + |l(y)|2 ) dx

|y (a)|2 = −Im m∞ (i). (|y(x)|2 + |l(y)|2 ) dx

378

Chapter 10. L-systems with Schr¨ odinger operator

These relations allow us to obtain some new inequalities (see [53], [135]). For instance, performing calculations for m∞ (i) of a symmetric Schr¨ odinger operator A˙ of the form (10.9) with q(x) = 0 in L2 [0, +∞) we obtain the sharp inequality

|y (0)| ≤ 2

− 14

0

∞

2

|y(x)| dx +

∞ 0

12 |y (x)| dx ,

2

(10.125)

where y(x), y (x) are absolutely continuous on any interval [0, b] ⊂ [0, +∞), y(0) = 0, and y ∈ L2 [0, +∞). Remark 10.7.7. One can prove that sup sin αy (a)+cos αy(a)=0

∞ a

|y (a)|2 1 = , (|y(x)|2 + |l(y)|2 ) dx l(y0 )|x=a + tan α (l(y0 )) |x=a

where y0 (x) is the unique solution of the boundary value problem ⎧ ⎪ ⎨ l (l(y)) + y = 0, y(a) = − tan α, y (a) = 1, ⎪ ⎩ y, l(y) ∈ D∗ . We conclude this section with the following general result. Theorem 10.7.8. An L-system Θ of the form ⎧ 1 ⎪ (Im h) 2 ⎪ ⎪ (Th − zI)x = [μδ(x − a) + δ (x − a)]ϕ− , ⎨ |μ − h| 1 ⎪ (Im h) 2 ⎪ ⎪ [μx(a) − x (a)], ⎩ ϕ+ = ϕ− − 2i |μ − h|

ϕ± ∈ C,

with an accretive Schr¨ odinger operator Th of the form (10.81) has the inverse Stieltjes impedance function VΘ (z) if and only if relation (10.114) holds. Proof. The proof immediately follows from Theorems 10.2.2 and 10.7.3.

10.8 Stieltjes-like functions and inverse spectral problems for systems with Schr¨ odinger operator In this section we are going to use the realization results for Stieltjes functions developed in Section 9.8 to obtain the solution of inverse spectral problem for L-systems with a Schr¨odinger operator of the form (10.81) in L2 [a, +∞) with non-self-adjoint boundary conditions

Th y = −y + q(x)y, q(x) = q(x), Im h = 0 . (10.126) y (a) = hy(a),

10.8. Stieltjes-like functions and inverse spectral problems

379

In particular, we will show that if a non-decreasing function σ(t) is the spectral function of non-negative self-adjoint boundary value problem

Aθ y = −y + q(x)y, y (a) = θy(a), and satisﬁes conditions ∞

∞ dσ(t) = ∞,

0

0

then, for every γ ≥ 0, a Stieltjes function V (z) = γ + 0

dσ(t) < ∞, 1+t

∞

dσ(t) , t−z

can be realized in a unique way as the impedance function VΘ (z) of an accretive L-system Θ with some Schr¨odinger operator Th . Let H = L2 [a, +∞) and l(y) = −y +q(x)y where q is a real locally summable function. Consider a symmetric operator with defect indices (1, 1),

˙ = −y + q(x)y, By (10.127) y (a) = y(a) = 0, together with its non-negative self-adjoint extension of the form

Bθ y = −y + q(x)y, y (a) = θy(a).

(10.128)

A non-decreasing function σ(t) deﬁned on (−∞, +∞) is called the distribution ˙ Bθ ), where Bθ is a self-adjoint extension of symfunction of an operator pair (B, ˙ metric operator B with deﬁciency indices (1,1) in a Hilbert space H, if the formulas ϕ(t) = U f (x),

f (x) = U −1 ϕ(t),

(10.129)

establish a one-to-one isometric correspondence U between Lσ2 (R) and H. Moreover, this correspondence is such that the operator Bθ is unitarily equivalent to the operator Λσ ϕ(t) = tϕ(t), (ϕ(t) ∈ Lσ2 (R)) (10.130) in Lσ2 (R) while symmetric operator B˙ in (10.127) is unitarily equivalent to the symmetric operator ⎧ ⎫ +∞ ⎨ ⎬ Λ˙ σ ϕ(t) = tϕ(t), Dom(Λ˙ σ ) = ϕ(t), tϕ(t) ∈ Lσ2 (R) | ϕ(t)dσ(t) = 0 . ⎩ ⎭ −∞

(10.131) We are going to introduce a class of Stieltjes-like functions structure similar to that of S0 (R) of Section 9.8 but dealing with scalar functions only.

380

Chapter 10. L-systems with Schr¨ odinger operator

Deﬁnition 10.8.1. A scalar function V (z) is said to be a member of the class SL0 (R) if it admits the integral representation V (z) = γ + 0

∞

γ ∈ (−∞, +∞) ,

dσ(t) , t−z

(10.132)

where the non-decreasing function σ(t) satisﬁes the conditions 0

∞

dσ(t) = ∞,

0

∞

dσ(t) < ∞. 1+t

(10.133)

Consider the following subclasses of SL0 (R). Deﬁnition 10.8.2. A scalar function V (z) ∈ SL0 (R) belongs to the class SL0 (R, K) if ∞ dσ(t) = ∞. t 0 Deﬁnition 10.8.3. A scalar function V (z) ∈ SL0 (R) belongs to the class SL01(R, K) if ∞ dσ(t) < ∞. t 0 Consider an operator Λ˙ σ of the form (10.131) in L2 [0, +∞). Let T σ be a σ quasi-self-adjoint extension of Λ˙ σ and let H+ = Dom(Λ˙ ∗σ ). Then the following theorem describes the realization of the class SL0 (R). Theorem 10.8.4. Let V (z) ∈ SL0 (R). Then it can be realized by a minimal model L-system Λ Kσ 1 ΘΛ = , (10.134) σ σ H+ ⊂ Lσ2 [0, +∞) ⊂ H− C where Λ = Re Λ + iK σ (K σ )∗ is a (∗)-extension of an operator T σ such that Λ ⊃ T σ ⊃ Λ˙ σ , Λ˙ σ is deﬁned via (10.131), and K σ c = c · 1, (K σ )∗ x = (x, 1), σ σ c ∈ C, 1 ∈ H− , x(t) ∈ H+ . Proof. We start by applying the general realization Theorem 6.5.2 to a HerglotzNevanlinna function V (z) and obtain a rigged L-system of the form (10.134) such that V (z) = VΘΛ (z). Following the steps for construction of the model L-system described in Theorem 6.5.2, we note that all the components of the system are described as in the statement of the theorem. Also, the real part Re Λ is a self-adjoint bi-extension of Λ˙ σ that has a quasi-kernel Λσ of the form (10.130). According to Theorem 6.6.7, operator Λ˙ σ of the form (10.131) is a prime operator. Thus, Lsystem ΘΛ of the form (10.134) is minimal. In addition we can observe that for the system ΘΛ of the form (10.134), the σ function η(λ) ≡ 1 belongs to H− . To conﬁrm this we need to show that (x, 1)

10.8. Stieltjes-like functions and inverse spectral problems

381

σ deﬁnes a continuous linear functional for every x ∈ H+ . It was shown in Theorem 6.5.2 that

4

4 c1 c2 t σ H+ = Dom(Λ˙ σ ) , c1 , c2 ∈ C. (10.135) 1 + t2 1 + t2 σ Consequently, every vector x ∈ H+ has three components x = x1 + x2 + x3 according to the decomposition (10.135) above. Obviously, (x1 , 1) and (x2 , 1) yield convergent integrals while (x3 , 1) boils down to ∞ t dσ(t). 1 + t2 0

To see the convergence of the above integral we notice that t t−1 1 1 1 = + ≤ + . 2 2 2 1+t (1 + t )(t + 1) 1 + t 1+t 1+t The integrals taken of the last two expressions on the right side converge due to (6.52) and (10.133), and hence so does the integral of the left side. Thus, (x, 1) σ σ deﬁnes a continuous linear functional for every x ∈ H+ , and hence 1 ∈ H− . The σ σ σ σ state space of the L-system ΘΛ is H+ ⊂ L2 [0, +∞) ⊂ H− , where H+ = Dom Λ˙ ∗σ . At this point we are ready to state and prove the main realization result of this section. Theorem 10.8.5. Let V (z) ∈ SL0 (R) and the function σ(t) be the distribution func˙ Bθ ) of the form (10.127), (10.128). Then there exist tion of an operator pair (B, a unique Schr¨ odinger operator Th (Im h > 0) of the form (10.126), an operator A given by (10.11), an operator K as in (10.17), and a minimal L-system A K 1 Θ= , (10.136) H+ ⊂ L2 [a, +∞) ⊂ H− C of the form (10.18) such that V (z) = VΘ (z). Proof. It follows from the deﬁnition of the distribution function above that there is an operator U deﬁned in (10.129) establishing a one-to-one isometric correspondence between Lσ2 [0, +∞) and L2 [a, +∞) while providing for unitary equivalence between the operator Bθ and the operator of multiplication by an independent variable Λσ of the form (10.130). Let us consider the L-system ΘΛ of the form (10.134) constructed in the proof of Theorem 10.8.4. Applying Theorem 6.6.10 on unitary equivalence to the isometry U deﬁned in (10.129) we obtain a triplet of isometric operators U+ , U , and U− , where σ ∗ ∗ U+ = U H+ , U− = U+ . σ This triplet of isometric operators maps the rigged Hilbert space H+ ⊂ Lσ2 [0, +∞) σ σ ⊂ H− into another rigged triplet H+ ⊂ L2 [a, +∞) ⊂ H− . Moreover, U+ is an

382

Chapter 10. L-systems with Schr¨ odinger operator

σ ∗ ∗ isometry from H+ = Dom(Λ˙ ∗σ ) onto H+ = Dom(B˙ ∗ ), and U− = U+ is an isometry σ from H+ onto H− . This is true since the operator U provides unitary equivalence between the symmetric operators B˙ and Λ˙ σ . Now we construct an L-system A K 1 Θ= , H+ ⊂ L2 [a, +∞) ⊂ H− C −1 where K = U− K σ and A = U− ΛU+ is a (∗)-extension of operator T = U T σ U −1 ˙ This system is clearly minimal due to the unitary equivasuch that A ⊃ T ⊃ B. lence between the operator B˙ and prime operator Λ˙ σ . The real part Re A contains the quasi-kernel Bθ . This construction of A is unique due to Theorem 4.3.9 on the uniqueness of a (∗)-extension for a given quasi-kernel. On the other hand, all (∗)˙ Bθ ) must take the form (10.11) for some values of extensions based on a pair (B, parameters h and μ. Consequently, our function V (z) is realized by the L-system Θ given by (10.136) and

V (z) = VΘΛ (z) = VΘ (z).

Theorem 10.8.6. The operator Th in Theorem 10.8.5 is accretive if and only if ∞ dσ(t) γ2 + γ + 1 ≥ 0. (10.137) t 0 The operator Th is α-sectorial for some α ∈ (0, π/2) if and only if the inequality (10.137) is strict. In this case the exact value of angle α can be calculated by the formula tan α =

γ2 +

∞ dσ(t) t 0 ∞ γ 0 dσ(t) t

+1

.

(10.138)

Proof. It was shown in Theorem 10.3.1 that for the L-system Θ in (10.136), described in the previous theorem, the operator Th is accretive if and only if the function Vh (z) of the form (10.32) is holomorphic in Ext[0, +∞) and satisﬁes the inequality (10.33), i.e., 1 + Vh (−0) Vh (−∞) ≥ 0. Here WΘ (z) is the transfer function of (10.136). According to Theorem 10.3.1, the operator Th is α-sectorial for some α ∈ (0, π/2) if and only if the inequality (10.33) is strict while the exact value of angle α can be calculated by the formula (10.34). Using Theorem 10.8.5 and equation (5.41) yields WΘ (z) = (I − iV (z)J)(I + iV (z)J)−1 . By direct calculations one obtains ∞ 1 − i γ + 0 dσ(t) t+1 WΘ (−1) = ∞ dσ(t) , 1 + i γ + 0 t+1

∞ 1+i γ + 0 −1 WΘ (−1) = ∞ 1−i γ + 0

dσ(t) t+1 dσ(t) t+1

.

10.8. Stieltjes-like functions and inverse spectral problems

383

Using the notations a= 0

∞

dσ(t) t+1

and

∞

b= 0

dσ(t) , t

and performing straightforward calculations we obtain Vh (−0) =

a−b 1 + ab

and

Vh (−∞) =

a−γ . 1 + aγ

(10.139)

Substituting (10.139) into (10.34) and doing the necessary steps we get ∞ dσ(t) γ2 + γ 0 +1 1 + bγ cot α = = . ∞ dσ(t)t b−γ 0

(10.140)

t

Taking into account that b − γ > 0, we combine (10.33), (10.34) with (10.140) and this completes the proof of the theorem. Corollary 10.8.7. The operator Th in Theorem 10.8.5 is accretive if and only if 1 + V (−0) V (−∞) ≥ 0.

(10.141)

The operator Th is α-sectorial for some α ∈ (0, π/2) if and only if the inequality (10.141) is strict. In this case the exact value of angle α can be calculated by the formula V (−∞) − V (−0) tan α = . (10.142) 1 + V (−0) V (−∞) Proof. Taking into account that V (−0) = γ + 0

∞

dσ(t) , t

and VΘ (−∞) = γ, we use (10.137) and (10.138) to obtain (10.141) and (10.142). We note that for the remainder of this chapter the accretive operator Th is extremal if and only if it is not α-sectorial for any α ∈ (0, π/2) (see Remark 9.5.17). It also directly follows from Theorem 9.8.2 that an L-system Θ of the form (10.136) is accretive if and only if the function V (z) belongs to the class S0 (R). Now let us consider V (z) ∈ SL0 (R) satisfying the conditions of Theorem 10.8.5. According to Theorem 10.8.5 there exists a minimal L-system Θ given by (10.136) with unique Schr¨odinger operator Th (Im h > 0) of the form (10.126). In the remainder of the section we will derive the formulas for calculation of the boundary parameter h in Th as well as a real parameter μ that is used in construction (10.11) of the operator A of the L-system Θ realization. An elaborate investigation of these formulas will show the dynamics of the restored parameters

384

Chapter 10. L-systems with Schr¨ odinger operator

h and μ in terms of the changing free term γ from the integral representation (10.132) of the function V (z). We consider two major cases. ∞ Case 1. In the ﬁrst case we assume that 0 dσ(t) < ∞. This means that our t function V (z) belongs to the class SL01 (R, K). In what follows we write ∞ dσ(t) b= and m = m∞ (−0). t 0 Suppose that b ≥ 2. Then the quadratic inequality (10.137) implies that, for all γ such that > = √ √ −b − b2 − 4 −b + b2 − 4 γ ∈ −∞, ∪ , +∞ , (10.143) 2 2 the restored operator Th is accretive. Clearly, this operator is extremal accretive if √ −b ± b2 − 4 γ= . 2 In particular if b = 2, then γ = −1 and the function ∞ dσ(t) V (z) = −1 + t−z 0 is realized using an extremal accretive Th . Now suppose that 0 < b < 2. For every γ ∈ (−∞, +∞) the restored operator Th will be accretive and α-sectorial for some α ∈ (0, π/2). Consider a function V (z) deﬁned by (10.132). Conducting realizations of V (z) by operators Th for diﬀerent values of γ ∈ (−∞, +∞) we notice that the operator Th with the largest angle of sectorialilty occurs when γ = −(b/2) and the angle is found according to the formula b α = arctan . 1 − b2 /4 This follows from formula (10.138), the fact that γ 2 + γ b + 1 > 0 for all γ, and the formula 2 b b2 2 γ +γb+1= γ + + 1− . 2 4 Now we will focus on the description of the parameter h in the restored operator Th . It was shown in Theorem 10.7.4 that the quasi-kernel Aˆ of the realizing Lsystem Θ from Theorem 10.8.5 takes a form of (10.120) or more speciﬁcally

: = −y + qy Ay , y (a) = ξy(a)

ξ=

μRe h − |h|2 . μ − Re h

10.8. Stieltjes-like functions and inverse spectral problems

385

On the other hand, since σ(t) is also the spectral function of the operator pair ˙ Bθ ), we can conclude that Aˆ is equal to the operator Bθ of the form (10.128). (B, This connection allows us to obtain θ=ξ=

μRe h − |h|2 . μ − Re h

(10.144)

Assuming that h = u + iv, we will use (10.144) to derive the formulas for u and v in terms of γ. First, to eliminate parameter μ, we notice that (10.20) and (5.41) imply WΘ (λ) =

μ − h m∞ (λ) + h 1 − iV (z) = . 1 + iV (z) μ − h m∞ (λ) + h

Passing to the limit when z → −∞ and taking into account that V (−∞) = γ and utilizing (10.39), we obtain μ−h 1 − iγ = . (10.145) 1 + iγ μ−h Let us write p=

1 − iγ . 1 + iγ

(10.146)

Solving (10.145) for μ and using (10.146) yields μ=

¯ h − ph . 1−p

Substituting this value into (10.144) after simpliﬁcation produces u + iv − p(u − iv)u − (u2 + v 2 )(1 − p) = θ. u + iv − p(u − iv) − u(1 − p) After straightforward calculations aiming to represent the numerator and denominator of the last equation in standard form, one obtains the relation u − γ v = θ.

(10.147)

It was shown in Theorem 10.3.3 that the α-sectorialilty of the operator Th and (10.34) lead to Im h v tan α = = . (10.148) Re h + m∞ (−0) u + m∞ (−0) Combining (10.147) and (10.148) one obtains u − γ(u tan α + m∞ (−0) tan α) = θ,

386

Chapter 10. L-systems with Schr¨ odinger operator

h1

h2 q -

c 2

q

-m

q+

u

c 2

Figure 10.4: b > 2

-1

2

-b - b -4 2

b2 - 4

-b + 2

g

0

Figure 10.5: γ interval

or u=

θ + γm∞ (−0) tan α . 1 − γ tan α

But tan α is also determined by (10.138). Direct substitution of tan α =

b 1 + γ(γ + b)

into the above equation yields u=θ+

θ + m∞ (−0) bγ . 1 + γ2

Using the short notation m = m∞ (−0) and ﬁnalizing calculations we get h = u + iv,

γ[θ + m]b , 1 + γ2

u=θ+

v=

[θ + m]b . 1 + γ2

(10.149)

At this point we can use (10.149) to provide analytical and graphical interpretation of the parameter h in the restored operator Th . Let c = (θ + m)b. Again we consider three subcases.

10.8. Stieltjes-like functions and inverse spectral problems Subcase 1. b > 2

387

Using basic algebra we transform (10.149) into c 2 c2 (u − θ)2 + v − = . 2 4

(10.150)

Since in this case the parameter γ belongs to the interval in (10.143), we can see that h traces the highlighted part of the circle in Figure 10.4 as γ moves in the direction from −∞ towards +∞. We also notice that the removed point (θ, 0) corresponds to the value of γ = ±∞ while √the points h1 and h2 √ 2 2 correspond to the values γ1 = −b− 2 b −4 and γ2 = −b+ 2 b −4 , respectively.

h

a

-m

q -

c 2

q

q+

u

c 2

Figure 10.6: b < 2 Subcase 2. b < 2 For every γ ∈ (−∞, +∞) the restored operator Th will be accretive and α-sectorial for some α ∈ (0, π/2). As we have mentioned above, the operator Th achieves the largest angle of sectorialilty when γ = − 2b . In this particular case (10.149) becomes h = u + iv,

u=

θ(4 − b2 ) − 2b2 m , 4 + b2

v=

4(θ + m)b . 4 + b2

(10.151)

The value of h from (10.151) is marked in Figure 10.6. Subcase 3. b = 2 The behavior of parameter h in this case is depicted in Figure 10.7. It shows that in this case the function V (z) can be realized using an extremal accretive Th when γ = −1. The value of the parameter h according to (10.149) then becomes h = u + iv,

u = −m,

v = θ + m.

Clockwise direction of the circle again corresponds to the change of γ from −∞ to +∞ and the marked value of h occurs when γ = −1. Now we consider the second case.

388

Chapter 10. L-systems with Schr¨ odinger operator

h

-m =q -

c 2

q

q+

c 2

u

Figure 10.7: b = 2 ∞ Case 2. Here we assume that 0 dσ(t) = ∞. This means that our function t V (z) belongs to the class SL0 (R, K) and b = ∞. According to Theorem 10.8.6 and formulas (10.137) and (10.138), the restored operator Th is accretive if and only if γ ≥ 0 and α-sectorial if and only if γ > 0. It directly follows from (10.138) that the exact value of the angle α is then found via tan α =

1 . γ

The latter implies that the restored operator Th is extremal if γ = 0. This means that a function V (z) ∈ SL0 (R, K) is realized by an L-system with an extremal operator Th if and only if ∞ dσ(t) V (z) = . t−z 0

On the other hand, since γ ≥ 0, the function V (z) is a Stieltjes function of the class S0K (R). Applying Theorem 9.8.14 we conclude that V (z) admits realization by an accretive L-system θ of the form (6.36) with Re A containing the Kre˘in-von Neumann extension AK as a quasi-kernel. Here AK is deﬁned by (10.87). This yields θ = −m∞ (−0) = −m. As in the beginning of the previous case we derive the formulas for u and v, where h = u + iv. Using (10.144) and (10.147) leads to 2 +v 2 ) θ = μu−(u , μ−u (10.152) u = θ + γv. Solving this system for u and v leads to u=

θ + μγ 2 , 1 + γ2

v=

(μ − θ)γ . 1 + γ2

(10.153)

10.8. Stieltjes-like functions and inverse spectral problems

389

Combining (10.152) and (10.153) gives −m + μγ 2 , 1 + γ2

u=

v=

(m + μ)γ . 1 + γ2

(10.154)

To proceed, we ﬁrst notice that our function V (z) satisﬁes the conditions of Theorem 10.7.4. Indeed, the inequality μ≥

(Im h)2 + Re h, m∞ (−0) + Re h

turns into

v2 + u, u−m if one uses θ = −m and the ﬁrst equation in (10.152). Applying Theorem 10.7.4 yields ⎛ ⎞2 μ=

∞ 0

dσ(t) Im h = 1 + t2 |μ − h|2

⎜ ⎜ sup ⎜ ⎝y∈Dom(AK ) ∞

⎟ ⎟ . 12 ⎟ ⎠ (|y(x)|2 + |l(y)|2 ) dx |μy(a) − y (a)|

a

Taking into account that μy(a) − y (a) = (μ + m)y(a) and setting c1/2 =

sup

∞ y∈Dom(AK )

|y(a)|

12 , (|y(x)|2 + |l(y)|2 ) dx

(10.155)

a

we obtain Im h (μ + m)2 c = |μ − h|2

∞ 0

dσ(t) . 1 + t2

(10.156)

Considering that Im h = v and combining (10.156) with (10.154) we use straightforward calculations to get μ = −m + Let ξ=

1 c

1 γc

0

∞

∞ 0

dσ(t) . 1 + t2

dσ(t) . 1 + t2

(10.157)

ξ . γ

(10.158)

Then the last relation becomes μ = −m +

390

Chapter 10. L-systems with Schr¨odinger operator

x

h x 2

-m-

x 2

-m

-m+

x 2

u

Figure 10.8: b = ∞

Figure 10.9: γ ≥ 0

Applying (10.158) on (10.154) yields u = −m +

γξ , 1 + γ2

v=

ξ , 1 + γ2

γ ≥ 0.

(10.159)

Following the previous case approach we transform (10.159) into 2 ξ ξ2 = . (u + m)2 + v − 2 4 The connection between the parameters γ and h in the accretive restored operator Th is depicted in Figures 10.8 and 10.9. As we can see h traces the highlighted part of the circle clockwise in Figure 10.8 as γ moves from 0 towards +∞. As we mentioned earlier the restored operator Th is extremal if γ = 0. In this case formulas (10.159) become u = −m,

v = ξ,

γ = 0,

(10.160)

where ξ is deﬁned by (10.157). Now once we have described all the possible outcomes for the restored accretive operator Th , we can concentrate on the state-space operator A of the L-system (10.136). We recall that A is deﬁned by formulas (10.11) and besides the parameter

10.8. Stieltjes-like functions and inverse spectral problems

391

h above contains also parameter μ. We will obtain the behavior of μ in terms of the components of our function V (z) the same way we treated the parameter h. As before we consider two major cases ∞dividing them into subcases when necessary. Case 1. Assume that b = 0 dσ(t) < ∞. In this case our function V (z) t belongs to the class SL01(R, K). First we will obtain the representation of μ in terms of u and v, where h = u + iv. We recall that μ=

¯ h − ph , 1−p

where p is deﬁned by (10.146). By direct computations we derive that p=

1 − γ2 2γ − i, 2 1+γ 1 + γ2

and ¯= h − ph

1−p=

2γ 2 2γ + i, 2 1+γ 1 + γ2

2γ 2 2γ 2γ 2 u+ v + v+ u i. 1 + γ2 1 + γ2 1 + γ2 1 + γ2

Plugging the last two equations into the formula for μ above and simplifying we obtain 1 (10.161) μ = u + v. γ We recall that during the present case u and v parts of h are described by the formulas (10.149). Once again we elaborate in three subcases. Subcase 1. b > 2 As we have shown this above, the formulas (10.149) can be transformed into an equation of the circle (10.150). In this case the parameter γ belongs to the interval in (10.143), the accretive operator Th corresponds to the values of h shown in the bold part of the circle in Figure 10.4 as γ moves from −∞ towards +∞. Substituting the expressions for u and v from (10.149) into (10.161) and simplifying we get μ=θ+

(θ + m)b . γ

The connection between values of γ and μ is depicted in Figure 10.10. We note that μ = 0 when γ = − (θ+m)b . Also, the endpoints θ γ1 =

−b −

√

b2 − 4

2

and

γ2 =

−b +

√ b2 − 4 , 2

of γ-interval (10.143) are responsible for the μ-values μ1 = θ +

(θ + m)b γ1

and

μ2 = θ +

(θ + m)b . γ2

392

Chapter 10. L-systems with Schr¨odinger operator

Figure 10.10: b > 2 The values of μ that are acceptable parameters of operator A of the restored L-system make the bold part of the hyperbola in Figure 10.10. It follows from Theorem 9.8.2 that the operator A of the form (10.11) is accretive if and only if γ ≥ 0 and thus μ sweeps the right branch on the hyperbola. We note that Figure 10.10 shows the case when −m < 0, θ > 0, and θ > −m. Other possible cases, such as (−m < 0, θ < 0, θ > −m), (−m < 0, θ = 0), and (m = 0, θ > 0) require corresponding adjustments to the graph shown in Figure 10.10. Subcase 2. b < 2 For every γ ∈ (−∞, +∞) the restored operator Th will be accretive and α-sectorial for some α ∈ (0, π/2). As we have mentioned above, the operator Th achieves the largest angle of sectorialilty when γ = − 2b . In this particular case (10.149) becomes h = u + iv,

u=

θ(4 − b2 ) − 2b2 m , 4 + b2

v=

4(θ + m)b . 4 + b2

Substituting γ = −b/2 into (10.161) we obtain μ = −(θ + 2m).

(10.162)

This value of μ from (10.162) is marked in Figure 10.11. The corresponding operator A of the realized L-system is based on these values of parameters h and μ.

10.8. Stieltjes-like functions and inverse spectral problems

393

Figure 10.11: b < 2 and b = 2

Subcase 3. b = 2 The behavior of parameter μ in this case is also shown in Figure 10.11. It was shown above that in this case the function V (z) can be realized using an extremal accretive Th when γ = −1. The values of the parameters h and μ then become h = u + iv,

u = −m,

v = θ + m,

μ = −(θ + 2m).

The value of μ above is marked on the left branch of the hyperbola and occurs when γ = −1 = −b/2. ∞ Case 2. Again we assume that 0 dσ(t) = ∞. Hence V (z) ∈ SL0 (R, K) and t b = ∞. As we mentioned above the restored operator Th is accretive if and only if γ ≥ 0 and α-sectorial if and only if γ > 0. It is extremal if γ = 0. The values of u, v, and μ were already calculated and are given in (10.159) and (10.158), respectively. That is u = −m +

γξ , 1 + γ2

v=

ξ , 1 + γ2

μ = −m +

ξ , γ

γ ≥ 0,

where ξ is deﬁned in (10.157). Figure 10.12 gives a graphical representation of this case. Only the right-hand bold branch of the hyperbola shows the values of μ in the case b = ∞. If m = 0, then ξ μ= γ

394

Chapter 10. L-systems with Schr¨ odinger operator

Figure 10.12: b = ∞

and the graph should be adjusted accordingly. In the case when γ = 0 and Th is extremal we have u = −m,

μ = ∞,

v = ξ,

h = −m + iξ,

and according to (10.11) we have A = −y + q(x)y + [(−m + iξ)y(a) − y (a)]δ(x − a),

(10.163)

which is the state-space operator of the realized L-system. Example. Consider a function i V (z) = √ . z A direct check conﬁrms that V (z) is a Stieltjes function. Let us also consider a function σ(λ) such that σ(λ) = 0 for λ ≤ 0 and 1 λ i σ(λ) = C + lim Im √ dx, for λ > 0. y→0 π 0 x + iy One can verify [207] that σ (λ) =

1 √ for λ > 0. π λ

By direct calculations we have that

∞

V (z) = 0

dσ(t) i = √ , t−z z

10.8. Stieltjes-like functions and inverse spectral problems and that

∞

0

dσ(t) = t

0

∞

395

dt = ∞. πt3/2

It is also clear that the constant term in the integral representation (9.141) is zero, i.e., γ = 0. ˙ AK ), where Let us consider an operator pair (A,

˙ = −y , Ay (10.164) y(0) = y (0) = 0, is a symmetric operator and

AK y = −y , y (0) = 0,

is a self-adjoint one. Let us assume that σ(t) satisﬁes our deﬁnition of spectral ˙ AK ) given in Section 10.8 (we will prove it a distribution function of the pair (A, bit later). Operating under this assumption, we proceed to restore parameters h and μ and apply formulas (10.159) for the values γ = 0 and m = −θ = 0. This yields u = 0. To obtain v we ﬁrst ﬁnd the value of ∞ dσ(t) 1 =√ , 2 1 + t 2 0 √ and then use formula (10.155) to get the value of c. This yields c = 1/ 2. Consequently, 1 ∞ dσ(t) ξ= = 1, c 0 1 + t2 and hence h = vi = i. From (10.158) we have that μ = ∞ and (10.163) becomes A y = −y + [iy(0) − y (0)]δ(x).

(10.165)

The operator Th in this case is

Th y = −y , y (0) = iy(0).

The channel vector g of the form (10.15) then equals g = δ(x), satisfying Im A =

A − A∗ = KK ∗ = (., g)g, 2i

and channel operator Kc = cg, (c ∈ C) with K ∗ y = (y, g) = y(0).

(10.166)

396

Chapter 10. L-systems with Schr¨odinger operator

The real part of A,

Re A y = −y − y (0)δ(x),

contains the self-adjoint quasi-kernel

: = −y , Ay y (0) = 0. The L-system with Schr¨ odinger operator of the form (10.18) that realizes V (z) can now be written as A K 1 , Θ= C H+ ⊂ L2 [0, +∞) ⊂ H− where A and K are deﬁned above via (10.165) and (10.166), respectively. Now we can back up our assumption on σ(t) to be the spectral distribution ˙ AK ). Indeed, calculating the function VΘ (z) for the Lfunction of the pair (A, system Θ above directly via formula (10.21) with μ = ∞ and comparing the result to V (z) gives the exact value of h = i. It is known that operator A˙ in (10.164) is prime. On the other hand, by Theorem 10.8.4, V (z) can be realized as an impedance function of a model system ΘΛ of the form (10.134), whose symmetric operator Λ˙ σ is also prime (see Theorem 6.6.7). Hence, we can apply Theorem 6.6.10 on bi-unitary equivalence that provides, with the unitary mapping U , for a deﬁnition of the spectral distribution function. Thus we conﬁrm that σ(t) is the ˙ AK ). spectral distribution function of the pair (A, All the derivations above can be repeated for a Stieltjes-like function i V (z) = γ + √ , z

−∞ < γ < +∞,

γ = 0

with very minor changes. In this case the restored values for h and μ are described as follows: 1 1 γ , v= , μ= . h = u + iv, u = 1 + γ2 1 + γ2 γ The dynamics of changing h according to changing γ is depicted in Figure 10.8 where the circle has its center at the point i/2 and radius of 1/2. The behavior of μ is described by a hyperbola μ = 1/γ (see Figure 10.12 with θ = 0). In the case when γ > 0 our function becomes Stieltjes and the restored L-system Θ is accretive. The operators A and K of the restored L-system are given according to the formulas (10.11) and (10.17), respectively.

10.9 Inverse Stieltjes-like functions and inverse spectral problems for systems with Schr¨odinger operator In this section we are going to use the realization technique and results developed for inverse Stieltjes functions in Section 9.9 to obtain a solution of the inverse spectral problem for L-systems with Schr¨odinger operator of the form (10.126).

10.9. Inverse Stieltjes-like functions and inverse spectral problems

397

Deﬁnition 10.9.1. A scalar Herglotz-Nevanlinna function V (z) is called an inverse Stieltjes-like function if it has an integral representation ∞ ∞ 1 1 dτ(t) V (z) = α + − dτ (t), < ∞, (10.167) t−z t t + t2 0 0 similar to (9.180) but with an arbitrary (not necessarily non-positive) constant α. We are going to introduce a new class of realizable scalar inverse Stieltjes-like functions whose structure is similar to that of S0−1 (R) of Section 9.9. Deﬁnition 10.9.2. An inverse Stieltjes-like function V (z) belongs to the class SL−1 0 (R) if it admits an integral representation ∞ 1 1 V (z) = α + − dτ(t), (10.168) t−z t 0 where non-decreasing function τ (t) satisﬁes the conditions ∞ ∞ dτ (t) dτ(t) = ∞, < ∞. t + t2 0 0

(10.169)

Consider the following subclasses of SL−1 0 (R). −1 Deﬁnition 10.9.3. A function V (z) ∈ SL−1 0 (R) is a member of the class SL0 (R, F ) if ∞ dτ(t) = ∞. (10.170) t 0 −1 Deﬁnition 10.9.4. A function V (z) ∈ SL−1 0 (R) is a member of the class SL01 (R, F ) if ∞ dτ(t) < ∞. (10.171) t 0

The following theorem describes the realization of the class SL−1 0 (R). Theorem 10.9.5. Let V (z) ∈ SL−1 0 (R). Then it can be realized as an impedance function of a minimal L-system. Proof. We start by applying the general realization Theorem 6.5.2 to a HerglotzNevanlinna function V (z) and obtain an L-system Λ Kτ 1 ΘΛ = , (10.172) τ τ H+ ⊂ Lτ2 [0, +∞) ⊂ H− C such that V (z) = VΘΛ (z). Following the steps for construction of the model Lsystem described in Theorem 6.5.2, we note that Λ = Re Λ + iK τ (K τ )∗

398

Chapter 10. L-systems with Schr¨ odinger operator

is a (∗)-extension of an operator T τ such that Λ ⊃ T τ ⊃ Λ˙ τ where Λ˙ τ is deﬁned in (10.131). The real part Re Λ is a self-adjoint bi-extension of Λ˙ τ that has a quasi-kernel Λτ of the form (10.130). It was also shown in Section 9.9 that the operator Λ possess the accumulative property (9.181). The operator K τ in the above L-system is deﬁned by K τ c = c · 1,

(K τ )∗ x = (x, 1)

c ∈ C,

τ τ 1 ∈ H− , x(t) ∈ H+ .

τ In addition we can observe that the function η(λ) ≡ 1 belongs to H− . To conﬁrm this we need to show that (x, 1) deﬁnes a continuous linear functional for every τ x ∈ H+ . It was shown in Theorem 6.5.2 that

4

4 c1 c2 t τ ˙ H+ = Dom(Λτ ) , c1 , c2 ∈ C. (10.173) 1 + t2 1 + t2 τ Consequently, every vector x ∈ H+ has three components x = x1 + x2 + x3 according to the decomposition (10.173) above. Obviously, (x1 , 1) and (x2 , 1) yield convergent integrals while (x3 , 1) boils down to ∞ t dτ (t). 1 + t2 0

The convergence of the latter is guaranteed by the deﬁnition of a inverse Stieltjesτ τ like function. The state-space of the L-system ΘΛ is H+ ⊂ Lτ2 [0, +∞) ⊂ H− , where τ H+ = Dom Λ˙ ∗τ . According to Theorem 6.6.7 the operator Λ˙ τ of the form (10.131) is a prime operator. Thus, L-system ΘΛ of the form (10.172) is minimal. At this point we are ready to state and prove the main realization result of this section. Theorem 10.9.6. Let V (z) ∈ SL−1 0 (R) and the function τ (t) be the distribution ˙ Bθ ) of the form (10.127), (10.128). Then there exist a unique function of a pair (B, Schr¨ odinger operator Th (Im h > 0) of the form (10.126), an operator A given by (10.11), an operator K as in (10.17), and an L-system A K 1 Θ= , (10.174) H+ ⊂ L2 [a, +∞) ⊂ H− C of the form (10.18) such that V (z) = VΘ (z). Proof. It follows from the deﬁnition of the distribution function above that there is an operator U deﬁned in (10.129) establishing a one-to-one isometric correspondence between Lτ2 [0, +∞) and L2 [a, +∞) while providing for unitary equivalence between the operator Bθ and the operator of multiplication by independent variable Λτ of the form (10.130). Let us consider the L-system ΘΛ of the form (10.172) constructed in the proof of Theorem 10.9.5. Applying Theorem 6.6.10 on unitary equivalence to the

10.9. Inverse Stieltjes-like functions and inverse spectral problems

399

isometry U deﬁned in (10.129) we obtain a triplet of isometric operators U+ , U , and U− , where τ ∗ ∗ U+ = U H+ , U− = U+ . τ This triplet of isometric operators will map the rigged Hilbert space H+ ⊂ τ τ ⊂ H− into the triplet H+ ⊂ L2 [a, +∞) ⊂ H− . Moreover, U+ is an τ ∗ ∗ isometry from H+ = Dom(Λ˙ ∗τ ) onto H+ = Dom(B˙ ∗ ), and U− = U+ is an isomτ etry from H+ onto H− . This is true since the operator U provides the unitary equivalence between the symmetric operators B˙ and Λ˙ τ . Now we construct an L-system A K 1 Θ= , H+ ⊂ L2 [a, +∞) ⊂ H− C

Lτ2 [0, +∞)

−1 where K = U− K τ and A = U− ΛU+ is a (∗)-extension of operator T = U T τ U −1 ˙ The real part Re A contains the quasi-kernel Bθ . This such that A ⊃ T ⊃ B. construction of A is unique due to Theorem 4.3.9 on the uniqueness of a (∗)extension for a given quasi-kernel. On the other hand, all (∗)-extensions based ˙ Bθ ) must take form (10.11) for some values of parameters h and μ. on a pair (B, Consequently, our function V (z) is realized by the L-system Θ of the form (10.174) and V (z) = VΘΛ (z) = VΘ (z).

The theorem below gives the criteria for the operator Th of the realizing L-system to be accretive. Theorem 10.9.7. Let V (z) ∈ SL−1 0 (R) satisfy the conditions of Theorem 10.9.6. Then the operator Th in the conclusion of Theorem 10.9.6 is accretive if and only if ∞ dτ (t) 2 α −α + 1 ≥ 0. (10.175) t 0 The operator Th is φ-sectorial for some φ ∈ (0, π/2) if and only if the inequality (10.175) is strict. In this case the exact value of angle φ can be calculated by the formula tan φ =

α2 −

∞ dτ (t) 0 ∞ t α 0 dτt(t)

+1

.

(10.176)

Proof. It was shown in Theorem 10.3.1 that for the L-system Θ in (10.174) described in the previous theorem the operator Th is accretive if and only if the function Vh (z) of the form (10.32) is holomorphic in Ext[0, +∞) and satisﬁes the inequality (10.33). Here WΘ (z) is the transfer function of (10.174). According to Theorem 10.3.1, the operator Th is φ-sectorial for some φ ∈ (0, π/2) if and only if the inequality (10.33) is strict while the exact value of angle φ can be calculated by the formula (10.34), i.e., cot φ =

1 + Vh (−0) Vh (−∞) . |Vh (−∞) − Vh (−0)|

(10.177)

400

Chapter 10. L-systems with Schr¨ odinger operator According to Theorem 10.9.6 and equation (6.48) WΘ (z) = (I − iV (z)J)(I + iV (z)J)−1 .

By direct calculations one obtains ∞ (t) ∞ 1 − i α − 0 dτ 1+i α− 0 t+t2 −1 WΘ (−1) = ∞ (t) , WΘ (−1) = ∞ 1 + i α − 0 dτ 1 − i α− 0 2 t+t Using the notations

c=α−

0

∞

dτ(t) t + t2

and

d =α−

0

∞

dτ (t) t+t2 dτ (t) t+t2

.

dτ (t) , t

and performing straightforward calculations we obtain WΘ (−1) =

1−ic , 1+ic

WΘ (−∞) =

1− id , 1+ id

and

c−α c−d and Vh (−∞) = . (10.178) 1 +cα 1 + cd Substituting (10.178) into (10.177) and performing the necessary steps we get ∞ α2 − α 0 dτt(t) + 1 1 + αd cot φ = = . (10.179) ∞ dτ (t) α−d Vh (−0) =

0

t

Taking into account that α − d > 0 we combine (10.175), (10.177) with (10.179) and this completes the proof of the theorem. Now let us consider V (z) ∈ SL−1 0 (R) satisfying the conditions of Theorem 10.9.6. According to Theorem 10.9.6 there exists a minimal L-system Θ of the form (10.174) with unique Schr¨odinger operator Th (Im h > 0) of the form (10.126). For the rest of this section we will derive the formulas for calculation of the boundary parameter h in Th as well as a real parameter μ that is used in construction (10.11) of the operator A of the realizing L-system Θ. An elaborate investigation of these formulas is going to show the dynamics of the restored parameters h and μ in terms of the changing free term α from the integral representation (10.168) of the function V (z). Below we will derive the formulas for calculation of the boundary parameter h in the restored Schr¨ odinger operator Th of the form (10.126). We consider two major cases. ∞ dτ (t) Case 1. In the ﬁrst case we assume that 0 < ∞. This means that our t function V (z) belongs to the class SL−1 (R, F ). In what follows we write 01 ∞ dτ (t) b= and m = m∞ (−0). t 0

10.9. Inverse Stieltjes-like functions and inverse spectral problems

401

Suppose that b ≥ 2. Then the quadratic inequality (10.175) implies that, for all α such that > = √ √ b − b2 − 4 b + b2 − 4 α ∈ −∞, ∪ , +∞ , (10.180) 2 2 the restored operator Th is accretive. Clearly, this operator is extremal accretive if √ b ± b2 − 4 α= . 2 In particular if b = 2, then α = 1 and the function ∞ 1 1 V (z) = 1 + − dτ(t) t−z t 0 is realized using an extremal accretive Th . Now suppose that 0 < b < 2. Then for every α ∈ (−∞, +∞) the restored operator Th will be accretive and φ-sectorial for some φ ∈ (0, π/2). Consider a function V (z) deﬁned by (10.168). Conducting realizations of V (z) by operators Th for diﬀerent values of α ∈ (−∞, +∞) we notice that the operator Th with the largest angle of sectoriality occurs when α = b/2 and is found according to the formula b φ = arctan . 1 − b2 /4 This follows from the formula (10.176), the fact that α2 − α b + 1 > 0 for all α, and the formula 2 b b2 α2 − α b + 1 = α − + 1− . 2 4 Now we will focus on the description of the parameter h in the restored operator Th . It was shown in Theorem 10.7.5 that the quasi-kernel Aˆ of the realizing Lsystem Θ from Theorem 10.9.5 takes the form

: = −y + q(x)y, μRe h − |h|2 Ay η= . y (a) = ηy(a), μ − Re h On the other hand, since τ (t) is also the distribution function of the non-negative self-adjoint operator, we can conclude that Aˆ equals the operator Bθ of the form (10.128). This connection allows us to obtain θ=η=

μRe h − |h|2 . μ − Re h

Assuming that h = u + iv,

(10.181)

402

Chapter 10. L-systems with Schr¨ odinger operator

we will use (10.181) to derive the formulas for u and v in terms of γ. First, to eliminate parameter μ, we notice that (10.20) and (5.41) imply WΘ (λ) =

μ − h m∞ (λ) + h 1 − iV (z) = . 1 + iV (z) μ − h m∞ (λ) + h

(10.182)

Passing to the limit in (10.182) when λ → −∞ and taking into account that V (−∞) = α − b and m∞ (−∞) = ∞ we obtain μ−h 1 − i(α − b) = . 1 + i(α − b) μ−h Let us set p=

1 − i(α − b) . 1 + i(α − b)

(10.183)

(10.184)

Solving (10.183) for μ and using (10.184) yields μ=

¯ h − ph . 1−p

Substituting this value of μ in (10.181) and simplifying we obtain u + iv − p(u − iv)u − (u2 + v 2 )(1 − p) = θ. u + iv − p(u − iv) − u(1 − p) After straightforward calculations aiming to represent the numerator and denominator of the last equation in standard form, one obtains the relation u − (α − b) v = θ.

(10.185)

It was shown in Theorem 10.3.3 that the φ-sectorialilty of the operator Th and (10.34) lead to Im h v tan φ = = . (10.186) Re h + m∞ (−0) u + m∞ (−0) Combining (10.185) and (10.186) one obtains u − (α − b)(u tan φ + m∞ (−0) tan φ) = θ, or u=

θ + (α − b)m∞ (−0) tan φ . 1 − (α − b) tan φ

But tan φ is also determined by (10.176). Direct substitution of tan φ =

b 1 + α(α − b)

10.9. Inverse Stieltjes-like functions and inverse spectral problems

403

h1

h2 q -

c 2

q

-m

q+

u

c 2

a

0

b -

b 2

2

1

- 4

b + b2 - 4 2

Figure 10.13: b > 2

into the above equation yields θ + m∞ (−0) b(α − b) u=θ+ . 1 + (α − b)2 Using the short notation and ﬁnalizing calculations we get h = u + iv,

u=θ+

(α − b)[θ + m]b , 1 + (α − b)2

v=

[θ + m]b . 1 + (α − b)2

(10.187)

At this point we can use (10.187) to provide analytical and graphical interpretation of the parameter h in the restored operator Th . Let c = (θ + m)b. Again we consider three subcases. Subcase 1. b > 2

Using basic algebra we transform (10.187) into c 2 c2 (u − θ)2 + v − = . 2 4

(10.188)

Since in this case the parameter α belongs to the interval in (10.180), we can see that h traces the highlighted part of the circle in Figure 10.13 as α moves from −∞ towards +∞. We also notice that the removed point (θ, 0) corresponds to the value of α = ±∞ while the points h1 and h2 correspond to √ √ b− b2 −4 b+ b2 −4 the values α1 = and α2 = , respectively (see Figure 10.13). 2 2

404

Chapter 10. L-systems with Schr¨odinger operator

h

-m

f q -

c 2

q

c 2

q+

u

Figure 10.14: b < 2

Subcase 2. b < 2 For every α ∈ (−∞, +∞) the restored operator Th will be accretive and φ-sectorial for some φ ∈ (0, π/2). As we have mentioned above, the operator Th achieves the largest angle of sectorialilty when α = 2b . In this particular case (10.187) becomes h = u + iv,

u=θ−

2(θ + m)b2 , 4 + b2

v=

4(θ + m)b . 4 + b2

(10.189)

The value of h from (10.189) is marked in Figure 10.14. Subcase 3. b = 2 The behavior of parameter h in this case is depicted in Figure 10.15. It shows that in this case the function V (z) can be realized using an extremal accretive Th when α = 1. The value of the parameter h according to (10.187) then becomes h = u + iv,

u = −m,

v = θ + m.

Clockwise direction of the circle again corresponds to the change of α from −∞ to +∞ and the marked value of h occurs when α = 1. Now we consider the second case. ∞ Case 2. Here we assume that 0 dτt(t) = ∞. This means that our function V (z) belongs to the class SL−1 0 (R, F ) and b = ∞. According to Theorem 10.9.7 and formulas (10.175) and (10.176), the restored operator Th is accretive if and only if α ≤ 0 and φ-sectorial if and only if α < 0. It directly follows from (10.176) that the exact value of the angle φ is then found from 1 tan φ = − . α

(10.190)

The latter implies that the restored operator Th is extremal if α = 0. This means that a function V (z) ∈ SL−1 0 (R, F ) is realized by an L-system with an extremal

10.9. Inverse Stieltjes-like functions and inverse spectral problems

405

h

-m =q -

c 2

q

q+

c 2

u

Figure 10.15: b = 2

operator Th if and only if

∞

V (z) = 0

1 1 − t−z t

dτ (t).

On the other hand, since α ≤ 0, the function V (z) is an inverse Stieltjes function −1 of the class S0,F (R). Applying realization Theorem 9.9.7, we conclude that V (z) admits realization by an accumulative L-system Θ of the form (6.36) with Re A containing the Friedrichs extension AF as a quasi-kernel. Here AF is deﬁned by (10.73). This yields μu − (u2 + v 2 ) θ= = ∞, μ−u and hence μ = u. As at the beginning of the previous case, we derive the formulas for u and v, where h = u + iv. Assuming that α = 0 and using (10.186) and (10.190) one obtains u+m u = μ, v = − . (10.191) α To proceed, we ﬁrst notice that our function V (z) satisﬁes the conditions of Theorem 10.7.5. Indeed, since V (z) ∈ S0−1 (R), we can use Theorem 10.7.3 and obtain inequality (10.114) that is required for Theorem 10.7.5 to hold. Applying Theorem 10.7.5 yields ⎛ ∞ 0

dτ (t) Im h = 2 1+t |μ − h|2

⎞2

⎜ ⎜ ⎜ sup ⎝y∈Dom(AF ) ∞

⎟ ⎟ . ⎟ ⎠ 2 2 (|y(x)| + |l(y)| ) dx |μy(a) − y (a)|

1 2

a

Taking into account that for the case of AF we have y(a) = 0 and hence |μy(a) −

406

Chapter 10. L-systems with Schr¨odinger operator

v d ¥

d t (t ) 1 + t 2

ò 0

u

0

a 0 Figure 10.16: b = ∞ y (a)| = |y (a)| we set d1/2 =

|y (a)|

sup

∞ y∈Dom(AF )

(|y(x)|2

+

12 ,

(10.192)

|l(y)|2 ) dx

a

and obtain Im h d= |μ − h|2

∞ 0

dτ (t) . 1 + t2

(10.193)

Since Im h = v and μ = u (see (10.191)), we solve (10.193) for v to get v = ∞ 0

d dτ (t) 1+t2

.

Consequently, equations (10.191) describing h = u + iv take the form αd u = −m + ∞ dτ (t) , 0

1+t2

v = ∞ 0

d dτ (t) 1+t2

.

(10.194)

The equations (10.194) above provide parametrical equations of the straight horizontal line shown in Figure 10.16. The connection between the parameters α and h in the accretive restored operator Th is depicted in bold.

10.9. Inverse Stieltjes-like functions and inverse spectral problems

407

Figure 10.17: b > 2

As we mentioned earlier, the restored operator Th is extremal if α = 0. In this case formulas (10.194) become u = −m,

v = ∞ 0

d dτ (t) 1+t2

.

Now once we have described all possible outcomes for the restored accretive operator Th , we can concentrate on the main operator A of the L-system (10.174). We recall that A is deﬁned by formulas (10.11) and besides the parameter h above contains also parameter μ. We will obtain the behavior of μ in terms of the components of our function V (z) in the same way as we treated the parameter h. As before we consider two major cases ∞ dividing them into subcases when necessary. Case 1. Assume that b = 0 dτt(t) < ∞. In this case our function V (z) belongs to the class SL−1 01 (R, F ). First we will obtain the representation of μ in terms of u and v, where h = u + iv. We recall that ¯ h − ph μ= , 1−p where p is deﬁned by (10.184). By direct computations we derive that p=

1 − (α − b)2 2(α − b) − i, 1 + (α − b)2 1 + (α − b)2

1−p=

2(α − b)2 2(α − b) + i, 1 + (α − b)2 1 + (α − b)2

408

Chapter 10. L-systems with Schr¨odinger operator

and

¯= h − ph

2(α − b)2 2(α − b) u+ v 1 + (α − b)2 1 + (α − b)2 2 2(α − b) + v+ u i. 1 + (α − b)2 1 + (α − b)2

Plugging the last two equations into the formula for μ above and simplifying we obtain v . (10.195) μ=u+ α−b We recall that during the present case u and v parts of h are described by the formulas (10.187). Once again we elaborate in three subcases. Subcase 1. b > 2 As we have shown before, the formulas (10.187) can be transformed into an equation of the circle (10.188). In this case the parameter α belongs to the interval in (10.180), the accretive operator Th corresponds to the values of h shown in the bold part of the circle in Figure 10.13 as α moves from −∞ towards +∞. Substituting the expressions for u and v from (10.187) into (10.195) and simplifying we get μ=θ+

(θ + m)b . α−b

The connection between values of α and μ is depicted in Figure 10.17. . Also, the endpoints We note that μ = 0 when α = − mb θ α1 =

b−

√

b2 − 4 2

and

α2 =

b+

√

b2 − 4 , 2

of α-interval (10.180) are responsible for the μ-values μ1 = θ +

(θ + m)b α1

and

μ2 = θ +

(θ + m)b . α2

The values of μ that are acceptable parameters of operator A of the restored L-system with an accretive operator Th make the bold part of the hyperbola in Figure 10.17. It follows from Theorems 9.9.4 and 9.9.5 that the operator A of the form (10.11) is accumulative if and only if α ≤ 0 and thus μ belongs to the part of the left branch on the hyperbola when α ∈ (−∞, 0]. We note that Figure 10.17 shows the case when −m < 0, θ > 0, and θ > −m. Other possible cases, such as (−m < 0, θ < 0, θ > −m), (−m < 0, θ = 0), and (m = 0, θ > 0) require corresponding adjustments to the graph shown in Figure 10.17.

10.9. Inverse Stieltjes-like functions and inverse spectral problems

409

Figure 10.18: b < 2 and b = 2 Subcase 2. b < 2 For every α ∈ (−∞, +∞) the restored operator Th is accretive and φ-sectorial for some φ ∈ (0, π/2). As we have mentioned above, the operator Th achieves the largest angle of sectorialilty when α = 2b . In this particular case (10.187) becomes (10.189). Substituting α = b/2 and (10.189) into (10.195) we obtain μ = −(θ + 2m). (10.196) This value of μ from (10.196) is marked in Figure 10.18. The corresponding operator A of the realizing L-system is based on these values of parameters h and μ. Subcase 3. b = 2 The behavior of parameter μ in this case is also shown in Figure 10.18. It was shown above that in this case the function V (z) can be realized using an extremal accretive operator Th when α = 1. The values of the parameters h and μ then become h = u + iv,

u = −m,

v = θ + m,

μ = −(θ + 2m).

The value of μ above is marked on the left branch of the hyperbola and occurs when α = 1 = b/2. ∞ Case 2. Again we assume that 0 dτt(t) = ∞. Hence V (z) ∈ SL−1 0 (R, F ) and b = ∞. As we mentioned above the restored operator Th is accretive if and only if

410

Chapter 10. L-systems with Schr¨odinger operator

Figure 10.19: b = ∞

α ≤ 0 and φ-sectorial if and only if α < 0. It is extremal if α = 0. The values of u and v, were already calculated and are given in (10.194). In particular, the value for μ is given by αd μ = u = −m + ∞ dτ (t) . (10.197) 0

1+t2

where d is deﬁned in (10.192). Figure 10.19 gives a graphical representation of this case. The left bold part of the line corresponds to the values of μ that yield an accumulative realizing L-system. If m = 0, then the line passes through the origin and the graph should be adjusted accordingly. In the case when α = 0 and Th is extremal we have μ = m. Example. Consider a function √ V (z) = i z. A direct check conﬁrms that V (z) is an inverse Stieltjes function. Let us also consider a function τ (λ) such that τ (λ) = 0 for λ ≤ 0 and 1 y→0 π

τ (λ) = C + lim

λ

+ Im i x + iy dx.

0

One can verify (see also [207]) that τ (λ) =

1√ λ for λ > 0. π

10.9. Inverse Stieltjes-like functions and inverse spectral problems

411

By direct calculations we have that ∞ √ 1 1 V (z) = − dτ(t) = i z, t − z t 0 and that

∞

0

dτ (t) = t

∞

0

dt √ = ∞. π t

It is also clear that the constant term in the integral representation (10.167) is zero, i.e., α = 0. ˙ AF ) with a symmetric operator of the Let us consider an operator pair (A, form (10.9) given by

˙ = −y , Ay (10.198) y(0) = y (0) = 0, and a self-adjoint operator

AF y = −y , y(0) = 0.

We assume that τ (t) satisﬁes our deﬁnition of distribution function of the pair ˙ AF ) given in Section 10.8 (we are going to show it later). Operating under (A, this assumption, we proceed to restore parameters h and μ and apply formulas (10.194) for the values α = 0 and m = m∞ (−0) = 0. This yields u = 0. To obtain v we ﬁrst ﬁnd the value of ∞ dτ (t) 1 =√ , 2 1 + t 2 0 √ and then use formula (10.192) to get the value of d. This yields d = 1/ 2. Consequently, d v = ∞ dτ (t) = 1, 0

1+t2

and hence h = vi = i. From (10.197) we have that μ = 0 and (10.11) becomes A y = −y − [iy(0) + y (0)]δ (x). The operator Th in this case is

(10.199)

Th y = −y , y (0) = iy(0).

The channel vector g of the form (10.15) then equals g = δ (x), satisfying Im A =

A − A∗ = KK ∗ = (., g)g, 2i

and channel operator Kc = cg, (c ∈ C) with K ∗ y = (y, g) = −y (0).

(10.200)

412 The real part of A,

Chapter 10. L-systems with Schr¨odinger operator

Re A y = −y − y(0)δ (x),

contains the self-adjoint quasi-kernel

: = −y , Ay y(0) = 0. An L-system with Schr¨odinger operator of the form (10.18) that realizes V (z) can now be written as A K 1 Θ= , H+ ⊂ L2 [0, +∞) ⊂ H− C where A and K are deﬁned above via (10.199) and (10.200), respectively. Now we can back up our assumption on τ (t) to be the spectral distribution function of the ˙ AF ). Indeed, calculating the function VΘ (z) for the L-system Θ above pair (A, directly via formula (10.21) with μ = 0 and comparing the result to V (z) gives the exact value of h = i. It is known that operator A˙ in (10.198) is prime. On the other hand, we know that V (z) can be realized as an impedance function of a model system ΘΛ of the form (10.134), whose symmetric operator Λ˙ τ is also prime (see Section 10.4). Hence, we can apply Theorem 6.6.10 on bi-unitary equivalence that, with the unitary mapping U , provides for the deﬁnition of a spectral distribution function. Thus we conﬁrm that τ (t) is the spectral distribution function of the ˙ AF ). pair (A, All the derivations above can be repeated for an inverse Stieltjes-like function √ −∞ < α < +∞, V (z) = α + i z, with very minor changes. In this case the restored values for h and μ are described as follows: h = u + iv, u = α, v = 1, μ = α. The dynamics of changing h according to changing α is depicted in Figure 10.16 where the horizontal line has a y-intercept of 1. The behavior of μ is described by a sloped line μ = α (see Figure 10.19 with m = 0). In the case when α ≤ 0 our function becomes inverse Stieltjes and the restored L-system Θ is accretive. The operators A and K of the restored L-system are given according to formulas (10.11) and (10.17), respectively.

Chapter 11

Non-self-adjoint Jacobi Matrices and System Interpolation In this chapter we consider a new type of solutions of the Nevanlinna-Pick interpolation problem for the class of scalar Herglotz-Nevanlinna functions, the so-called, explicit system solutions that are impedance functions of the Liv˘sic canonical systems. The conditions of the existence and uniqueness of solutions are presented in terms of interpolation data. We derive an exact formula for the angle of sectoriality of the corresponding state-space operator in the explicit system solution. The criterion for this operator to be accretive but not α-sectorial for any angle α ∈ (0, π/2) is obtained in terms of interpolation data and the classic Pick matrices. We ﬁnd conditions on interpolation data when the explicit system solution is generated by the Liv˘sic canonical dissipative system whose state-space operator is a non-self-adjoint, prime dissipative Jacobi matrix with a rank-one imaginary part. These results are based on a new model for prime, bounded, dissipative operators with rank-one imaginary part. In addition, an inverse spectral problem for ﬁnite non-self-adjoint Jacobi matrices with rank-one imaginary part is solved. We show that any ﬁnite sequence of non-real numbers in the open upper half-plane is the set of eigenvalues (counting multiplicity) of some dissipative non-self-adjoint Jacobi matrix with rank-one imaginary part. The algorithm of reconstruction of the unique Jacobi matrix from its non-real eigenvalues is presented.

Y. Arlinskii et al., Conservative Realizations of Herglotz–Nevanlinna Functions, Operator Theory: Advances and Applications, DOI 10.1007/978-3-7643-9996-2_11, © Springer Basel AG 2011

413

414

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

11.1 Systems with Jacobi matrices The three-diagonal matrices of the form ⎛ b1 a 1 0 0 ⎜a1 b2 a2 0 ⎜ ⎜ 0 a 2 b3 a 3 J =⎜ ⎜· · · · ⎜ ⎝· · · · · · · · ⎛

and

b1 ⎜a1 J =⎜ ⎝0 ·

a1 b2 a2 ·

0 a2 b3 ·

· · · · · 0

· · · · · an−1

0 0 a3 ·

0 · 0 · 0 · · ·

· · · ·

⎞

⎟ ⎟ ⎟ ⎟, ⎟ ⎟ an−1 ⎠ bn

(11.1)

⎞ · ·⎟ ⎟, ·⎠ ·

(11.2)

where ak > 0, and bk are real numbers for all k ≥ 1 are called self-adjoint Jacobi matrices [2]. We call the matrices of the form (11.1) or (11.2), with ak > 0 for k ≥ 1, and bk are real numbers for k ≥ 2, and Im b1 > 0, dissipative ﬁnite (respect., semi-inﬁnite) Jacobi matrices. Let a linear space Cn of columns be equipped with the usual inner product n ) (x, y) = xk yk and let l2 (N) be the Hilbert space of square summable sequences k=1

x = {x1 , x2 , . . . , xk , . . .} which we consider as semi-inﬁnite vector-columns with the inner product given by ∞ # (x, y) = xk yk . k=1

By {δk } we denote the canonical orthonormal basis in Cn (l2 (N)). An (n × n) complex Jacobi matrix J determines a linear operator in the Hilbert space Cn by means of the matrix product J · x. For the semi-inﬁnite case we also assume that sup{|ak | + |bk |} < ∞.

(11.3)

k

Then this condition is necessary and suﬃcient for boundedness of the Jacobi operator in l2 (N) deﬁned as J · x, where J is a semi-inﬁnite complex Jacobi matrix. The complex semi-inﬁnite Jacobi matrix determines a compact operator in l2 (N) if and only if lim bk = lim ak = 0. k→∞

k→∞

Suppose that (11.3) is fulﬁlled. In this case

||J || ≤ 3 max sup{|ak |}, sup{|bk |} . k

k

11.1. Systems with Jacobi matrices Because

J k δ1

k+1

415

k J δ1 m = 0,

= ak ak−1 . . . a1 ,

m ≥ k + 2,

(11.4)

and ak = 0, the vectors δ1 , J δ1 , . . . , J k δ1 , . . . are linearly independent. Let A be a self-adjoint operator in a separable Hilbert space H and let E(t) be its resolution of identity. We say that the operator A has a simple spectrum if there exists a nonzero vector g in H such that the linear span of all vectors of the form E(Δ)g, where Δ runs through all intervals of R, is dense in H [3]. In general, a vector g ∈ H is called a cyclic vector for a densely deﬁned operator A if c. l. s.{g, Ag, . . . , Ak g, . . .} = H. A cyclic vector g is called normalized if g = 1. In particular, the vector δ1 above is a cyclic vector for the operator J in l2 (N). The next two theorems are well known [239]. Theorem 11.1.1. If A is a self-adjoint operator with a simple spectrum in a Hilbert space H, then there exists a cyclic vector for A. Theorem 11.1.2. If A is a self-adjoint operator with a simple spectrum in a Hilbert space H, then there exists an orthonormal basis in which the matrix of the operator A is a self-adjoint Jacobi matrix. Proof. Let χ ∈ H, ||χ|| = 1 be a cyclic vector for A. Deﬁne Hk = c. l. s.{χ, Aχ, . . . , Ak−1 χ},

k ≥ 1.

Then dim Hk ≤ k for all k ≥ 1. Suppose that n is a minimal natural number such that dim Hn+1 < n + 1. Since ξ is a cyclic vector for A, we get Hn+1 = Hn . Hence dim H = n. Clearly, dim H = ∞ if and only if dim Hk = k for all natural k. Suppose dim H = ∞. Set N1 = H1 = {λχ,

λ ∈ C},

Nk = Hk Hk−1 ,

k ≥ 2.

Then dim Nk = 1. We can ﬁnd the system of vectors {χk }k≥1 such that χ 1 = χ ∈ N1 ,

χk ∈ Nk ,

||χk || = 1,

(χk+1 , Aχk ) > 0,

k ≥ 1.

The system {χk }k≥1 forms an orthonormal basis in H. This system in fact is obtained from the system {Ak−1 χ}k≥1 by the Gram-Schmidt orthonormalization procedure. Because AHk ⊂ Hk+1 , we get Aχk ⊥ χj

for all j ≥ k + 2.

The self-adjointness of A yields Aχk ⊥ χj for j ≤ k − 2 (k ≥ 3). Hence, with respect to the orthonormal basis {χk }k≥1 the matrix of the operator A takes the three-diagonal form. Since (Aχk , χk+1 ) > 0 for k ≥ 1, this matrix is a self-adjoint Jacobi matrix of the form (11.2). The above construction is valid also in the case dim H < ∞.

416

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

Lemma 11.1.3. Let T be a dissipative operator with a rank-one imaginary part and let g be a vector in H such that 2 Im T h = (h, g)g,

h ∈ H.

Then T is prime if and only if the vector g is cyclic for the real part Re T . Proof. Suppose that T is prime. Then (5.7) holds. Let us prove that g is a cyclic vector for Re T . Let H = c. l. s. {(Re T )n g, n = 0, 1, . . .} = H. Then H and H = H H are invariant with respect to Re T . Since H ⊂ Ker Im T , it follows that Re T H = T H = T ∗ H , T ∗n H ⊂ H and Im T T ∗n H = 0 for all n = 0, 1, . . . . Now from (5.7) we obtain that H = {0}, i.e., g is a cyclic vector for Re T . Conversely, suppose that the vector g is cyclic for Re T , i.e., H = H. Let the subspace Hs be deﬁned by the right-hand side of (5.7). Then T (H Hs ) = Re T (H Hs ), and H Hs as well as Hs reduces Re T . Because g ∈ Hs , we get that H ⊂ Hs . It follows that Hs = H, i.e., T is a prime operator. Theorem 11.1.4. Let H be a separable Hilbert space and let T be a bounded, prime, dissipative operator in H with a rank-one imaginary part. Then there exists an orthonormal basis in H in which the matrix of the operator T is a bounded dissipative Jacobi matrix. Proof. The operator T takes the form Re T + iIm T and Im T = l(·, χ)χ, where χ ∈ Ran(Im T ), ||χ|| = 1, l > 0. Since T is a bounded prime operator, then, by Theorem 5.2.1, χ is a cyclic vector for T . Consequently, χ is a cyclic vector for bounded self-adjoint operator Re T . By Stone’s Theorem 11.1.2, there exists an orthonormal basis {χk }k≥1 , χ1 = χ in H such that the matrix of Re T is a selfadjoint Jacobi matrix. It follows from (T χ, χ) = (Re T χ, χ) + il that the matrix of T with respect to {χk }k≥1 is a dissipative Jacobi matrix. Let J ∗ be the adjoint matrix to a Jacobi matrix J and let 1 1 Re J = (J + J ∗ ), Im J = (J − J ∗ ), 2 2i be the Hermitian components of J . One has for the (n × n) case ⎛ ⎞ Re b1 a1 0 0 0 · · · ⎜ a1 b2 a 2 0 0 · · · ⎟ ⎜ ⎟ ⎜ 0 a 2 b3 a 3 0 · · · ⎟ ⎟, Re J = ⎜ ⎜ · · · · · · · · ⎟ ⎜ ⎟ ⎝ · · · · · · · an−1 ⎠ · · · · · 0 an−1 bn

11.2. The Stone theorem and its generalizations ⎛ Im b1 ⎜ 0 ⎜ ⎜ 0 Im J = ⎜ ⎜ · ⎜ ⎝ · 0 and for the semi-inﬁnite case ⎛ Re b1 ⎜ a1 Re J = ⎜ ⎝ 0 · ⎛ Im b1 ⎜ 0 Im J = ⎜ ⎝ 0 ·

0 0 0 · · 0

a1 b2 a2 · 0 0 0 ·

0 a2 b3 ·

⎞ · · 0 · · 0⎟ ⎟ · · 0⎟ ⎟, · · ·⎟ ⎟ · · ·⎠ · · 0

· · · · · ·

0 0 0 · · 0

0 0 a3 ·

0 0 0 0 0 0 · ·

417

· · · ·

0 0 0 · 0 0 0 ·

· · · ·

· · · · · · · ·

⎞ · ·⎟ ⎟, ·⎠ · ⎞ · ·⎟ ⎟. ·⎠ ·

In addition, for every x ∈ Cn (l2 (N)), Im J x = (x, g) g = KK∗ , where g=

+

Im b1 δ1 ,

Kc = cg,

K∗ x = (x, g),

c ∈ C.

(11.5)

The Liv˘sic canonical system of the form (5.6) with ﬁnite Jacobi matrix J as a state-space operator J K 1 Δ= , (11.6) Cn C and, respectively with semi-inﬁnite Jacobi matrix as a state-space operator, J K 1 Δ= (11.7) l2 (N) C is called the Liv˘sic system in Jacobi form.

11.2 The Stone theorem and its generalizations Let A be a bounded self-adjoint operator with simple spectrum in a separable Hilbert space H and let χ be a normalized (χ = 1) cyclic vector for A. Deﬁnition 11.2.1. The function m(z) = ((A − zI)−1 χ, χ),

z ∈ ρ(A),

is called the Weyl function (or the m-function) of A.

418

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

The Weyl function of A is a Herglotz-Nevanlinna function holomorphic on C \ [−||A||, ||A||]. It has the Taylor expansion m(z) = −

∞ #

(An−1 χ, χ)z −n,

|z| > ||A||.

n=1

In particular lim zm(z) = −1.

z→∞

Let E(t) be a resolution of identity for A. Then the Weyl function m(z) admits the integral representation dσ(t) m(z) = , t−z R

where σ(t) = (E(t)χ, χ) and dσ is a probability measure with a compact support supp(dσ) ⊆ [−||A||, ||A||]. It follows from the deﬁnition of the Weyl function and the Hilbert identity that ¯ m(z) − m(ζ) = ((A − zI)−1 χ, (A − ζI)−1 χ), z − ζ¯

z, ζ ∈ ρ(A).

(11.8)

Theorem 11.2.2. Let A1 and A2 be two bounded self-adjoint operators in Hilbert spaces H1 and H2 , respectively. Let χ1 and χ2 be normalized cyclic vectors for A1 and A2 . If the corresponding Weyl functions m1 (z) and m2 (z) coincide in a neighborhood of inﬁnity, then there exists a unitary operator U ∈ [H1 , H2 ] such that U A1 = A2 U and U χ1 = χ2 . Proof. Let m1 (z) = m2 (z) for all z in a neighborhood G of inﬁnity. From (11.8) we get that ((A1 − zI)−1 χ1 , (A1 − ζI)−1 χ1 )H1 = ((A2 − zI)−1 χ2 , (A2 − ζI)−1 χ2 )H2 for z, ζ ∈ G. Since χ1 and χ2 are cyclic vectors for A1 and A2 , respectively, we get the relations c.l.s.{(A1 − zI)−1 χ1 , z ∈ G} = H1 ,

c.l.s.{(A2 − zI)−1 χ2 , z ∈ G} = H2 .

L1 = span{(A1 − zI)−1 χ1 , z ∈ G},

L2 = span{(A2 − zI)−1 χ2 , z ∈ G}.

Let

Deﬁne on L1 a linear operator U as # # U ck (A1 − zk I)−1 χ1 = ck (A2 − zk I)−1 χ2 .

11.2. The Stone theorem and its generalizations

419

Then U L1 = L2 and U is an isometry. Because L1 and L2 are dense in H1 and H2 , respectively, the operator U has unitary continuation on H1 . Since zU (A1 − zI)−1 χ1 = z(A2 − zI)−1 χ2 , for all z ∈ G, we get U χ1 = −U ( lim z(A1 − zI)−1 χ1 ) = − lim z(A2 − zI)−1 χ2 = χ2 . z→∞

z→∞

Furthermore, ) ) U A) ck (A1 − zk I)−1 χ1 = U ck) χ1 + zk (A1 − zk I)−1χ1 1 = ckχ I)−1 χ2 = A2 ck (A2 − zk I)−1 χ2 )2 + zk (A2 − zk−1 = A2 U ck (A1 − zk I) χ1 .

Hence, U A1 = A2 U.

Deﬁnition 11.2.3. A Herglotz-Nevanliina function m(z) belongs to the class N0 if m(z) is holomorphic in the neighborhood of inﬁnity and lim zm(z) = −1.

z→∞

A function m(z) ∈ N0 has the integral representation dσ(t) m(z) = , t−z R

where dσ is a probability measure with compact support. The asymptotic expansion m(z) = −z −1 − b1 z −2 − (b21 + a21 )z −3 + o(z −3 ), z → ∞, holds, where

b1 =

a21 + b21 =

tdσ(t), R

z→∞

t2 dσ(t).

R

We can recover the numbers b1 and b1 = − lim z 2 (m(z) + z −1 ),

a21

from the function m(z),

a21 = − lim z 3 (m(z) + z −1 + b1 z −2 ) − b21 . z→∞

Notice that if supp(dσ) ⊂ [−C, C], then |b1 | ≤ C and |a1 | ≤ C. In the following we assume that a1 > 0. Deﬁne the function m1 (z) via the formula m1 (z) =

1 b1 − z − m−1 (z) . a21

Clearly, −m−1 (z) is a Herglotz-Nevanlinna function and −m−1 (z) = z − b1 − a21 z −1 + o(z −1 ),

z → ∞.

420

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

If supp(dσ) ⊂ [−C, C], then m(z) > 0 for z < −C and m(z) < 0 for z > C. Hence −m−1 (z) is holomorphic on C \ [−C, C]. It follows that m1 is Herglotz-Nevanlinna function, holomorphic on C \ [−C, C] and limz→∞ zm1 (z) = −1. Thus, m1 belongs to the class N0 . Repeating such a procedure, we obtain ﬁnite or inﬁnite sequences of real numbers {bk }k≥1 and positive numbers {ak }k≥1 . Moreover, |bk | ≤ C and |ak | ≤ C for all k. The process terminates on the N -th step if and only if aN = 0, i.e., m(z) is a rational function of the form m(z) =

N #

ck , μ −z n=1 k

where μ1 , .) . . , μN are distinct real numbers and c1 , . . . , cN are positive numbers N such that n=1 ck = 1. Thus we have obtained the following representation of m(z) by means of the continued fraction −1

m(z) =

−a21

z − b1 + z − b2 +

.

(11.9)

−a22

z − b3 + . . . +

−a2n−1 z − bn + . . .

Using the sequences {bk } and {ak }, we construct a Jacobi matrix J of the form (11.1) or (11.2). Theorem 11.2.4. Let m(z) ∈ N0 and let {bk } and {ak } be corresponding parameters of m(z). Then the Weyl function mJ (z) := ((J − zI)−1 δ1 , δ1 ), where J is a self-adjoint Jacobi matrix of the form (11.1) or (11.2), coincides with m(z). Proof. Recall that the Schur-Frobenius formula −1 PH1 (A − zI)−1 H1 = −zI + A11 − A12 (A22 − zI)−1 A21 , written for

A=

A11 A21

A12 A22

(11.10)

H1 H1 : ⊕ → ⊕ H2 H2

holds for z ∈ ρ(A) ∩ ρ(A22 ). Here PH1 is the orthogonal projection in H onto H1 . Applying (11.10) to J gives 1 b1 − z − m−1 J (z) = mJ1 (z), 2 a1

11.2. The Stone theorem and its generalizations

421

where mJ1 (z) is the Weyl function of the Jacobi matrix J1 obtained from J by crossing out the ﬁrst row and the ﬁrst column. It follows that the parameters corresponding to mJ (z) are the sequences {bk } and {ak }. Hence, the continued fraction expansion of mJ (z) is the same as for m(z) (see (11.9)). Therefore, mJ (z) = m(z). Theorem 11.2.2 and Theorem 11.2.4 yield the Stone theorem. Theorem 11.2.5. Let A be a bounded self-adjoint operator with simple spectrum. Then A is unitarily equivalent to the self-adjoint operator given by a Jacobi matrix of the form (11.1) or (11.2). The following is a generalization of the Stone Theorem. Theorem 11.2.6. Let H be separable Hilbert space and T be a bounded prime dissipative operator in H with a rank-one imaginary. Then T is unitarily equivalent to the operator given by a Jacobi matrix of the form (11.1) or (11.2). Proof. Let g ∈ H be such that 2 Im T h = (h, g)g, h ∈ H. According to Theorem 5.1.2, there is the Liv˘sic canonical system T K 1 Θ= , H C √ where Kc = c(g/ 2), c ∈ C. Let WΘ (z) be the transfer function of the system Θ of the form (5.17). Then 1 WΘ (z) = 1 − i (T − zI)−1 g, g , z ∈ ρ(T ). It follows that

lim z(WΘ (z) − 1) = i||g||2 .

z→∞

Let c = g2 . Then VΘ (z) = i

WΘ (z) − 1 1 = (Re T − zI)−1 g, g , WΘ (z) + 1 2

We set m(z) =

2 VΘ (z). g2

Then m(z) is the Weyl function of Re T . Hence there exists a unique Jacobi matrix ⎛ ⎞ b a1 0 0 0 · · · ⎜a1 b2 a2 0 0 · · ·⎟ ⎟ H=⎜ ⎝ 0 a2 b3 a3 0 · · ·⎠ , · · · · · · · · 1 When we deal with the Liv˘ sic canonical system whose input-output space is C, the transfer and impedance operator-functions WΘ (z) and VΘ (z) applied to an element c ∈ C can be considered as scalar functions multiplied by a scalar c, i.e., WΘ (z)c = WΘ (z) · c and VΘ (z)c = VΘ (z) · c.

422

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

with real entries b, b2 , b3 , . . . and positive entries a1 , a2 , . . . such that m(z) = (H − zI)−1 δ1 , δ1 . Note that (11.3) holds because H deﬁnes a bounded operator in l2 (N). Moreover, the entries of H can be found by means of the continued fraction expansion −1

m(z) =

−a21

z−b+ z − b2 +

. −a22

z − b3 + . . . + Let

⎛ b + i||g||2 /2 ⎜ a1 J =⎜ ⎝ 0 · ⎛ ||g||2 /2 ⎜ 0 = H + i⎜ ⎝ 0 ·

a1 b2 a2 · 0 0 0 ·

0 a2 b3 ·

0 0 a3 ·

0 0 0 0 0 0 · ·

0 0 0 · · · · ·

· · · ·

−a2n−1 z − bn + . . . · · · ·

· · · · ⎞ · ·⎟ ⎟. ·⎠ ·

⎞ · ·⎟ ⎟ ·⎠ ·

(11.11)

Suppose ΘJ is the Liv˘sic canonical system constructed via Theorem 5.1.2 with J as the state-space operator. Then the transfer function of ΘJ is WΘJ (z) = 1 − i||g||2 (J − zI)−1 δ1 , δ1 . Since Re J = H and m(z) = (H − zI)−1 δ1 , δ1 , we get that WΘJ (z) = WΘ (z). Because the matrix J is prime, we can apply Theorem 5.4.3 and conclude that the operator T is unitarily equivalent to J . An application of the Schur-Frobenius formula (11.10) to J given by (11.11) shows that the entries of J can be also found using the continued fraction expansion −1 M (z) = , −a21 z − b1 + −a22 z − b2 + −a2n−1 z − b3 + . . . + z − bn + . . . where M (z) = ((J − zI)−1 δ1 , δ1 ) = and β = lim (iz(1 − W (z))) . z→∞

i (W (z) − 1), β

11.2. The Stone theorem and its generalizations

423

Theorem 11.2.7. Any Herglotz-Nevanlinna function of the form b 1 V (z) = dσ(t), a t−z such that σ(t) is a non-negative, non-decreasing function on ﬁnite interval [a, b], can be realized in the form V (z) = i

WΔ (z) − 1 , WΔ (z) + 1

where WΔ (z) is the transfer function of the Liv˘sic canonical system in Jacobi form (11.6), (11.7). Proof. Consider an auxiliary system Θ of the form (5.37) constructed in Theorem 5.5.1, A K 1 Θ= , (11.12) L2 ([a, b], dσ) C where the state-space operator A is given by (Af )(t) = tf (t) + 2i

b

f (t)dσ(t),

(11.13)

a

and Kc = ch, c ∈ C,

K ∗ f = (f, h)h, f ∈ L2 ([a, b], dσ),

h = h(t) = 1.

Applying Theorem 5.5.1 we have VΘ (z) = K ∗ (Re A − zI)−1 K = V (z), and V (z) = i

WΘ (z) − 1 WΘ (z) + 1

(11.14)

where WΘ (z) is a transfer function of an auxiliary system Θ of the form (11.12). The system in Jacobi form (11.6), (11.7) is a minimal system since the vector δ1 is a cyclic vector (see (5.8)) for Jacobi matrix J (11.1), (11.2) and J is a prime operator. It is established in Theorem 11.2.6 that any bounded, dissipative, prime operator A with a rank-one imaginary part is unitarily equivalent to a Jacobi matrix of the form (11.1), (11.2). The auxiliary model operator A (11.13) is prime, since vector h = h(t) = 1 is a cyclic vector for (Re Af )(t) = tf (t) in L2 ([a, b], dσ) and therefore this vector is cyclic for A. By the above mentioned theorem, auxiliary model operator A is unitarily equivalent to the Jacobi matrix J . Therefore, there exists a unitary operator U from L2 ([a, b], dσ) onto Cn (respectively l2 (N)) such that UAU −1 = J ,

424

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

√ Obviously, (Uh)(t) = eiφ g for h = h(t) = 1, g = Im b1 δ1 , and thus UKc = Kφ c = ceiφ g, c ∈ C. Considering a new unitary operator U = e−iφ U we get that U AU −1 = J and U h = g. Hence systems Θ and Δ of the form J K 1 Δ= , (11.15) Cn C or of the form (respectively with semi-inﬁnite Jacobi matrix as a state-space operator) J K 1 Δ= , l2 (N) C are unitarily equivalent and have the same transfer functions, i.e., WΔ (z) = WΘ (z). The representation (11.14) completes the proof of the theorem.

11.3 Inverse spectral problems for ﬁnite dissipative Jacobi matrices Let J be a dissipative (n × n) Jacobi matrix of the form (11.1) or (11.2). Then the corresponding operator in Cn is a prime dissipative operator with rank-one imaginary part. Therefore, the matrix J has only non-real eigenvalues with positive imaginary parts. The next theorem establishes that the n arbitrary non-real numbers, counting algebraic multiplicity taken from the open upper half-plane, determine uniquely some dissipative (n × n) Jacobi matrix with rank-one imaginary part. Theorem 11.3.1. Suppose that z1 , . . . , zn are not necessarily distinct complex numbers with positive imaginary parts. Then there exists a unique (n × n) dissipative Jacobi matrix whose eigenvalues (counting algebraic multiplicity) coincide with {zk }nk=1 . Proof. Let W (z) =

n * z − z¯k . z − zk

k=1

Then lim z(W (z) − 1) = 2i

z→∞

Let c =

)n k=1

n #

Im zk .

k=1

Im zk and deﬁne m(z) =

i W (z) − 1 . c W (z) + 1

(11.16)

11.3. Inverse spectral problems for ﬁnite dissipative Jacobi matrices

425

The Herglotz-Nevanlinna function m(z) has the expansion in the neighborhood of inﬁnity 1 b b2 + a21 1 m(z) ∼ − − 2 − +O , z z z3 z4 and determines a probability measure supported at n points. By Theorem 11.2.4 there exists a unique self-adjoint (n × n) Jacobi matrix ⎛ ⎞ b a1 0 0 · · · ⎜ a1 b 2 a2 0 · · · ⎟ ⎜ ⎟ ⎜ 0 a2 b3 a3 · · · ⎟ ⎟, H =⎜ ⎜· · · · · · · ⎟ ⎜ ⎟ ⎝· · · · · · an−1 ⎠ · · · · 0 an−1 bn such that m(z) = (H − zI)−1 δ1 , δ1 . Let ⎛ b + ic a1 ⎜ a1 b2 ⎜ ⎜ 0 a 2 J =⎜ ⎜ · · ⎜ ⎝ · · · ·

0 a2 b3 · · ·

0 0 a3 · · ·

· · · · · · · · · · 0 an−1

· · · ·

⎞

⎛

c 0 ⎟ ⎜0 0 ⎟ ⎜ ⎟ ⎜ ⎟ = H + i ⎜0 0 ⎟ ⎜· · ⎟ ⎜ ⎝· · an−1 ⎠ bn · ·

0 0 0 0 0 0 · · · · · ·

· · · · · · · · · · 0 0

⎞ · ·⎟ ⎟ ·⎟ ⎟. ·⎟ ⎟ ·⎠ 0

Then (Im J ) x = (x, g) g,

x ∈ Cn ,

√ where g = c δ1 ∈ Cn . By Theorem 5.1.2, operator J can be included in the system √ Δ of the form (11.6) with the channel operator K deﬁned by (11.5) using g = c δ1 . Then the transfer function WΔ (z) has a form WΔ (z) = 1 − 2i c (J − zI)−1 δ1 , δ1 , while the impedance function is VΔ (z) = i

WΔ (z) − 1 = c (H − zI)−1 δ1 , δ1 = c m(z). WΔ (z) + 1

From (11.16) we get VΔ (z) = i

W (z) − 1 . W (z) + 1

Therefore, WΔ (z) = W (z). Hence, the eigenvalues of J coincide with {zk }, counting algebraic multiplicity. Example. Let us construct a (3×3) dissipative Jacobi matrix with eigenvalues z1 = i of multiplicity 2 and z2 = 2i (of multiplicity 1). Then the transfer function WΔ (z)

426

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

of the system Δ of the form (11.6) with J as a state-space operator (constructed in the above proof) is 2 z+i z + 2i WΔ (z) = . z−i z − 2i Then lim z(WΔ (z) − 1) = 4i. Note that 2Im z1 + Im z2 = 4. Let z→∞

m(z) =

i WΔ (z) − 1 i (z + i)2 (z + 2i) − (z − i)2 (z − 2i) −2z 2 + 1 = = . 4 WΔ (z) + 1 4 (z + i)2 (z + 2i) + (z − i)2 (z − 2i) 2z 3 − 10z

Then

−1 . 9/2 z− 1/2 z− z + + It follows that b1 = 4i, b2 = b3 = 0, a1 = 9/2, a2 = 1/2 and the Jacobi matrix J takes the form ⎛ ⎞ 4i √32 0 ⎜ √1 ⎟ 0 J = ⎝ √32 . 2⎠ 1 √ 0 0 2 m(z) =

Example. In order to construct a dissipative (n × n) Jacobi matrix with eigenvalue z0 = x0 + iy0 of algebraic multiplicity n (y0 > 0), it is suﬃcient to construct an (n × n) Jacobi matrix Jn with eigenvalue z0 = i of algebraic multiplicity n. The transfer function WΔ (z) of the system Δ of the form (11.6) with Jn as a statespace operator (constructed in the proof of Theorem 11.3.1) is n z+i WΔ (z) = . z−i The corresponding Weyl function of the real part of Jn is mn (z) =

i (z + i)n − (z − i)n . n (z + i)n + (z − i)n

Because traceJn = in, we get Im b1 = n. Expanding the function mn (z) via continued fractions we obtain the real part of Jn . In particular, m2 (z) =

m4 (z) =

−1 , 1 z− z

m3 (z) =

−1 , 8/3 z− 1/3 z− z

−1 , −5 z+ − 4/5 z+ − 1/5 z+ z

m5 (z) =

−1 . −8 z+ − 7/5 z+ − 16/35 z+ − 1/7 z+ z

11.4. Reconstruction of a dissipative Jacobi matrix from its triangular form 427 Therefore, 2i 1 J2 = , 1 0 √ ⎛ 4i 5 √ ⎜ 5 0 ⎜ J4 = ⎜ √2 ⎝ 0 5 0 0

0 √2 5

0

√1 5

⎞

0 0⎟ ⎟ ⎟, √1 ⎠ 5 0

⎛

3i

⎜ √ J3 = ⎝ 2√ 2 3 0 ⎛ 5i ⎜ 2√2 ⎜√ ⎜ 5 J5 = ⎜ ⎜ 0 ⎜ ⎝ 0 0

√ 2√ 2 3

0 √1 3 √ 2√ 2 5

0

√ √7 5

0 0

0

⎞

⎟ √1 ⎠ , 3 0

0

√ √7 5

0 √4 35

0

0

0

0

0

√4 35

0

√1 7

0 √1 7

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

0

Note that the eigenvalues of the real part of Jn are solutions of the equations n z+i = −1, z−i and given by λk = − cot π+2πk , k = 0, 1 . . . , n − 1. 2n

11.4 Reconstruction of a dissipative Jacobi matrix from its triangular form Let z1 , z2 , . . . , zn be (not necessarily distinct) complex numbers with positive imaginary parts. According to Theorem 11.3.1 there exists a unique dissipative (n × n) Jacobi matrix with one-dimensional imaginary part, ⎛ ⎞ b1 a 1 0 0 · · · ⎜a1 b2 a2 0 · · · ⎟ ⎜ ⎟ ⎜ 0 a 2 b3 a 3 · · · ⎟ ⎜ ⎟, J =⎜ · · · · · · ⎟ ⎜· ⎟ ⎝· · · · · · an−1 ⎠ · · · · 0 an−1 bn whose eigenvalues coincide with {zk }nk=1 counting algebraic multiplicity. On the other hand, by Theorem 5.6.4, such a matrix is unitarily equivalent to the triangular matrix of the form ⎛ ⎞ z1 iβ1 β2 · · · iβ1 βn ⎜0 z2 · · · iβ2 βn ⎟ ⎟, J = ⎜ (11.17) ⎝· · · · · · ⎠ 0 0 · · · zn where zk = αk + i

βk2 , 2

Im zk =

βk2 , 2

βk > 0,

k = 1, . . . , n.

(11.18)

428

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

It follows from (11.17) that ⎛

β12 ⎜ β1 β2 2 Im J = ⎜ ⎝ · β1 βn x = (x, g ) g, and that 2 Im J

β1 β2 β22 · β2 βn

· · · ·

· · · ·

· · · ·

⎞ β1 βn β2 βn ⎟ ⎟, · ⎠ βn2

x ∈ Cn , where ⎛ ⎞ β1 ⎜ β2 ⎟ ⎜ ⎟ ⎜ · ⎟ ⎟ g = ⎜ ⎜ · ⎟. ⎜ ⎟ ⎝ · ⎠ βn

(11.19)

Let U be a unitary matrix ⎛

u11 ⎜ u21 U =⎜ ⎝ · un1

· · · ·

u12 u22 · un2

· · · ·

⎞ · u1n · u2n ⎟ ⎟, · · ⎠ · unn

such that U J = J U,

√

U g = g,

(11.20)

where g = 2Im b1 δ1 . Next we present an algorithm which allows us to ﬁnd the Jacobi matrix J and the unitary matrix U satisfying (11.20). It follows from (11.20) that kg , k = 1, 2, . . . , n. U J k g = (J) (11.21) Since U is a unitary matrix, we have g2 = g2 , and taking into account (11.5) and (11.19) we obtain g2 = 2Im b1 = β12 + β22 + . . . + βn2 = g2 . Thus Im b1 =

n #

Im zk .

(11.22)

k = 1, 2, . . . , n.

(11.23)

k=1

Since U g = g , we have √ Im zk uk1 = , n ) Im zj j=1

11.4. Reconstruction of a dissipative Jacobi matrix from its triangular form 429 Relations (11.20) and (11.21) yield g , g = (U J g, U g) = (J g, g) . J Taking into account (11.1), (11.5), (11.17) and (11.19), we get ⎞ ⎛ n ⎛ √ ⎞ ) 2 βk β1 ⎟ b1 √2 Im b1 ⎜ z1 + i k=2 ⎜a1 2 Im b1 ⎟ ⎜ ⎟ n ⎜ ⎟ ⎜ ⎟ ) 2 ⎜ ⎟ ⎜ βk β2 ⎟ 0 ⎜ ⎟ ⎜ z2 + i ⎟ k=3 ⎟ , J g = ⎜ ⎟. Jg = ⎜ · ⎜ ⎟ ⎜ ⎟ · ⎜ ⎟ ⎜ ⎟ · ⎜ ⎟ ⎜ ⎟ · ⎜ ⎟ ⎝ ⎠ · ⎝ ⎠ · 0 zn βn

(11.24)

Therefore, (J g, g) = 2 b1 Im b1 and

g, g = J

n #

⎛ ⎝zj + i

j=1

n #

⎞ βk2 ⎠ βj2 .

k=j+1

g, g , we get from (11.18) and (11.22) that Since (J g, g) = J n )

b1 =

n )

zj + i

j=1

k=j+1 n )

βk2

n )

Im zj ,

Re b1 =

Im zk

k=1

k=1

Re zk Im zk n )

. Im zk

k=1

( (2 g, we get J g2 = ( g ( Since U is unitary and U J g = J (J ( . This equality and (11.24) yield , ,2 , n , n # # , , 2 2, ,zj + i |b1 |2 (2 Im b1 ) + a21 (2 Im b1 ) = β k , βj , , , j=1 , k=j+1 and hence ⎛

, ,2 , n , n ) ) , , 2 ⎜ z + i β , j k , Im zj ⎜ j=1 , , k=j+1 ⎜ a1 = ⎜ − n ) ⎜ Im zk ⎝ k=1

, ,2 ⎞1/2 ,) , n n ) , , zj + i βk2 Im zj , ⎟ , ,j=1 , ⎟ k=j+1 ⎟ ⎟ . n 2 ⎟ ) ⎠ Im zk k=1

430

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

Recall that by (11.4), (J m−1 g)m = am−1 am−2 · · · a1

+

2 Im b1 ,

and (J m−1 g)m+1 = (J m−1 g)m+2 = · · · = (J m−1 g)n = 0, for m = 1, . . . , n. Let m ≥ 2. Suppose that (J m−1 g)k are already known, where k = 1, 2 . . . , m − 1. Then from (11.1) we obtain ⎛ ⎞ b (J m−1 g) +a (J m−1 g) 1 m−1

1 1 m−1

2 m−1

⎜ a1 (J m−1 g)1 +b2 (J m−1 g)2 +a2 (J m−1 g)3 ⎟ g)2 +b3 (J g)3 +a3 (J g)4 ⎟ ⎜ a2 (J ⎜ ⎟ · ⎜ ⎟ · ⎜ ⎟ · J m g = ⎜ am−1 (J m−1 g)m−1 +bm (J m−1 g)m ⎟ . ⎜ ⎟ ⎜ am (J m−1 g)m ⎟ ⎜ ⎟ 0 ⎝ ⎠ · · · 0

Consequently

J g, J m

m−1

g =

n #

(J m g)j (J m−1 g)j

j=1 m−1 #

=

(J m g)j (J m−1 g)j + am−1 (J m−1 g)m−1 + bm (J m−1 g)m (J m−1 g)m .

j=1

From

m g, (J)m−1g , J m g, J m−1 g = (J) √ = am−1 am−2 · · · a1 2 Im b1 we get a linear equation with respect

and (J m−1 g)m to bm : m−1 #

(J m g)j (J m−1 g)j + am−1 (J m−1 g)m−1 + bm (J m−1 g)m (J m−1 g)m

j=1

m−1g . = (J)mg , (J)

This equation can be solved since the coeﬃcient (J m−1 g)m is non-zero by (11.4). ( (2 ( ( 2 In order to ﬁnd am we use the relation J m g = ((J)mg ( which takes the form m−1 # j=1

, ,2 |(J m g)j |2 + ,am−1 (J m−1 g)m−1 + bm (J m−1 g)m , ( (2 ( ( + a2m (J m−1 g)2m = ((J)mg( .

11.5. System Interpolation and Sectorial Operators

431

Solving for a2m we can ﬁnd am . According to (11.23), the elements {uk1 }nk=1 are expressed by means of g gives the following linear system with rez1 , z2 , . . . , zn . The equality U J g = J spect to the second column {uk2 }nk=1 of the matrix U : ⎧ g)1 , u11 (J g)1 + u12 (J g)2 = (J ⎪ ⎪ ⎪ ⎪ ⎨ u (J g) + u (J g) = (J g)2 , 21 1 22 2 ⎪ ... ... ⎪ ⎪ ⎪ ⎩ g )n . un1 (J g)1 + un2 (J g)2 = (J

√ Since (J g)2 = a1 2 Im b1 = 0, one can ﬁnd {uk2 }nk=1 . By induction, the equality m−1g enables us to ﬁnd {uk m }n for m ≤ n. Finally, the relation U J m−1 g = (J) k=1 n−1 −1 J = U JU , and the already known entries {ak }n−1 k=1 and {bk }k=1 allow us to ﬁnd the entry bn . In conclusion, we provide formulas for the reconstruction of a (2 × 2) dissipative Jacobi matrix J with a rank-one imaginary part from its eigenvalues z1 , z2 , and the corresponding unitary matrix U : √ ⎛ Re z1 Im z1 +Re z2 Im z2 ⎞ + i(Im z1 + Im z2 ) ω Im z1 Im z2 Im z1 +Im z2 ⎠, J =⎝ √ Re z1 Im z2 +Re z2 Im z1 ω Im z1 Im z2 Im z1 +Im z2 2 Re z1 −Re z2 where ω = + 1 and Im z1 +Im z2 ⎛@ U = ⎝@

Im z1 Im z1 +Im z2 Im z2 Im z1 +Im z2

@ −

Im z2 Im z1 +Im z2

@

Im z1 Im z1 +Im z2

⎞ ⎠.

The presented algorithm of reconstruction of the unique dissipative nonself-adjoint Jacobi matrix, with a rank-one imaginary part having given non-real numbers as its eigenvalues, allows us to recover the set of tri-diagonal matrices with the same non-real eigenvalues.

11.5 System Interpolation and Sectorial Operators We begin this section with the following deﬁnition of interpolation systems. Deﬁnition 11.5.1. Let Θ be the Liv˘sic canonical system of the form (5.6) and VΘ (z) be its impedance function deﬁned by (5.28),(5.30). Then Θ is called an interpolation system (solution) in the Nevanlinna-Pick interpolation problem for the data {z ∈ C+ , = 1, . . . m} and {v ∈ [E, E], Im v ≥ 0, = 1, . . . m} if VΘ (z ) = v ,

= 1, . . . m.

(11.25)

432

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

We will be interested in constructing, when possible, the Liv˘sic canonical scattering system with given interpolation data whose state-space operator is αsectorial. We will also focus on systems with the state-space H of minimal dimension. The following uniqueness result explains the importance of this last requirement. Theorem 11.5.2. Let {z , = 1, . . . m} and {v , = 1, . . . m} be the interpolation data (as in Deﬁnition 11.5.1). Let Θ1 and Θ2 be two minimal scattering interpolation systems for this data such that T1 K 1 I T2 K2 I Θ1 = , Θ2 = , H1 E H2 E dim(E) < ∞, H1 and H2 are ﬁnite dimensional (of dimensions n1 and n2 respectively), and ker(K1 ) = ker(K2 ) = {0}. Assume further that c. l. s.{Ran((T1 − z I)−1 K1 ), −1

c. l. s.{Ran((T2 − z I)

K2 ),

= 1, . . . , m} = H1 , = 1, . . . , m} = H2 ,

(11.26)

and that |z | > max (T1 , T2 ),

= 1, . . . m.

(11.27)

Then, the two systems are unitarily equivalent, n1 = n2 , and VΘ1 (z) = VΘ2 (z),

z ∈ ρ(T1 ) ∩ ρ(T2 ).

Proof. Condition (11.27) forces the points z to be in the resolvent sets of the operators T1 and T2 . Therefore the interpolation conditions VΘ1 (z ) = VΘ2 (z ) for = 1, . . . m can be rewritten as K1∗ (T1 − z I)−1 K1 = K2∗ (T2 − z I)−1 K2 ,

= 1, . . . m,

(11.28)

that is, WΘ1 (z ) = WΘ2 (z ), = 1, . . . m. Moreover, we claim that ((T1 − z I)−1 K1 φ, (T1 − zj I)−1 K1 φ)H1 = ((T2 − z I)−1 K2 φ, (T2 − zj I)−1 K2 φ)H2 (11.29) for any φ ∈ E and all , j = 1, . . . m. Indeed, taking into account that Im T1 = K1 K1∗ ,

Im T2 = K2 K2∗ ,

we have that (T1 − z I)−1 − (T1∗ − z¯j I)−1 = (T1∗ − z¯j I)−1 ((T1∗ − z¯j I) − (T1 − z I)) (T1 − z )−1 = −2i(T1∗ − z¯j )−1 K1 K1∗ (T1 − z I)−1 + (z − z¯j )(T1∗ − z¯j )−1 (T1 − z I)−1 , (11.30)

11.5. System Interpolation and Sectorial Operators

433

and the same relation holds for T2 . Equations (11.28) and (11.30) lead to K1∗ (T1∗ − z¯j I)−1 (T1 − z )−1 K1 = K2∗ (T2∗ − z¯j I)−1 (T2 − z )−1 K2 ,

(11.31)

and hence to (11.29). Due to (11.31), we can deﬁne an operator U via U (T1 − z I)−1 K1 φ = (T2 − z I)−1 K2 φ,

(φ ∈ E,

= 1, . . . , m).

(11.32)

This operator is isometric and in fact unitary, because of the range conditions (11.26). Rewriting (11.28) as K1∗ U ∗ U (T1 − zj I)−1 K1 φ = K2∗ (T2 − zj I)−1 K2 φ, and taking into account the deﬁnition of U , we obtain that K1∗ U ∗ U (T1 − zj I)−1 K1 φ = K2∗ (T2 − zj I)−1 K2 φ. The range conditions (11.26) then leads to K1∗ U ∗ = K2∗ . We now show that U T1 = T2 U . Condition (11.27) allows us to write the power expansions (T1 − zj I)−1 = −

I T1 T2 − 2 − 13 − · · · , zj zj zj

(T2 − zj I)−1 = −

I T2 T2 − 2 − 23 − · · · , zj zj zj

and

for j = 1, . . . m. The deﬁnition (11.32) of U gives us I T1 T12 I T2 T22 U − − 2 − 3 − · · · K1 = − − 2 − 3 − · · · K2 , zj zj zj zj zj zj and therefore U K1 − +U zj

T1 T2 − 2 − 13 − · · · zj zj

K2 K1 = − + zj

T2 T2 − 2 − 23 − · · · zj zj

K2 .

Since K2 = U K1 , we obtain T1 I T1 T12 T2 I T2 T22 U − − 2 − 3 − · · · K1 = − − 2 − 3 − · · · K2 , zj zj zj zj zj zj zj zj so that U T1 (T1 −zj I)−1 K1 = T2 (T2 −zj I)−1 K2 , for j = 1, · · · m. This last equation can be rewritten as U T1 U −1 U (T1 − zj I)−1 K1 = T2 (T2 − zj I)−1 K2 ,

j = 1, · · · m.

The range conditions (11.26) and the deﬁnition of U lead then to U T1 U −1 = T2 , which concludes the proof.

434

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

Now we focus on bounded α-sectorial operators whose deﬁnition was given in Section 9.5. It follows from (9.104) that when the bounded operator T is αsectorial, the exact value of the angle α can be computed via the formula tan α =

|Im (T ξ, ξ)| . ξ∈Dom(T ) Re (T ξ, ξ) sup

(11.33)

When the state space is ﬁnite dimensional, the angle α is given by another formula:

Theorem 11.5.3. Let Θ=

T K H

I E

,

be the Liv˘sic canonical system with accretive state-space operator T (with both H and E ﬁnite dimensional) and invertible channel operator K. Then T is α-sectorial if and only if the limit VΘ (−0) =

lim

x∈R− ,x→0

VΘ (x)

is ﬁnite. In this case, the angle α is given by the formula tan α = VΘ (−0). Proof. Assume that the operator T is α–sectorial. It follows from (11.33) that tan α =

((Im T )ξ, ξ) ξ∈Ran(Re T ) ((Re T )ξ, ξ) sup

=

((Im T )(Re T )−1/2 f, (Re T )−1/2 f ) ||f ||2 f ∈Ran(Re T )

=

||K ∗ (Re T )−1/2 f ||2 ||f ||2 f ∈Ran(Re T )

sup sup

= ||K ∗ (Re T )−1/2 ||2 = ||(Re T )−1/2 K||2 . On the other hand (VΘ (−0)g, g)E = lim((Re T − xI)−1 Kg, Kg) = ||(Re T )−1/2 Kg||2 , x↑0

g ∈ E.

Hence, ||VΘ (−0)|| = ||(Re T )−1/2 K||2 = tan α. If ||VΘ (−0)|| < ∞, then Ran(Im T ) = Ran(K) ⊆ Ran(Re T ). Therefore T is an α-sectorial operator and, as before, tan α = ||VΘ (−0)||. Obviously, the operator T is accretive but not α-sectorial for any α ∈ (0, π/2) if and only if lim VΘ (x) = ∞. x→0−

11.5. System Interpolation and Sectorial Operators

Theorem 11.5.4. Let Θ=

T K H

I E

435

,

be the Liv˘sic canonical system with a bounded state-space operator T . If the operator T is α-sectorial, then for every set of non-real points z1 , . . . , zp ∈ C+ and every set of vectors h1 , . . . , hp ∈ E, the following inequalities are valid: p # Vθ (zk ) − Vθ∗ (z ) hk , h ≥ 0, (11.34) zk − z¯ E k,=1

p p # # zk Vθ (zk ) − z¯ Vθ∗ (z ) hk , h ≥ (cot α) (Vθ∗ (z )Vθ (zk )hk , h )E . zk − z¯ E

k,=1

k,=1

(11.35) Proof. Writing Vθ (z) = K ∗ (Re T − zI)−1 K we obtain Vθ (zk ) − Vθ∗ (z ) zk − z¯

∗ 1 K (Re T − zk I)−1 K − K ∗ (Re T − z¯ I)−1 K zk − z¯ K ∗ ((Re T − z I)−1 )∗ (Re T − zk I)−1 K.

= =

We set

ξk = (Re T − zk I)−1 Khk ,

and ξ=

n #

(k = 1, 2, . . . , n),

ξk .

(11.37)

k=1

Then n #

k,=1

=

=

Vθ (zk ) − Vθ∗ (z ) h k , h zk − z¯ E n # k,=1 n #

(K ∗ (Re T − z¯ I)−1 (Re T − zk I)−1 Khk , h )E (ξk , ξ )H = (ξ, ξ)H ≥ 0.

k,=1

Since the operator T is α–sectorial, then cot α · |(Im T ξ, ξ)| ≤ (Re T ξ, ξ). It follows from (11.36) and (11.37) that n # k,=1

(Re T ξk , ξ ) ≥ cot α ·

(11.36)

# (Im T ξk , ξ ). k,

436

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

Since Im T = KK ∗ , we get # (Re T (Re T − zk I)−1 Khk , (Re T − z I)−1 Kh ) k,=1

≥ cot α ·

# (KK ∗ (Re T − zk I)−1 Khk , (Re T − z I)−1 Kh ), k,

which leads to # (K ∗ (Re T − z¯ I)−1 Re T (Re T − zk I)−1 Khk , h )E k,=1

≥ cot α ·

#

(K ∗ (Re T − z¯ I)−1 KK ∗ (Re T − zk I)−1 Khk , h )E .

k,=1

Since K ∗ (Re T − z¯ I)−1 Re T (Re T − zk I)−1 K = K∗

z¯ (Re T − z¯ I)−1 − zk K ∗ (Re T − zk I)−1 K, z¯ − zk

we obtain (11.35).

11.6 The Liv˘sic interpolation systems in the Pick form In this section we consider the Nevanlinna-Pick interpolation problem in the class N of scalar Herglotz-Nevanlinna functions. We set vk − v¯j zk vk − z¯j v¯j P= , Q= . (11.38) zk − z¯j k,j=1,...m zk − z¯j k,j=1,...m Theorem 11.6.1. Let {zk , k = 1, . . . , m} ∈ C+ and {vk , k = 1, . . . , m} ∈ C+ be system interpolation data for which the matrix P in (11.38) is strictly positive.2 Then there exists a scattering interpolation system Θ with the state space of dimension m that is a solution of the interpolation problem for this data. The state-space operator of Θ is α-sectorial if and only if zk vk − z¯j v¯j ≥ cot α (vk v¯j )k,j=1,...m . (11.39) zk − z¯j k,j=1,...m Proof. Consider the space Cm with the inner product (ξ, η)Cm = η ∗ Pξ. P 2A

matrix is called strictly positive if it is non-negative and invertible.

11.6. The Liv˘sic interpolation systems in the Pick form

437

= P−1 Q is self–adjoint with respect to this It is easily seen that the operator A inner product. Set ⎛ ⎞ v¯1 ⎜ ⎟ g = P−1 ϕ, ϕ = ⎝ ... ⎠ . (11.40) v¯m The operators T

+ i(· , g )Cm g , A P

=

= P−1 Q, A

c · g

Kc =

(11.41) (11.42)

K ∗ and therefore deﬁne the Liv˘sic canonical scattering system satisfy Im T = K 1 T K Θ= . Cm C P Let

∗ −1 ∗ (Re T − zI)−1 K. VΘ K=K (z) = K (A − zI)

We are going to check that VΘ (zk ) = vk , (Q − zk P)j,k =

(11.43)

k = 1, . . . m. We have

zk vk − z¯j v¯j vk − v¯j − zk = v¯j . zk − z¯j zk − z¯j

(11.44)

It follows from (11.44) that the k-th column of the matrix Q − zk P is equal to ϕ − zk Im )ek = g (where ek denotes the (m × 1) column deﬁned in (11.40). Hence, (A vector whose all entries are equal to 0, besides the k–th one, which is equal to 1). Therefore, VΘ (zk )

− zk Im )−1 g, g)Cm = (ek , g)Cm = (Pek , P−1 g)Cm = ((A P P P = (Pek , P−1 g)Cm = (ek , g)Cm = vk .

Hence we proved that the Liv˘sic scattering system of the form (11.41)-(11.42) is the interpolation system for the Nevanlinna-Pick interpolation problem. If the operator T is α-sectorial, it follows from Theorem 11.5.4 that (11.39) holds. Conversely, let us suppose that (11.39) holds. We prove that the operator T is α-sectorial. Condition (11.39) implies that for an arbitrary vector ξ = (ξj ) ∈ Cm , we have the inequality cot α

m #

vk v¯j ξk ξj∗

k,j=1

i.e., cot α

m # 1

m # zk vk − z¯j v¯j ≤ ξk ξj∗ , zk − z¯j k,j=1

vk ξk

m # 1

∗ vj ξj

≤ (Qξ, ξ)Cm .

438

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

This inequality can be rewritten as m m ∗ # # ξ)Cm . cot α vk ξk vj ξj ≤ (Aξ, P 1

Besides, we have

(11.45)

1

(ξ, g )Cm = (Pξ, P−1 ϕ)Cm = (ξ, ϕ)Cm . Q

(11.46)

ξ)Cm , or It follows from (11.45) and (11.46) that cot α |(ξ, g )Cm |2 ≤ (Aξ, P P cot α |Im (T ξ, ξ)Cm | ≤ Re (T ξ, ξ)Cm . P P

Thus the operator T is α-sectorial. The system

= Θ

T Cm P

K

1 C

(11.47)

deﬁned by (11.41) and (11.42) constructed in the proof with operators T and K of Theorem 11.6.1, we call the Liv˘sic system in the Pick form. Theorem 11.6.2. Let z1 , . . . , zm ∈ C+ and v1 , . . . , vm ∈ C+ be the interpolation data. If the Liv˘sic canonical scattering system T K 1 Θ= , Im T = KK ∗ H C is the interpolation system for the Nevanlinna-Pick interpolation problem with this data and dim H = m, zk are not eigenvalues of the state-space operator T , and c. l. s.{(T − zk I)−1 KC,

k = 1, . . . , m} = H,

(11.48)

then the matrix P is strictly positive. The given system is unitarily equivalent to of the form (11.47) and the system Θ c. l. s.{Ran((T − zk I)−1 K),

k = 1, . . . , m} = Cm P .

Proof. Set VΘ (z) = K ∗ (ReT − zI)−1 K. Then, VΘ (zk ) − VΘ∗ (zj ) zk − z¯j

= K ∗ (ReT − z¯j I)−1 (ReT − zk I)−1 K = K ∗ (ReT − z¯j I)−1 KK −1 (K −1 )∗ K ∗ (ReT − zk I)−1 K = x2 VΘ (zj )∗ VΘ (zk ),

where we have set K −1 (K −1 )∗ = x2 I, (K −1 denotes the inverse of K on its range). Since the function VΘ (z) solves the interpolation problem associated to the interpolation data, we have vk − v¯j = x2 v¯j vk . zk − z¯j

(11.49)

11.6. The Liv˘sic interpolation systems in the Pick form

439

Now we are going to show that the matrix P is nonsingular. Let ξ = (ξj ) ∈ Cm be in the kernel of P. Then, m # vk − v¯j

zk − z¯j

k=1

ξk = 0,

j = 1, . . . m.

Equation (11.49) then implies that m #

x2 v¯j vk ξk = 0,

j = 1, . . . m,

k=1

and hence

)m k=1

vk ξk = 0, i.e., m #

)m k=1

K ∗ (Re T − zk I)−1 Kξk = 0. Thus, we have

(Re T − zk I)−1 Kξk = 0.

(11.50)

k=1

The operator T has a one-dimensional imaginary part. Therefore T can be represented in the form T = Re T + i ( · , g) g. (11.51) It follows from (11.48) that the vectors xk = 2(T − zk I)−1 g,

k = 1, . . . m,

are linearly independent. Hence (11.51) yields that xk = (2 − i(xk , g))(Re T − zk I)−1 g. Set Then we have

ω(z) = 1 − 2i((T − zI)−1 g, g). xk = (1 + ω(zk ))(Re T − zk I)−1 g.

Since the zk are not the eigenvalues of T , we have |ω(zk )| > 1. We may assume that K(1) = g and therefore (11.50) implies that m # k=1

ξk xk = 0. 1 + ω(zk )

ξk Since the vectors xk are linearly independent, we obtain that 1+ω(z = 0, and k) therefore ξk = 0, that is, P is nonsingular. Therefore we can consider the Liv˘sic in the Pick form (11.47). Since Θ is also an interpolation system we have system Θ (11.25). Thus VΘ k = 1, . . . m (zk ) = VΘ (zk ) = vk ,

440

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

∗ (Re T − zk I)−1 K for k = 1, . . . m. Hence we obtain and K ∗ (Re T − zk I)−1 K = K that WΘ (zk ) = WΘ k = 1, . . . m, (zk ), that is

∗ (T − zk I)−1 K, K ∗ (T − zk I)−1 K = K

k = 1, . . . m.

The operator U deﬁned by U (T − zk I)−1 K = (T − zk I)−1 K

(11.52)

= U K. We leave it to the reader is isometric from H onto CPm and satisﬁes K are unitarily to check that (11.52) implies that U T = T U and therefore Θ and Θ equivalent. Hence VΘ (z) = VΘ (z) and the theorem is proved. Theorem 11.6.3. Let zk and vk be the interpolation data with zk , vk ∈ C+ , k = 1, . . . , m, for which the matrix P of the form (11.38) is strictly positive. Let also Θ be the Liv˘sic system in the Pick form (11.47) corresponding to this data. Assume that zkj , j = 1, . . . , p are eigenvalues of T . Then, vk1 = · · · = vkp = i.

(11.53)

in the Pick form (11.47) Proof. By Theorem 11.6.1 there exists the Liv˘sic system Θ that is the interpolation system for the data. Thus, VΘ (zk ) = vk ,

k = 1, . . . m,

−1 ∗ and VΘ K. Recall that (zk ) = K (Re T − zk I)

WΘ (z) = i

VΘ (z) − 1 . VΘ (z) + 1

(11.54)

It is proved in [91] that the spectrum of the operator T coincides with the set of singular points of WΘ (z). Therefore the zki are poles of WΘ (z) and limz→zkj WΘ (z) = ∞. Hence vkj = lim VΘ (z) = i, z→zkj

due to (11.54).

Below we have the converse statement. Theorem 11.6.4. Let z1 , . . . , zm and v1 , . . . , vm be the system interpolation data with zk , vk ∈ C+ , k = 1, . . . , m. Suppose that the matrix P of the form (11.38) is be the Liv˘sic system in the Pick form strictly positive and (11.53) holds. Let also Θ (11.47) for this data. Then zk1 , . . . , zkp are eigenvalues of the state-space operator T deﬁned by (11.41).

11.6. The Liv˘sic interpolation systems in the Pick form

441

Proof. If the matrix P is strictly positive, then Theorem 11.6.1 implies that the the Liv˘sic system in the interpolation Nevanlinna-Pick problem is solvable with Θ, Pick form (11.47). Thus, VΘ (zkj ) = i for j = 1, . . . p. Since WΘ (z) = we get lim WΘ (z) = lim

z→zkj

z→zkj

1 − iVΘ (z) , 1 + iVΘ(z)

1 − iVΘ (z) = ∞, 1 + iVΘ (z)

j = 1, . . . p.

This means that the zkj are the poles of WΘ (z). Therefore, they are eigenvalues of T . Theorem 11.6.5. Let zk , k = 1, . . . m and vk = i, k = 1, . . . m be the interpolation data with zk , vk ∈ C+ . Then there exists the Liv˘sic scattering system solution of the corresponding system interpolation problem with state-space of dimension m. This system solution is unique up to a unitary equivalence. Am z−z ∗ Proof. Set W (z) = k=1 z−zkk and ⎛ ⎞ √ √ √ √ z1 i 2Im z1 2Im z2 · · · i√2Im z1 √2Im zm ⎜ 0 z2 · · · i 2Im z2 2Im zm ⎟ ⎟. TΔ = ⎜ (11.55) ⎝ · ⎠ · · · 0 0 ··· zm Then, Im TΔ = (·, gΔ )gΔ with

⎛ √ √Im z1 ⎜ Im z2 ⎜ gΔ = ⎜ .. ⎝ √ . Im zm

⎞ ⎟ ⎟ ⎟. ⎠

Consider the Liv˘sic system ΘΔ =

TΔ Cm

KΔ

1 C

,

(11.56)

where KΔ (c) = c · gΔ , c ∈ C. The transfer function of the system ΘΔ is WΘΔ (z) = 1 − 2i((TΔ − zI)−1 gΔ , gΔ ) =

m * z − z¯k . z − zk

(11.57)

k=1

The impedance function VΘΔ (z) = i

WΘΔ (z) − 1 , WΘΔ (z) + 1

(11.58)

442

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

clearly satisﬁes the interpolation conditions VΘΔ (zk ) = i,

k = 1, . . . m.

Therefore the Liv˘sic scattering system (11.56) is the interpolation system for the stated Nevanlinna-Pick interpolation problem. Let the system T K 1 Θ= , dim H = m, Im T = KK ∗ , H C be the interpolation system in the Nevanlinna-Pick interpolation problem for the data zk , k = 1, . . . m and vk = i, k = 1, . . . m. Without loss of generality, we can assume that Im T = (· , g)g and K 1 = g. Then, K ∗ x = (x, g). Deﬁne VΘ (z) and WΘ (z) as above, with this choice of T and K. Obviously (since Θ is the interpolation system), VΘ (zk ) = i for k = 1, . . . m and lim

z→zkj

1 − iVΘ (z) = ∞, 1 + iVΘ (z)

j = 1, . . . m.

Therefore, each point zk is a pole of WΘ (z). Since dim H = m, then according to (5.46) (see also Section 5.6), zk are exactly the points of the spectrum of T and WΘ (z) =

m * z − z¯k = WΘΔ (z). z − zk

k=1

By Theorem 5.4.3 on unitary equivalence of systems, the systems Θ and ΘΔ are unitary equivalent and therefore VΘ (z) = VΘΔ (z). Corollary 11.6.6. Let zk , k = 1, . . . , m, be arbitrary points in the open upper half– plane. There exists a unique matrix U such that U ∗ PU = Im , U and

2i zk − z¯j

U

−1 zk + z¯j i + iΠ = TΔ U, zk − z¯j

2i zk − z¯j

−1

⎛ √

⎞ Im z1 −i ⎟ .. ⎝ . ⎠=⎜ ⎝ ⎠, .. √ . −i Im zm ⎛ −i ⎞

where

1 1 ··· 1 Π=

1 1 ··· 1

.. .

.. .

1 1 ··· 1

and TΔ is of the form (11.55).

,

(11.59)

11.6. The Liv˘sic interpolation systems in the Pick form Proof. The matrix P =

2i zk −¯ zj

443

is strictly positive and the function VΘΔ (z) de-

ﬁned by (11.58) satisﬁes VΘΔ (zk ) = i for k = 1, . . . , m. It has been shown that of the form (11.47) also generates a solution of this Nevanlinna-Pick the system Θ interpolation problem, of the form −1 VΘ g, g )Cm , (z) = ((A − zI) P

where = Re T = P−1 A Since for c ∈ C

zk + z¯j zk − z¯j

⎛

⎞ −i ⎜ ⎟ g = P−1 ⎝ ... ⎠ . −i

,

∗ (Re T − zI)−1 Kc = V (z) · c, K Θ

we get that VΘ (zk ) = i for k = 1, . . . , m and WΘΔ (z) = WΘ (z), where WΘΔ (z) has the form (11.57) and WΘ (z) has the form −1 ∗ WΘ K. (z) = I − 2iK (T − zI) m There exits thus a unique unitary mapping U from Cm such that P onto C

U T = TΔ U,

= KΔ . UK

(11.60)

The fact that U is an isometry imlies that U ∗ PU = Im .

(11.61)

Taking into account (11.55), (11.56), (11.41)–(11.42) we get that relations (11.60) and (11.61) coincide with (11.59). Theorem 11.6.7. Let zk and vk be the interpolation data with zk , vk ∈ C+ , k = 1, . . . , m. Assume that the matrix P of the form (11.38) is strictly positive and the be the Liv˘sic system matrix Q of the form (11.38) is non-negative. Let also Θ in the Pick form (11.47) for this data. Then the state-space operator T of Θ deﬁned by (11.41) is accretive. Moreover, T is accretive but not α-sectorial for any α ∈ (0, π/2) if and only if det Q = 0 and for the unitary operator U on Cm P for which ∗ = U P−1 QU ∗ = diag (0, . . . 0, λ1 , . . . , λq ), U AU (11.62) (with λi > 0, p elements equal to 0 and p + q = m) at least one from the ﬁrst p coordinates of the vector Ug with respect to the orthonormal basis of eigenvectors = P−1 Q is not equal to 0. of the self-adjoint operator A Proof. The accretiveness of T immediately follows from (11.41) and positivity of P and Q. Set −1 VΘ Q − zI)−1g , g)Cm , (z) = ((P P

444

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

and let T of the form (11.41) be accretive but not α-sectorial for any α ∈ (0, π/2) operator. From Theorem 11.5.3 we have that −1 lim VΘ Q + xI)−1g, g )Cm = ∞. (x) = lim ((P P

x→0−

x→0−

(11.63)

Assume that Q is invertible. Then, lim (P−1 Q + xI)−1 = Q−1 P,

x→0−

and the limit (11.63) would then be ﬁnite. Therefore, det Q = 0. Since the operator is self-adjoint, there exists a unitary operator U on Cm for which (11.62) holds. A P Suppose that all the p ﬁrst coordinates of the vector Ug with respect to this orthonormal basis are equal to 0, i.e., ⎛0⎞ ⎜ ... ⎟ ⎜0⎟ ⎟ Ug = ⎜ ⎜ ξ1 ⎟ , ⎝ . ⎠ .. ξq

with ξi = 0. We have VΘ (−0) =

lim ((P−1 Q + εI)−1g , g )Cm lim (U (P−1 Q + εI)−1 U ∗ Ug , Ug )Cm P P

ε→0

ε→0

−1

∗ −1

lim ((U (P Q + εI)U ) Ug , Ug )Cm P ⎛ 1 ⎞⎛ ⎞ ⎛ ⎞ 0 0 0 0 ε ⎟ ⎜ .. ⎟ ⎜ .. ⎟ ⎜ .. ⎜ ⎟⎜ . ⎟ ⎜ . ⎟ . ⎜ ⎟⎜ 0 ⎟ ⎜ 0 ⎟ = lim ⎜ 0 1 0 ⎟ ⎜ ξ1 ⎟ , ⎜ ξ1 ⎟ 0 ε ε→0 ⎜ ⎟⎝ . ⎠ ⎝ . ⎠ m 1 ⎝ ⎠ . C .. λ1 +ε . 1 =

ε→0

λq +ε

=

lim

ε→0

q # 1

ξq

ξq

1 |ξk |2 = ∞, λk + ε

since g = 0. This contradiction gives the proof of the necessity. Conversely, assume det Q = 0. With the notation already used above, assume that there is η = 0 such that ⎛ ⎞ .. ⎜ η. ⎟ ⎜ ⎟ ⎜ . ⎟ . ⎟ Ug = ⎜ ⎜ ξ.1 ⎟ . ⎜ ⎟ ⎝ . ⎠ .. ξq

11.6. The Liv˘sic interpolation systems in the Pick form

445

We have −1 | lim VΘ Q + εI)−1g , g )Cm (−ε)| = lim ((P P ε→0

ε→0

= ((U P−1 QU ∗ + εI)−1 Ug, Ug)Cm P

⎛

⎞ ⎛ ⎞ .. . . ⎟ ⎜ .. ⎟ ⎜ ⎜ η ⎟ ⎜ η ⎟ ⎜ . ⎟ ⎜ . ⎟ 1 1 1 1 ⎜ .. ⎟ , ⎜ .. ⎟ = lim diag ,..., , ,..., ⎟ ⎜ ⎟ ε→0 ε ε λ1 + ε λq + ε ⎜ ⎜ ξ1 ⎟ ⎜ ξ1 ⎟ Cm ⎝ . ⎠ ⎝ . ⎠ .. .. ξq

ξq

|η|2 ≥ lim = ∞. ε→0 ε Applying Theorem 11.5.3 we see that T is accretive but not α-sectorial for any α ∈ (0, π/2) . Theorem 11.6.8. Let {zk } and {vk } be the interpolation data with zk , vk ∈ C+ , k = 1, . . . , m. Suppose that matrices Q and P of the form (11.38) are strictly deﬁned in (11.41) is α-sectorial positive. Then, the state-space operator T of Θ and the angle α is given by ⎛ ⎞ v¯1 ⎜ ⎟ tan α = (Q−1 ϕ, ϕ)Cm , ϕ = ⎝ ... ⎠ . (11.64) v¯m Proof. By Theorem 11.5.3, tan α = =

VΘ (−0) = lim ((P−1 Q + xI)−1 g, g)Cm P x→0−

(Q

−1

Pg, g)Cm = (Q−1 Pg, Pg)Cm P

which allows us to conclude, since g = P−1 ϕ.

Corollary 11.6.9. Let {zk } and {vk } be an interpolation data with zk , vk ∈ C+ , k = 1, . . . , m for which the matrices P and Q are strictly positive. Then, Q≥

1 (Q−1 ϕ, ϕ)Cm

ϕ∗ ϕ.

Proof. The result follows from Theorems 11.6.1, 11.6.8 and inequality (11.39). Theorem 11.6.10. Let {zk } and {vk } be the interpolation data with zk , vk ∈ C+ , k = 1, . . . , m. Suppose also that the corresponding matrix Q of the form (11.38) is strictly positive. Assume that the Liv˘sic canonical scattering system T K 1 Θ= H C

446

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

is the interpolation system in the Nevanlinna-Pick interpolation problem with dim H = m for which the zk ∈ ρ(T ) and such that c. l. s.{(T − zk I)−1 KC,

k = 1, . . . , m} = H.

Then, the state-space operator T is α-sectorial and the value of the angle α is given by formula (11.64). Proof. Theorem 11.6.2 implies that the corresponding matrix P is strictly positive. The same theorem yields that Θ is unitarily equivalent to the system (11.41)(11.42). Theorem 11.6.8 implies then that the main operator T is α-sectorial and that the angle α is given by (11.64).

11.7 The Nevanlinna-Pick rational interpolation with distinct poles In this section we consider the Nevanlinna-Pick interpolation problem in the class Nr of rational Herglotz-Nevanlinna functions V (z) ∈ N with n distinct poles and V (∞) = 0. It is formulated as follows: Given n points z1 , . . . , zn ∈ C+ and n not necessarily distinct points v1 , . . . , vn ∈ C+ , ﬁnd a function V (z) ∈ Nr with n distinct poles such that V (zk ) = vk , k = 1, . . . , n. Let P=

vk − v¯j zk − z¯j

Q=

and k,j=1,...n

zk vk − z¯j v¯j zk − z¯j

(11.65) k,j=1,...n

be the Pick matrices corresponding to the interpolation data. Theorem 11.7.1. Let

Θ=

T H

K

1 C

,

(11.66)

be the Livsic canonical system with n-dimensional state space H and channel operator Kc = cg, c ∈ C, g ∈ H, K ∗ x = (x, g), x ∈ H. Then the following statements are equivalent: (i) the vector g is a cyclic vector for Re T ; (ii) for every distinct n numbers z1 , . . . , zn from ρ(Re T ) the vectors (Re T − z1 I)−1 g, . . . , (Re T − zn I)−1 g, are linearly independent;

11.7. The Nevanlinna-Pick rational interpolation with distinct poles

447

(iii) for every distinct non-real n numbers z1 , . . . , zn from the open (lower) halfplane and for numbers vk = ((Re T − zk I)−1 g, g), k = 1, . . . , n the corresponding Pick matrix P given by (11.65) is strictly positive. Proof. (i)⇒(ii): Let the vectors g, Re T g, . . . , Re T n−1 g be linearly independent. Suppose that the vectors h1 = (Re T − z1 I)−1 g, . . . , hn = (Re T − zn I)−1 g, are not linearly independent. Let then h1 , . . . , hm , m ≤ n−1 be a maximal linearly independent subsystem. Then the vector hn is a linear combination of h1 . . . . , hm , i.e., m # (0) hn = λk hk . k=1

Since T (T − ξI) we get

−1

ϕ = ϕ + ξ(T − ξI) g + zn hn =

−1

ϕ for every ϕ ∈ H and every ξ ∈ ρ(Re T ),

m # (0) (0) λk g + λk zk hk . k=1

It follows that

1−

m #

(0) λk

g=

k=1

Hence (1 −

m ) k=1

m #

(0)

(zk − zn )λk hk .

k=1

(0)

λk ) = 0 and, therefore, g =

Re T g =

m #

m ) k=1

(1)

λk hk . Since

(1)

λk (g + zk hk ),

k=1

we get that Re T g =

m #

(2)

λk hk .

k=1

Continuing this process we conclude that the vectors g, Re T g, . . . , Re T n−1 g are linear combinations of h1 , . . . , hm that yields a contradiction. Thus, (Re T − z1 I)−1 g, . . . , (Re T − zn I)−1 g, are linearly independent. (ii)⇒(i): Let the vectors (Re T − z1 I)−1 g, . . . , (Re T − zn I)−1 g be linearly independent. Because the subspace H0 := c.l.s{g, Re T g, . . . , Re T n−1 g},

448

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

reduces Re T , the vectors (Re T −z1 I)−1 g, . . . , (Re T −zn I)−1 g lie in H0 . Therefore, H0 = H. (ii) ⇐⇒ (iii). Let λ = (λ1 , . . . , λn ) ∈ Cn . Since ( n (2 n (# ( # ( −1 ( λk (T − zk I) g ( = λk λ¯j ((T − zk I)−1 g, (T − zj I)−1 g) ( ( ( k=1 n #

=

=

k,j=1 n # k,j=1

k,j=1

λk λ¯j λk λ¯j

((T − zk I)

−1

g, g) − ((T − zj∗ I)−1 g, g) zk − zj∗

vk − vj∗ = λ∗ Pλ, zk − zj∗

we get that P is non-negative. This matrix is invertible if and only if the vectors (Re T − z1 I)−1 g, . . . , (Re T − zn I)−1 g are linearly independent. Lemma 11.7.2. Any rational function V (z) from the class Nr of the form V (z) =

n # k=1

ak , tk − z

where t1 , . . . , tn are distinct real numbers and ak > 0, represented in the form

(11.67) k = 1, 2, . . . , n, can be

WΘ (z) − 1 V (z) = (Re T − zI)−1 g, g = i , WΘ (z) + 1 where WΘ (z) is a transfer function of the minimal system of the form T K 1 Θ= , Cn C

(11.68)

g is a cyclic vector for Re T , and Im T = (., g)g. Proof. Let δk = (0B . CD . . 01E 0 . . . 0),

k = 1, . . . , n,

k

be an orthonormal system of vectors from Cn . Deﬁne a self-adjoint operator Re T in Cn as Re T δk = tk δk , k = 1, . . . , n, n √ ) and let g = ak δk . Then k=1 −1

((Re T − zI)

g, g) =

n # k=1

ak , tk − z

z = tk , k = 1, . . . , n.

11.7. The Nevanlinna-Pick rational interpolation with distinct poles

449

It follows that V (z) = ((Re T − zI)−1 g, g). Moreover, the vectors g, Re T g, . . ., Re T n−1 g are linearly independent. Consider the system Θ of the form (11.68) where T = Re T + i(., g)g,

Kc = cg, c ∈ C,

K ∗ x = (x, g),

x ∈ Cn .

It was shown in Lemma 11.1.3 that vector g is cyclic for the real part Re T of operator T if and only if it is cyclic for the operator T . Therefore operator T is prime and the system Θ as a result of that is minimal. Denote by {zk , k = 1, . . . , n} and {vk , k = 1, . . . , n} interpolation data on the open upper half-plane C+ , and let the matrices P and Q be deﬁned by (11.65). Let also 1 T K = Θ , (11.69) CnP C be the system Liv˘sic system in the Pick form (written for this data) deﬁned by deﬁned by (11.41) and (11.42), respectively. (11.47) with operators T and K Theorem 11.7.3. Let {zk , k = 1, . . . , n} ∈ C+ and {vk , k = 1, . . . , n} ∈ C+ be in in the Pick form (11.69) is a minimal interpolation terpolation data. The system Θ system in the Nevanlinna-Pick interpolation problem and function −1 VΘ χ, χ (z) = (Re T − zI) is a rational function from the class Nr with n distinct poles and representation (11.67). Proof. We need to show that the function −1 VΘ χ, χ n = (Q − zP)−1 ϕ, ϕ Cn , (z) = (Re T − zI) CP

z ∈ ρ(Re T ) (11.70)

satisﬁes conditions VΘ (zk ) = vk , k = 1, . . . , n. We have that (Q − zk P)j,k =

zk vk − z¯j v¯j vk − v¯j − zk = v¯j . zk − z¯j zk − z¯j

Therefore the k-th column of the matrix Q − zk P is equal to the vector ϕ of the form (11.40) . Hence, (Re T − zk I)δk = χ (where δk denotes the (n × 1) column vector whose entries are all equal to 0, except the k-th one, which is equal to 1). Thus, V (zk ) = ((Re T − zk I)−1 χ, χ)CnP = (δk , χ)CnP = (Pδk , P−1 χ)CnP = (Pδk , P−1 χ)Cn = (δk , χ)Cn = vk . Theorem 11.7.1 implies that vector χ is cyclic for operator Re T and this operator has n distinct eigenvalues. Therefore function VΘ (z) is rational with n poles and belongs to the class Nr .

450

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

Theorem 11.7.4. The Nevanlinna-Pick interpolation problem in the class Nr has a solution if and only if the Pick matrix P (11.65) is strictly positive. Moreover, if the matrix P is strictly positive, then the Nevanlinna-Pick interpolation problem has the unique solution VΘ (z) of the form (11.70) that is represented by the minimal Livsic interpolation system in the Pick form (11.69). Proof. Suppose that function V (z) is a solution of Nevanlinna-Pick interpolation problem in the class Nr . By Lemma 11.7.2 function V (z) has a representation V (z) = ((Re T − zI)−1 g, g) = i

WΘ (z) − 1 , WΘ (z) + 1

where WΘ (z) is a transfer function of the minimal system of the form T K 1 Θ= , Cn C

(11.71)

(11.72)

and Kc = cg, K ∗ x = (x, g), x, g ∈ Cn , g is a cyclic vector for Re T. It follows from Theorem 11.7.1 that the Pick matrix P is invertible and non-negative. If matrix P is invertible and non-negative, then by Theorem 11.7.3 the function VΘ (z) of the form (11.70) is a solution of the Nevanlinna-Pick interpolation problem in the of the Livsic class Nr and is represented by the minimal interpolation system Θ type in the Pick form (11.69). Now we will establish the solution uniqueness of the Nevanlinna-Pick interpolation problem in the class Nr . Suppose that in addition to the solution VΘ (11.70), (11.69) there is another solution V (z). As we mentioned at the beginning of the proof, this solution can be represented by (11.71) V (z) = VΘ (z) = ((Re T − zI)−1 g, g), and is generated by the system Θ of the form (11.72). Then from the equality V (zk ) = VΘ (zk ) = VΘ k = 1, . . . , n follows (zk ), (Re T − zk I)−1 χ, χ n = (Re T − zk I)−1 g, g Cn = vk , CP (Re T − z¯k I)−1 χ, χ n = (Re T − z¯k I)−1 g, g Cn = v¯k , k = 1, . . . , n, CP

and

∗ (Re T − zk I)−1 χ, χ n − (Re T − zm I)−1 χ, χ n CP CP ∗ = (Re T − zk I)−1 g, g Cn − (Re T − zm I)−1 g, g Cn .

(11.73)

Applying the Hilbert identity for resolvents, we get ((Re T − zk I)−1 χ, (Re T − zm I)−1 χ)CnP = (Re T − zk I)−1 g, (T − zm I)−1 g Cn .

(11.74)

11.8. Examples

451

Let the operator U : CnP → Cn be deﬁned as U (Re T − zk I)−1 χ = (Re T − zk I)−1 g,

k = 1, . . . n.

Then from Theorem 11.7.1 and relation (11.74) we get that U is a unitary operator. From (11.73) we obtain U (Re T − zk I)−1 χ, g n = (Re T − zk I)−1 g, g)Cn = ((Re T − zk I)−1 χ, χ)CnP , C

for k = 1, . . . , n, and hence (U h, g)Cn = (h, χ)CnP for all h ∈ CnP . Therefore U χ = g. Furthermore (Re T )U (Re T − zk I)−1 χ = (Re T )(Re T − zk I)−1 g = g + zk (Re T − zk I)−1 g = U χ + zk U (Re T − zk I)−1 χ = U (Re T )(Re T − zk I)−1 χ. Finally, (Re T )U h = U Re T h for all h ∈ CnP , i.e., the operator Re T is unitarily equivalent to the operator Re T and moreover, VΘ (z) = VΘ (z) for all z ∈ ρ(Re T ) = ρ(Re T ). Theorem 11.7.5. Let z1 , . . . , zn ∈ C+ be n distinct points, and v1 , . . . , vn ∈ C+ be n not necessarily distinct points for which the Pick matrix P (11.65) is strictly in the Pick form (11.69) correspondpositive. Then the Livsi˘c canonical system Θ ing to the given data is unitarily equivalent to the system Δ of the form (11.15) with some (n × n) Jacobi matrix J (11.1) as a state-space operator of the system. of the form (11.69). By Theorem 11.7.4 Proof. Consider interpolation system Θ the system Θ is minimal and operator T is prime. Therefore by Theorem 5.6.4 this operator has only non-real eigenvalues in the upper half-plane. By Theorem 11.2.6 operator T is unitarily equivalent to an operator given by a Jacobi matrix J of the form (11.1) that can be obtained by the algorithm of reconstruction (see Section 11.4) from non-real eigenvalues of the matrix T . Consequently, we obtain that the is unitarily equivalent to some system Δ of the form (11.15). system Θ It follows from this theorem that the system Δ with the Jacobi matrix J as a state-space operator is a system solution of the Nevanlinna-Pick interpolation problem in the class Nr as well.

11.8 Examples Example. Consider the following interpolation data: z1 = 2i,

z2 = 3i,

v1 =

2i , 5

v2 =

3i . 10

452

Chapter 11. Non-self-adjoint Jacobi Matrices and System Interpolation

in the Pick form whose impedance function We will ﬁnd an interpolation system Θ is the solution of the Nevanlinna-Pick interpolation problem for the given inter is unitarily equivalent to the system Δ polation data. We will also show that Θ in Jacobi form with a non-self-adjoint Jacobi matrix (with a rank-one imaginary part) as a state-space operator of the system. We have 1 7 i 0 5 50 50 P= , Q= , 7 1 i − 50 0 50 10 and thus Re T = P−1 Q =

7i 5i −10i −7i

Furthermore, we have ∗ v1 −5i χ = P−1 = , v2∗ −10i

.

T =

−5i 72 i −6i −4i

.

Thus the corresponding interpolation system in the Pick form is equal to ⎛ ⎞ −5i 72 i 5i Kc = c · J = 1 ⎜ ⎟ −6i −4i −10i ⎜ ⎟ ⎜ ⎟ =⎜ Θ ⎟. ⎞ ⎜ C2⎛ 1 C ⎟ 7 ⎝ ⎠ ⎝

5 7 50

50 ⎠ 1 10

the Livsi˘c Applying the algorithm of reconstruction we get from the system Θ canonical system Δ with a non-self-adjoint Jacobi matrix as a state-space operator of the system. This system has the form ⎛ ⎞ i 1 1 Kc = c · J = 1 ⎠. 1 0 0 Δ=⎝ 2 C C i 1 where the state-space operator is the Jacobi matrix J = and 1 0 Re J =

0 1

1 0

,

g=

1 0

.

The corresponding impedance function that is the system solution is VΔ (z) = ((Re J − zI)−1 g, g) =

z . 1 − z2

Chapter 12

Non-canonical Systems In Chapter 6 we described the class N (R) of Herglotz-Nevanlinna functions in a ﬁnite-dimensional Hilbert space that can be realized as impedance functions of canonical L-systems. Since the class N (R) is substantially narrower than the set of all Herglotz-Nevanlinna functions, the problem of the general realization, or description of a new non-canonical type of systems that realize an arbitrary Herglotz-Nevanlinna function, remain open. In the present chapter we introduce the notion of a non-canonical impedance system (NCI-system) as well as a non-canonical L-system (NCL-system). We are going to show that any Herglotz-Nevanlinna function in ﬁnite-dimensional Hilbert space E can be realized as a transfer function of an NCI-system of the form

(D − zF+ )x = Kϕ− , (12.1) ϕ+ = K ∗ x, where D and F+ are self-adjoint operators acting from H+ into H− and in addition F+ is an orthogonal projector in H+ and H. In this case, the associated transfer function of the NCI-system is given by V (z) = K ∗ (D − zF+ )−1 K,

(12.2)

and every Herglotz-Nevanlinna function can be represented in the form (12.2). We also consider the following non-canonical L-systems or NCL-systems

(A − zF+ )x = KJϕ− , (12.3) ϕ+ = ϕ− − 2iK ∗ x. This system can be expressed via an array similar to the one of an L-system in (6.36), that is A F+ K J ΘF+ = . (12.4) H+ ⊂ H ⊂ H− E Y. Arlinskii et al., Conservative Realizations of Herglotz–Nevanlinna Functions, Operator Theory: Advances and Applications, DOI 10.1007/978-3-7643-9996-2_12, © Springer Basel AG 2011

453

454

Chapter 12. Non-canonical Systems

The additional ingredient in (12.4) is the operator F+ which is an orthogonal projection in H+ and H. The corresponding transfer function of an NCL-system is WΘF+ (z) = I − 2iK ∗ (A − zF+ )−1 KJ, and the impedance function of an NCL-system is VΘF+ (z) = K ∗ (Re A − zF+ )−1 K = i[WΘF+ (z) + I]−1 [WΘF+ (z) − I]J.

(12.5)

We will show that, after some restrictions on the linear coeﬃcient L in the integral representation +∞ 1 t V (z) = Q + zL + − dG(t), (12.6) t−z 1 + t2 −∞ any Herglotz-Nevanlinna function V (z) is realizable as the impedance function of an NCL-system of the form (12.3). We begin this chapter with a special case. In the following two sections, we give realization results for Herglotz-Nevanlinna functions of the form +∞ dG(t) V (z) = Q + Lz + , (12.7) −∞ t − z where Q = Q∗ , L ≥ 0, and G(t) is a non-decreasing function on R such that +∞ (dG(t)x, x)E < ∞, for all x ∈ E. (12.8) −∞

In particular, it is shown that each Herglotz-Nevanlinna function V (z) of the form (12.7) satisfying (12.8) gives rise to an F -system of the form M F K J ΘF = , (12.9) H E where the state-space operator M is now allowed to be unbounded. Moreover, the additional ingredient in (12.9) is the operator F which is an orthogonal projection in H. The system determined by (12.9) has properties that are analogous to those of canonical Livˇsic systems (5.6) in Chapter 5. The function VΘF (z) is obtained from (12.9) via an F -resolvent in the form VΘF (z) = K ∗ (D − zF )−1 K,

z ∈ C \ R,

where D = Re M . The transfer function WΘF (z) associated to the F -system (12.9) is deﬁned by WΘF (z) = 1 − 2iK ∗ (M − zF )−1 KJ, for all z ∈ C for which (M − zF )−1 exists, bounded, and deﬁned on entire H. Again the functions VΘF (z) and WΘF (z) are related via (12.5).

12.1. F -systems: deﬁnition and basic properties

12.1

455

F -systems: deﬁnition and basic properties

Let H be a Hilbert space with inner product (·, ·). Let A be a closed linear operator in H and let F be an orthogonal projection in H. Associated to the pair (A, F ) is the resolvent set ρ(A, F ), i.e., the set of all z ∈ C for which A − zF is boundedly invertible in H and (A−zF )−1 is deﬁned on entire H. The corresponding resolvent operator is deﬁned as (A − zF )−1 , z ∈ ρ(A, F ). When the resolvent set ρ(A, F ) is non-empty, the following analog of the resolvent identity holds: (A − zF )−1 − (A − wF )−1 = (z − w)(A − wF )−1 F (A − zF )−1 ,

(12.10)

z, w ∈ ρ(A, F ). Note that if A is self-adjoint, then ρ(A, F ) is symmetric with respect to the real axis. In the sequel an important role is played by the class of densely-deﬁned closed linear operators A in H which are of the form A = D + iR,

D = D∗ ,

R = R∗ ∈ [H, H].

(12.11)

Here D, and hence A, may be unbounded, but R is bounded. Therefore A∗ = D ∗ − iR∗ , which implies 2D = A + A∗ = 2Re A and 2iR = A − A∗ = 2iIm A, so that D = Re A, R = Im A, are the real part and the imaginary part of A, respectively. Lemma 12.1.1. Let A = D + iR be a closed linear operator in the Hilbert space H of the form (12.11), so that D = D∗ and R = R∗ ∈ [H, H]. Let F be an orthogonal projection in H. Assume that R ≥ 0 or that R ≤ 0. Then: (i) ker(A − zF ) = ker(A∗ − z¯F ) = ker(A) ∩ ker(F ) for every z ∈ C− or for every z ∈ C+ , respectively; (ii) if z ∈ ρ(A, F ) for some z ∈ C− or for some z ∈ C+ , then z ∈ ρ(A, F ) for every z ∈ C− or for every z ∈ C+ , respectively. Moreover, if A is self-adjoint or equivalently R = 0, then the statements (i) and (ii) hold for all z ∈ C− ∪ C+ . Proof. It suﬃces to give a proof for the case R ≥ 0. The case R ≤ 0 can be dealt with in an analogous manner. (i) Clearly, ker(A) ∩ ker(F ) ⊂ ker(A − zF ) for any z ∈ C. To see the converse, assume that x ∈ ker(A − zF ), z ∈ C− . Then 0 = ((A − zF )x, x) = (Dx, x) + i(Rx, x) − z(F x, x), and, since z ∈ C− , one concludes that (Rx, x) = (F x, x) = 0. Therefore, Rx = F x = 0 and consequently Ax = 0, so that x ∈ ker(A) ∩ ker(F ). Similarly one proves ker(A∗ − z¯F ) = ker(A) ∩ ker(F ). (ii) Assume that z0 ∈ ρ(A, F ) for some z0 ∈ C− , so that ker(A − z0 F ) = {0} and Ran(A − z0 F ) = H. It follows from (i) that ker(A − zF ) = {0} for every

456

Chapter 12. Non-canonical Systems

z ∈ C− . It remains to show that Ran(A − zF ) = H for every z ∈ C− . Again by (i) ker(A∗ − z¯F ) = {0}, and thus it is enough to prove that Ran(A − zF ) is closed for every z ∈ C− ; or equivalently, that for every z ∈ C− there is a positive constant c(z) such that (A − zF )x ≥ c(z)x for all x ∈ Dom(A). Assume the converse, that there is z ∈ C− such that (A − zF )xn → 0 for some sequence (xn ), xn = 1, xn ∈ Dom(A). Then ((A − zF )xn , xn ) = (Dxn , xn ) + i(Rxn , xn ) − z(F xn , xn ) → 0,

(12.12)

implies that (F xn , xn ) = F xn 2 → 0. Hence F xn → 0, Axn → 0, and consequently (A − z0 F )xn → 0, a contradiction to the assumption z0 ∈ ρ(A, F ). It is clear that if A is self-adjoint, then the assertions (i) and (ii) are true for every z ∈ C \ R. Corollary 12.1.2. Let A = D + iR and F be as in Lemma 12.1.1, and assume that R = R∗ ∈ [H, H] satisﬁes R ≥ 0 or R ≤ 0, respectively. Then: (i) ker(A) ⊂ ker(Re A); (ii) ρ(Re A, F ) = ∅ implies ρ(A, F ) = ∅. Proof. (i) Assume that Ax = 0 for some x ∈ Dom(A). Then ((Re A+ iR)x, x) = 0, which implies (Re Ax, x) = (Rx, x) = 0. Consequently, Rx = 0 and hence also Re Ax = 0. (ii) Assume that ρ(Re A, F ) = ∅. Then, by Lemma 12.1.1, C+ ∪ C− ⊂ ρ(Re A, F ). With z ∈ C− part (i) and Lemma 12.1.1 imply ker(A − zF ) = ker(A) ∩ ker(F ) ⊂ ker(Re A) ∩ ker(F ) = {0}. To see that z ∈ ρ(A, F ) consider a sequence (xn ), xn = 1, with (A − zF )xn → 0 as in the proof of Lemma 12.1.1. 1 Then (12.12) shows that F xn → 0 and R 2 xn → 0. Therefore also Rxn → 0 and Re Axn → 0. Consequently, (Re A − zF )xn → 0 and this implies (ii). Let M be a closed linear operator in H of the form (12.11) and let F be an orthogonal projection in H, K ∈ [E, H], and J be a bounded, self-adjoint, and unitary operator in E. Let also Im M = KJK ∗ and L2[0,τ0 ] (E) be the Hilbert space of E-valued functions equipped with an inner product τ0 (ϕ, ψ)L2[0,τ ] (E) = (ϕ, ψ)E dt, ϕ(t), ψ(t) ∈ L2[0,τ0 ] (E) . (12.13) 0

0

Consider the system of equations ⎧ dχ ⎨ iF dt + M χ(t) = KJψ− (t), ⎩ χ(0) = x ∈ H, ∗ ψ+ = ψ− − 2iK χ(t). We have the following lemma.

(12.14)

12.1. F -systems: deﬁnition and basic properties

457

Lemma 12.1.3. If for a given continuous ψ− (t) ∈ L2[0,τ0 ] (E) we have χ(t) ∈ H and ψ+ (t) ∈ L2[0,τ0 ] (E) satisfy (12.14), then a system of the form (12.14) satisﬁes the metric conservation law τ τ 2 2 2F χ(τ ) − 2F χ(0) = (Jψ− , ψ− )E dt − (Jψ+ , ψ+ )E dt, (12.15) 0 0 τ ∈ [0, τ0 ]. Proof. Following the algebraic steps of the proof of Lemma 5.1.1 and taking into account that F is a bounded operator and Im M = KJK ∗ , we obtain d 1 1 (F χ(t), F χ(t)) = (Jψ− (t), ψ− (t))E − (Jψ+ (t), ψ+ (t))E . dt 2 2 Taking into account that ψ± (t) ∈ L2[0,τ0 ] (E), we integrate both sides from 0 to τ ∈ [0, τ0 ] and multiply by 2 to obtain (12.15). Given an input vector ψ− = ϕ− eizt ∈ E, we seek solutions to the system (12.14) as an output vector ψ+ = ϕ+ eizt ∈ E and a state-space vector χ(t) = xeizt ∈ H. Substituting the expressions for ψ± (t) and χ(t) allows us to cancel exponential terms and convert the system (12.14) to the stationary form

(M − zF )x = KJϕ− , , z ∈ ρ(M, F ). (12.16) ϕ+ = ϕ− − 2iK ∗ x, Following the canonical case we can re-write (12.16) as an array and have the following deﬁnition. Deﬁnition 12.1.4. Let H and E be Hilbert spaces with dim E < ∞, and let M , F , K, and J be linear operators. The array M F K J ΘF = , (12.17) H E is called an F -system if: (i) M is of the form (12.11); (ii) J = J ∗ = J −1 ∈ [E, E]; (iii) Im M = KJK ∗ , where K ∈ [E, H]; (iv) F is an orthogonal projection in H; (v) the resolvent sets ρ(Re M, F ) and ρ(M, F ) are nonempty. The system (12.17) is called a scattering F -system if in (ii) J = I, in which case the state-space operator M in (12.17) is dissipative: Im M ≥ 0, as follows from (iii). Moreover, in this case the ﬁrst condition ρ(Re M, F ) = ∅ in (v) implies the second condition ρ(M, F ) = ∅ in (v), as follows from Corollary 12.1.2.

458

Chapter 12. Non-canonical Systems To each F -system in Deﬁnition 12.1.4 one can associate the impedance func-

tion

VΘF (z) = K ∗ (Re M − zF )−1 K,

z ∈ ρ(Re M, F ).

(12.18)

Lemma 12.1.5. Let ΘF be an F -system of the form (12.17). Then for all z, w ∈ ρ(Re M, F ), the function VΘF (z) in (12.18) satisﬁes VΘF (z) − VΘF (w) ¯ ∗ = (z − w)K ¯ ∗ (Re M − zF )−1 F (Re M − wF ¯ )−1 K.

(12.19)

If ρ(Re M, F ) is nonempty, then C+ ∪ C− ⊂ ρ(Re M, F ) and VΘF (z), z ∈ C \ R, is a Herglotz-Nevanlinna function. Proof. For each z, w ∈ ρ(Re M, F ) the resolvent identity (12.10) reads (Re M − zF )−1 − (Re M − wF ¯ )−1 = (z − w)(Re ¯ M − zF )−1 F (Re M − wF ¯ )−1 . The deﬁnition (12.18) implies (12.19). If ρ(Re M, F ) is nonempty, then according to Lemma 12.1.1, C \ R ⊂ ρ(Re M, F ). Clearly, VΘF (z)∗ = VΘF (¯ z ) when z ∈ ρ(Re M, F ). Moreover, it follows from (12.19) that VΘF (z) is (locally) holomorphic on ρ(Re M, F ) and has a non-negative imaginary in the upper half-plane, so that VΘF (z) is an operator-valued Herglotz-Nevanlinna function. To each F -system in Deﬁnition 12.1.4 one can also associate the transfer function WΘF (z) = I − 2iK ∗ (M − zF )−1 KJ, z ∈ ρ(M, F ). (12.20) Lemma 12.1.6. Let ΘF be an F -system of the form (12.17). Then for all z, w ∈ ρ(M, F ), the function WΘ (z) in (12.20) satisﬁes WΘF (z)JWΘ∗ F (w) − J = 2i(w ¯ − z)K ∗ (M − zF )−1 F (M ∗ − wF ¯ )−1 K, WΘ∗ F (w)JWΘF (z) − J = 2i(w ¯ − z)JK ∗ (M ∗ − wF ¯ )−1 F (M − zF )−1 KJ. In particular, if z ∈ ρ(M, F ), then WΘF (z) is J-unitary when z ∈ R, J-expansive when z ∈ C+ , and J-contractive when z ∈ C− . Proof. The property (iii) in Deﬁnition 12.1.4 shows that for all z, w ∈ ρ(M, F ), (M − zF )−1 − (M ∗ − wF ¯ )−1 = (M − zF )−1 [(M ∗ − wF ¯ ) − (M − zF )](M ∗ − wF ¯ )−1 = (z − w)(M ¯ − zF )−1 F (M ∗ − wF ¯ )−1 − 2i(M − zF )−1 KJK ∗ (M ∗ − wF ¯ )−1 . This identity together with (12.20) implies that WΘF (z)JWΘ∗ F (w) − J = [I − 2iK ∗(M − zF )−1 KJ ]J[I + 2iJK ∗ (M ∗ − wF ¯ )−1 K] − J = 2i(w ¯ − z)K ∗ (M − zF )−1 F (M ∗ − wF ¯ )−1 K.

12.1. F -systems: deﬁnition and basic properties

459

This proves the ﬁrst equality. Likewise, by using ∗ (M − zF )−1 − (M ∗ − wF ¯ )−1 = (z − w)(M ¯ − wF ¯ )−1 F (M ∗ − zF )−1

− 2i(M ∗ − wF ¯ )−1 KJK ∗ (M − zF )−1 ,

one proves the second identity.

The impedance function VΘF (z) deﬁned in (12.18) and the transfer function WΘF (z) deﬁned in (12.20) are closely connected. Lemma 12.1.7. Let ΘF be an F -system of the form (12.17). Then for all z ∈ ρ(Re M, F ) ∩ ρ(M, F ) the operators I + iVΘF (z)J and I + WΘF (z) are bounded and invertible. Moreover, VΘF (z) = i[WΘF (z) − I][WΘF (z) + I]−1 J = i[WΘF (z) + I]−1 [WΘF (z) − I]J,

(12.21)

and WΘF (z) = [I − iVΘF (z)J][I + iVΘF (z)J]−1 = [I + iVΘF (z)J]−1 [I − iVΘF (z)J].

(12.22)

Proof. The following identity with z ∈ ρ(M, F ) ∩ ρ(Re M, F ) (Re M − zF )−1 − (M − zF )−1 = i(M − zF )−1 Im M (Re M − zF )−1 , leads to K ∗ (Re M − zF )−1 K−K ∗ (M − zF )−1 K = iK ∗ (M − zF )−1 KJK ∗ (Re M − zF )−1 K. Now in view of (12.20) and (12.18) 2VΘF (z) + i(I − WΘF (z))J = (I − WΘF (z))VΘF (z), or equivalently, [I + WΘF (z)][I + iVΘF (z)J] = 2I.

(12.23)

Similarly, the identity (Re M − zF )−1 − (M − zF )−1 = i(Re M − zF )−1 Im M (M − zF )−1 with z ∈ ρ(M, F ) ∩ ρ(Re M, F ) leads to [I + iVΘF (z)J][I + WΘF (z)] = 2I.

(12.24)

The equalities (12.23) and (12.24) show that the operators are boundedly invertible and consequently one obtains (12.21) and (12.22).

460

Chapter 12. Non-canonical Systems

When M is a bounded operator and F = I, the F -system in Deﬁnition 12.1.4 reduces to the Livsi˘c canonical system studied in Chapter 5. The present situation with M , generally speaking, unbounded and F an orthogonal projection is of interest as it is phrased in pure Hilbert space terminology; it does not require any rigged Hilbert space triplets as in Chapter 6. If the system (12.17) is a scattering system (J = I), then it follows from (12.22) that WΘF (z) is a Schur function; it is holomorphic on C− and its values are contractions. In general, when (12.17) is not a scattering system, the domain of holomorphy of WΘF (z) need not be C− .

12.2 Multiplication theorems for F -systems Consider the two F -systems ΘF1 and ΘF2 of the form (12.17), deﬁned by M1 F1 K1 J ΘF1 = , (12.25) H1 E and ΘF2 =

M2 F2 H2

K2

J . E

(12.26)

Deﬁne the Hilbert space H by H = H1 ⊕ H2 ,

(12.27)

and let Pj be the orthoprojections from H onto Hj , j = 1, 2. Deﬁne the operators M , F , and K by M = M1 P1 + M2 P2 + 2iK1JK2∗ P2 ,

F = F1 P1 + F2 P2 ,

K = K1 + K2 . (12.28)

Theorem 12.2.1. Let ΘF1 be the F1 -system in (12.25) and let ΘF2 be the F2 -system in (12.26). Then the aggregate M F K J Θ= , (12.29) H E with H, M , F , and K, deﬁned by (12.27) and (12.28), is an F -system. Proof. Taking adjoints in (12.28) gives M ∗ = M1∗ P1 + M2∗ P2 − 2iK2∗ JK1∗ P1 ,

K ∗ = K1∗ P1 + K2∗ P2 ,

and therefore, M − M ∗ = (M1 − M1∗ )P1 + (M2 − M2∗ )P2 + 2iK1 JK2∗ P2 + 2iK2 JK1∗ P1 = 2iK1JK1∗ P1 + 2iK2 JK2∗ P2 + 2iK1JK2∗ P2 + 2iK2 JK1∗ P1 = 2i(K1 + K2 )J(K1∗ P1 + K2∗ P2 ) = 2iKJK ∗.

Clearly, F is an orthoprojection. Hence, the aggregate (12.29) with H, M , F , and K, deﬁned by (12.27) and (12.28), is an F -system.

12.2. Multiplication theorems for F -systems

461

The F -system Θ in (12.29) is called the coupling of the F1 -system ΘF1 and the F2 -system ΘF2 ; it is denoted by Θ = ΘF1 ΘF2 . Theorem 12.2.2. Let the F -system Θ be the coupling of the F1 -system ΘF1 and the F2 -system ΘF2 . Then the associated transfer functions satisfy WΘ (z) = WΘF1 (z)WΘF2 (z),

z ∈ ρ(M1 , F1 ) ∩ ρ(M2 , F2 ).

(12.30)

Proof. Let z ∈ ρ(M1 , F1 ) ∩ ρ(M2 , F2 ). Observe that M − zF = M1 P1 + M2 P2 + 2iK1 JK2∗ P2 − z(F1 P1 + F2 P2 ) = (M1 − zF1 )P1 + (M2 − zF2 )P2 + 2iK1 JK2∗ P2 .

Therefore,

(M − zF ) (M1 − zF1 )−1 P1 + (M2 − zF2 )−1 P2 −2i(M1 − zF1 )−1 K1 JK2∗ (M2 − zF2 )−1 P2

= P1 − 2iK1 JK2∗ (M2 − zF2 )−1 P2 + P2 + 2iK1 JK2∗ (M2 − zF2 )−1 P2 = P1 + P2 = I. A similar result is obtained when the factors are multiplied in the reverse order. Hence, it follows that z ∈ ρ(M, F ) and that (M − zF )−1 =(M1 − zF1 )−1 P1 + (M2 − zF2 )−1 P2 − 2i(M1 − zF1 )−1 K1 JK2∗ (M2 − zF2 )−1 P2 . Furthermore, (12.30) follows from WΘ (z) = I − 2iK ∗ (M − zF )−1 KJ = I − 2i(K1∗ P1 + K2∗ P2 ) (M1 − zF1 )−1 P1 + (M2 − zF2 )−1 P2 −2i(M1 − zF1 )−1 K1 JK2∗ (M2 − zF2 )−1 P2 (K1 + K2 )J = I − 2iK1∗ (M1 − zF1 )−1 K1 J − 2iK2∗ (M2 − zF2 )−1 K2 J + (2i)2 K1∗ (M1 − zF1 )−1 K1 JK2∗ (M2 − zF2 )−1 K2 J = [I − 2iK1∗ (M1 − zF1 )−1 K1 J][I − 2iK2∗ (M2 − zF2 )−1 K2 J] = WΘF1 (z)WΘF2 (z).

When F1 = I and F2 = I, the above results reduce to the multiplication Theorem 7.3.2 for L-systems. Let ΘF be an F -system of the form (12.17). Assume that H1 is a closed linear subspace of H, which is invariant with respect to M and F . Let P1 be the orthogonal projection from H onto H1 . Consider the aggregate ΘF1 deﬁned by M1 F1 K1 J ΘF1 = , (12.31) H1 E

462

Chapter 12. Non-canonical Systems

where M1 = M H1 ,

K1 = P1 K,

F1 = F H1 .

(12.32)

Since H1 is invariant under M and F , M1 ∈ [H1 , H1 ], and F1 is an orthoprojection. Moreover, Im M1 = K1 JK1∗ , and hence (12.31) is an F1 -system. The F1 -system ΘF1 is called the projection of ΘF onto the invariant subspace H1 , and is denoted by: ΘF1 = prH1 ΘF . Let P2 be the orthogonal projection from H onto H2 = H H1 . Deﬁne the aggregate ΘF2 M2 F2 K2 J ΘF2 = , (12.33) H H1 E where M2 = P2 M H2 ,

K2 = P2 K,

F2 = F H2 .

(12.34)

Since H2 = H H1 is invariant under F2 , then F2 is an orthoprojection. Moreover, Im M2 = K2 JK2∗ , so that (12.33) is an F2 -system. The F2 -system ΘF2 is called the projection of ΘF onto the orthogonal complement H2 = H H1 of the invariant subspace H1 ; it is denoted by ΘF2 = prHH1 ΘF . Theorem 12.2.3. Let ΘF be an F -system of the form (12.17). Assume that H1 is a closed linear subspace of H, which is invariant under M and F . Then ΘF is the coupling of the F1 -system (12.31) and the F2 -system (12.33), i.e., the projections of ΘF onto the invariant subspace H1 and its orthogonal complement H2 = HH1 , respectively. Moreover, WΘ (z) = WΘF1 (z)WΘF2 (z).

(12.35)

Proof. Consider the operators deﬁned in (12.32) and (12.34). Since H1 is invariant under M and F , it is clear that H2 = H H1 is invariant under M ∗ and F . Therefore one obtains M = P1 M P1 + P2 M P2 + P1 M P2 + P2 M P1 = P1 M P1 + P2 M P2 + P1 M P2 = M1 P1 + M2 P2 + P1 M P2 = M1 P1 + M2 P2 + 2iP1 Im M P2 = M1 P1 + M2 P2 + 2iP1 KJK ∗ P2 = M1 P1 + M2 P2 + 2iK1 JK2∗ , where the identities K1∗ = K ∗ P1 and K2∗ = K ∗ P2 have been used. Moreover, it is clear that F = F1 P1 + F2 P2 and K = K1 + K2 . Hence, ΘF has the factorization ΘF1 ΘF2 , where ΘF1 and ΘF2 are given by (12.31) and (12.33). The identity (12.35) is now a consequence of Theorem 12.2.2. Theorem 12.2.4. Each constant J-unitary operator W on a ﬁnite-dimensional Hilbert space E can be realized as a transfer function of some F -system of the form (12.17).

12.2. Multiplication theorems for F -systems

463

Proof. Assume that (−1) belongs to the resolvent set of the J-unitary operator W , and deﬁne Q = i[W − I][W + I]−1 J, so that

Q∗ = −iJ[W ∗ + I]−1 [W ∗ − I].

The operator Q is selfadjoint, since W ∗ JW − J = 0 implies Q − Q∗ = i[W − I][W + I]−1 J + iJ [W ∗ + I]−1 [W ∗ − I] = iJ [W ∗ + I]−1 {(W ∗ + I)J(W − I) + (W ∗ − I)J(W + I)} [W + I]−1 J = 2iJ [W ∗ + I]−1 (W ∗ JW − J)[W + I]−1 J = 0. Now assume also that 1 ∈ ρ(W ), so that Q is invertible. Deﬁne the Hilbert space H by H = E and let K : E → E = H be any bounded and boundedly invertible operator. Deﬁne the operator M by M = KQ−1 (I + iQJ)K ∗ , so that Im M = KJK ∗ . Obviously, the aggregate KQ−1 (I + iQJ)K ∗ 0 Θ0 = E

K

J , E

is an F -system with F = 0. By means of I + iQJ = 2[W + I]−1 , one obtains WΘ0 (z) = I − 2iK ∗ (M − zF )−1 KJ = I − 2iK ∗ M −1 KJ = I − 2iK ∗ (K ∗ )−1 (I + iQJ)−1 QK −1 KJ = I − 2i(I + iQJ)−1 QJ = (I + iQJ)−1 [(I + iQJ) − 2iQJ] = (I + iQJ)−1 (I − iQJ) = W. Hence, the theorem has been shown for the case that ±1 ∈ ρ(W ). Now consider the case of an arbitrary J-unitary operator W acting in a Hilbert space E. Since E is ﬁnite-dimensional, it is easy to see that W can be represented in the form W = W1 W2 , where Wj is a J-unitary operator in E and ±1 ∈ ρ(Wj ), j = 1, 2. By the previous case, each of the operators W1 and W2 can be realized as transfer operator-valued functions of two F -systems ΘF1 and ΘF2 , respectively, i.e., WΘF1 (z) = W1 ,

WΘF2 (z) = W2 .

Consider the coupling ΘF = ΘF1 ΘF2 of these operator systems, and apply the multiplication theorem. Then WΘF (z) = WΘF1 (z)WΘF2 (z) = W1 W2 = W. This completes the proof.

464

Chapter 12. Non-canonical Systems Theorem 12.2.2 and Theorem 12.2.4 imply the following result.

Corollary 12.2.5. Let ΘF be an F -system of the form (12.17). Let U and V be F , such that constant J-unitary operators on E. Then there exists an F -system Θ its transfer function takes the form WΘ F (z) = U WΘF (z)V. Example. As an instructive example, consider H = L2[0,a] with a > 1, E = C, and the operators D ∈ [H, H], F ∈ [H, H], and K ∈ [E, H] deﬁned by (Df )(t) = t f (t), (Kξ)(t) = ξ,

(F f )(t) = χ[0,1] (t),

t ∈ [0, a],

f ∈ L2[0,a] ,

ξ ∈ C = E.

Here χ[0,1] (t) denotes the characteristic function of the closed interval [0, 1]. Deﬁne the operator M by a ∗ (M f )(t) = (Df )(t) + i(KK f )(t) = tf (t) + i f (t)dt. 0 ∗

Then Im M = KJK with J = 1 and the aggregate M F K I ΘF = L2[0,a] C is an F -system. Since Re M = D, the function VΘF (z) in (12.18) is given by VΘ (z) = K ∗ (Re M − zF )−1 K = K ∗ (D − zF )−1 K a 1 −1 = ((D − zF ) 1, 1)L2 = dt [0,a] t − zχ [0,1] (t) 0 1 6 a 5 dt dt 1 = + = log 1 − + log(a) . t z 0 t−z 1 By the formula (12.22) we obtain the transfer function WΘ (z) =

1 − i[log(1 − 1/z) + log(a)] . 1 + i[log(1 − 1/z) + log(a)]

12.3 Realizations in the case of a compactly supported measure Let A be a self-adjoint operator in a Hilbert space H, and let L be a closed linear subspace of H. We recall that when A is bounded, the subspace L is said to be invariant under A if AL ⊂ L. When A is possibly unbounded, the subspace L is said to be invariant under A if for some point z ∈ C \ R, (A − zI)−1 L ⊂ L.

(12.36)

12.3. Realizations in the case of a compactly supported measure

465

Lemma 12.3.1. Let L be a subspace of H. If (12.36) holds for some z ∈ C \ R, then L reduces A = A∗ : A = A1 ⊕ A2 ,

Aj = A∗j in Hj , j = 1, 2,

H1 = L,

H2 = H L.

(12.37)

Proof. By means of the resolvent operator of A one can describe the graph of A for each z ∈ ρ(A) as follows: A = { {(A − zI)−1 h, (I + z(A − zI)−1 )h} : h ∈ H }. By assumption (A − zI)−1 L ⊂ L, and therefore also (A − z¯I)−1 L⊥ ⊂ L⊥ . This gives rise to the following restrictions of A in H1 and H2 , respectively: A1 = { {(A − zI)−1 h, (I + z(A − zI)−1 )h} : h ∈ L }, A2 = { {(A − z¯I)−1 h, (I + z¯(A − z¯I)−1 )h} : h ∈ H L }. − Now Aj as a restriction of A is symmetric, say, with defect numbers (n+ j , nj ) in Hj , j = 1, 2. Since A1 ⊕ A2 = A and A is self-adjoint one concludes that + − − (n+ 1 + n2 , n1 + n2 ) = (0, 0). Consequently, A1 and A2 are (graphs of) self-adjoint operators.

Lemma 12.3.2. Let D be a bounded self-adjoint operator in a Hilbert space H and let F be an orthogonal projector in H with dim ker(F ) < ∞. Then the resolvent set ρ(D, F ) is non-empty if and only if ker(D) ∩ ker(F ) = {0},

(12.38)

in which case C \ R ⊂ ρ(D, F ). Proof. Let H = Ran(F ) ⊕ ker(F ) and let D = (Dij )2i,j=1 and F = (Fij )2i,j=1 be decomposed accordingly, so that F11 = I and Fij = 0 otherwise. Then z ∈ ρ(D, F ) if and only if the operator ∗ D22 − D12 (D11 − z)−1 D12

is boundedly invertible. Since dim ker(F ) < ∞, this is equivalent to ∗ ker D22 − D12 (D11 − z)−1 D12 = {0}. ∗ Assume that D22 f = D12 (D11 − z)−1 D12 f for some f = (I − F )g. Then the resolvent identity applied to (D11 − zI)−1 yields ∗ (z − z¯) D12 (D11 − z¯I)−1 (D11 − zI)−1 D12 f, f ker(F ) = 0.

For z ∈ C \ R, this implies (D11 − zI)−1 D12 f = 0 and thus D12 f = 0. But then also D22 f = 0, i.e., f ∈ ker(D) ∩ ker(F ). The converse statement is obvious.

466

Chapter 12. Non-canonical Systems

Lemma 12.3.3. Let D be a bounded self-adjoint operator and F be an orthogonal projection with dim ker(F ) < ∞ in a Hilbert space H. Let also ρ(D, F ) = ∅ and K be a bounded operator from a ﬁnite-dimensional Hilbert space E into H. Then there exists an interval [a, b] ⊂ R, such that C \ [a, b] ⊂ ρ(D, F ), and the function V (z) = K ∗ (D − zF )−1 K, z ∈ ρ(D, F ) admits the integral representation b dG(t) V (z) = Q + zL + , Q = Q∗ , L ≥ 0, (12.39) a t−z where Q is a self-adjoint operator, L is a non-negative operator, and G(t) is a non-decreasing operator-valued function in E. Proof. Deﬁne the subspaces H1 , H2 , and H3 by H1 = F H,

H2 = P (I − F )H,

H3 = (I − P )(I − F )H,

where P denotes the orthogonal projection onto ker((I − F )D (I − F )H), so that H = H1 ⊕ H2 ⊕ H3 . This gives rise to the following orthogonal operator matrix representations for D and F : D D D I 0 0 11 12 13 ∗ D22 0 0 00 . D = D12 , F = ∗ D13

0

0

0 00

Since dim ker(F ) < ∞ and ρ(D, F ) = ∅, we can apply Lemma 12.3.2 to obtain (12.38). The latter implies that D22 is invertible and therefore the following inverse exists and is bounded: −1 D11 −zI D12 ∗ H11 (z)−1 = D12 D22 (12.40) 0 0 I −1 ˆ − zI)−1 ( I −D12 D22 −1 ∗ = 0 D22 + −D−1 D12 (D ), 22

ˆ denotes the Schur complement of (Dij )2 where D i,j=1 , ˆ = D11 − D12 D−1 D21 , D 22

ˆ =D ˆ ∗. D

(12.41)

By assumption (Hij (z))2i,j=1 := (D − zF ) is boundedly invertible for z ∈ ρ(D, F ). The invertibility of H11 (z) in (12.40) now implies that also −1 ∗ ˆ U (z) = H21 (z)H11 (z)H12 (z) = D13 (D − zI)−1 D13 ,

(12.42)

ˆ in (12.41) implies has a bounded inverse for z ∈ ρ(D, F ). The boundedness of D −1 ˆ that (D−z) is holomorphic outside a compact interval [a, b] of the real line. Since the operator U (z) in (12.42) has a bounded inverse, ker(D13 ) = {0}. Moreover, ˆ = (D ˆ ij )2 the inverse of U (z) is holomorphic on C \ [a, b]. In fact, if H1 and D i,j=1 are decomposed orthogonally according to Ran(D13 ), one can write −1 −∗ U (z)−1 = D13 (T (z) − z)D13 ,

12.3. Realizations in the case of a compactly supported measure

467

where ˆ 11 − D ˆ 12 (D ˆ 22 − z)−1 D ˆ∗ . T (z) = D 12 The function T (z) is holomorphic on C \ [a, b], since the numerical ranges satisfy ˆ 22 ) ⊂ W (D). ˆ Now the analog of the inverse formula the obvious inclusion W (D in (12.40), when applied to (D − zF ) = (Hij (z))2i,j=1 , implies that also V (z) is holomorphic on C \ [a, b], i.e., C \ [a, b] ⊂ ρ(V ). Therefore, by Stieltjes’ inversion formula the measure in the Nevanlinna integral representation of V (z) is supported on [a, b], in which case this integral representation reduces to (12.39). It was shown in Lemma 12.1.5 that the function VΘF (z) associated with the F -system ΘF in (12.17) is a Herglotz-Nevanlinna function, assuming that ρ(Re M, F ) = ∅. However, more can be said when Re M and F have a “bounded non-commuting part” in the sense to be speciﬁed in the next theorem. Theorem 12.3.4. Let ΘF be an F -system of the form (12.17) such that dim ker(F ) < ∞. Assume that ρ(Re M, F ) is nonempty, and let VΘF (z) be its impedance function deﬁned by (12.18). Then the following statements hold: (i) If there is an orthogonal projection P commuting with F and (Re M − zI)−1 for some z ∈ C \ R, such that Ran(P ) ⊂ Ran(F ) and the restriction Re M ker(P ) is bounded, then VΘF (z) is an operator-valued Herglotz-Nevanlinna function with the integral representation VΘF (z) = Q + Lz +

+∞

−∞

dG(t) , t−z

z ∈ ρ(M, F ),

(12.43)

where Q = Q∗ , L ≥ 0, and G(t) is a non-decreasing operator-valued function on R satisfying (12.8). (ii) If, in particular, F (Re M − zI)−1 = (Re M − zI)−1 F for some z ∈ C \ R, then the integral representation of V (z) is of the form (12.43) with L = 0. Proof. Let P be an orthogonal projection satisfying the assumptions in (i). Then, by Lemma 12.3.1, Re M admits an orthogonal decomposition of the form (12.37), where M2 = Re M ker(P ) is bounded by the assumptions. Decomposing K = K1 ⊕ K2 according to H = Ran(P ) ⊕ ker(P ) one obtains VΘF (z) = K1∗ (M1 − zI)−1 K1 + K2∗ (M2 − zF2 )−1 K2 ,

(12.44)

where F2 = (I − P )F is an orthogonal projection in ker(P ). Now dim ker(F2 ) < ∞ and according to Lemma 12.3.3 the second summand in (12.44) has the integral representation K2∗ (M2 − zF2 )−1 K2 = Q2 + L2 z +

a

b

dG2 (t) , t−z

a, b ∈ R.

(12.45)

468

Chapter 12. Non-canonical Systems

Clearly, the ﬁrst summand in (12.44) has the integral representation +∞ +∞ dG1 (t) K1∗ (M1 − zI)−1 K1 = , (dG1 (t)x, x) < ∞, x ∈ E. (12.46) t−z −∞ −∞ Hence, (12.43) follows from (12.45) and (12.46) with Q = Q2 , L = L2 , and G(t) = G1 (t) + G2 (t). To prove (ii) observe that the decomposition Re M = M1 ⊕ M2 for Re M in Lemma 12.3.1 gives (12.44) with F2 = (I − F )F = 0. Since ρ(Re M, F ) = ∅, the operator M2 must be invertible, and therefore (Re M − zF )−1 = (M1 − zI)−1 ⊕ M2−1 , and

VΘF (z) = K1∗ (M1 − zI)−1 K1 + K2∗ M2−1 K2 .

Now one obtains (12.43) with Q = K2∗ M2−1 K2 and L = 0.

In the remaining part of this section the converse to Theorem 12.3.4 will be established. Lemma 12.3.5. Let Q and L ≥ 0 be self-adjoint operators in E, dim E < ∞. Then the following representations hold: Q = K1∗ (D1 − zF1 )−1 K1 ,

z ∈ ρ(D1 , F1 ) (= C),

(12.47)

zL = K2∗ (D2 − zF2 )−1 K2 ,

z ∈ ρ(D2 , F2 ) (= C),

(12.48)

where Kj is an injective operator from E into a Hilbert space Hj , dim Hj < ∞, Dj is a self-adjoint operator in Hj , and Fj is an orthogonal projection in Hj , j = 1, 2. Proof. The representation (12.47) is straightforward to show if Q is invertible and one can take H1 = E, K = I, D = Q−1 , F = 0. In the general case write Q = Q1 +Q2 with two invertible self-adjoint operators Q1 and Q2 . Let H1 = E ⊕E and introduce in H1 the operators −1 I Q1 0 K1 = , D1 = , F1 = 0. 0 Q−1 I 2 Then K1 is an injective operator from E into H1 = E ⊕ E, D1 is self-adjoint and invertible, and F1 = 0 is an orthogonal projection in H1 . Clearly, ρ(D1 , F1 ) = C and it is straightforward to check that K1∗ (D1 − zI)−1 K1 = Q1 + Q2 = Q. This proves (12.47). Now consider the function zL. Let H2 be the Hilbert space E ⊕ E and deﬁne in H2 the operators 1 0 iI 0 0 L2 K2 = , D2 = , F2 = , −iI 0 0 I kI

12.3. Realizations in the case of a compactly supported measure

469

where k ∈ R\{0}. Then K2 is an injective operator from E into H2 , D2 is selfadjoint and F2 is an orthogonal projection in H2 . Moreover, ρ(D2 , F2 ) = C, and it is easy to check that 1

1

K2∗ (D2 − zF2 )−1 K2 = zL + iL 2 k − ikL 2 = zL.

Theorem 12.3.6. Each Herglotz-Nevanlinna function V (z) in a Hilbert space E, n = dim E < ∞, of the form +∞ dG(t) V (z) = Q + zL + −∞ t − z where Q = Q∗ , L ≥ 0, and G(t) is a non-decreasing operator-valued function on R satisfying (12.8), can be represented in the form V (z) = K ∗ (D − zF )−1 K,

z ∈ C \ R ⊂ ρ(D, F ),

(12.49)

where K is an operator from E into H, D is a self-adjoint operator in a Hilbert space H, and F is an orthogonal projection in H with dim ker(F ) < ∞. Moreover, if L = 0, then the operators D and F can be selected such that (D − zI)−1 , z ∈ C \ R, and F commute. Proof. Consider the Hilbert space H3 = L2G (E) of measurable vector functions f = f (t) with +∞ f 2 = (dG(t)f (t), f (t)) < ∞. −∞

Deﬁne the vector functions ωj (t) ≡ ej , j = 1, . . . , n, where {ej }, j = 1, . . . , n, is an orthonormal basis in E. The integrability property (12.8) guarantees that ωj ∈ H3 . Let K3 : E → H3 be the linear operator determined by K3 ej = ωj (t), j = 1, . . . , n. Let D3 = t be the multiplication operator by the independent variable in H3 , and let F3 = I. Then dG(t) K3∗ (D3 − zF3 )−1 K3 = . R t−z Now let the Hilbert space H be deﬁned by H = H1 ⊕ H2 ⊕ H3 , where H1 and H2 are as in Lemma 12.3.5. In addition, using the operators introduced in the proof of Lemma 12.3.5, deﬁne ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ K1 D1 0 0 F1 0 0 K = ⎝K2 ⎠ , D = ⎝ 0 D2 0 ⎠ , F = ⎝ 0 F2 0 ⎠ . (12.50) K3 0 0 D3 0 0 F3 Then K is an operator from E into H, D is a self-adjoint operator and F is an orthogonal projection in H. Moreover, it is easy to check that (12.49) is satisﬁed. Observe, that if L = 0, then the deﬁnitions of K, D, and F can be reduced to K ∗ = (K1∗ , K3∗ ), D = D1 ⊕ D3 , and F = F1 ⊕ F3 , in which case the commutativity of D and F follows from F1 = 0, F3 = I.

470

Chapter 12. Non-canonical Systems

The next result shows that each Nevanlinna function of the form (12.43) can be realized via an F -system (12.17) with J = I, i.e., by means of a scattering F -system. Theorem 12.3.7. For each Herglotz-Nevanlinna function acting on a Hilbert space E, dim E < ∞, of the form

+∞

V (z) = Q + zL + −∞

dG(t) , t−z

where Q = Q∗ , L ≥ 0, and G(t) satisﬁes (12.8), there is a scattering F -system ΘF of the form (12.17) with dim ker(F ) < ∞, whose transfer function WΘF (z) is deﬁned and holomorphic on C− and such that V (z) = VΘF (z) = i[WΘF (z) − I][WΘF (z) + I]−1 . Proof. By Theorem 12.3.6 the function V (z) can be represented via an F -resolvent in the form (12.49). Let H, K, D, and F be as in Theorem 12.3.6. Deﬁne in H a dissipative operator M by M = D + iKK ∗. Then D = Re M and by construction C \ R ⊂ ρ(D, F ). Now Corollary 12.1.2 shows that ρ(M, F ) = ∅ and hence C− ⊂ ρ(M, F ), cf. Lemma 12.1.1. This gives rise to an F -system ΘF satisfying all the properties required in Deﬁnition 12.1.4. Moreover, the transfer function WΘF (z) associated to ΘF in (12.20) is holomorphic on C− . It remains to apply Lemma 12.1.7 to VΘF (z) = V (z) and WΘF (z) with z ∈ C− ⊂ ρ(M, F ) ∩ ρ(D, F ).

12.4 Deﬁnitions of NCI-systems and NCL-systems Now we can give the proper deﬁnitions for NCI-systems and NCL-systems. Deﬁnition 12.4.1. Let A˙ be a closed symmetric operator in a Hilbert space H and ˙ The system of let H+ ⊂ H ⊂ H− be the rigged Hilbert space associated with A. equations

(D − zF+ )x = Kϕ− , (12.51) ϕ+ = K ∗ x, where E is a ﬁnite-dimensional Hilbert space is called a non-canonical impedance system or NCI-system if: ˙ (i) D ∈ [H+ , H− ] is a t-self-adjoint bi-extension of a symmetric operator A; (ii) K ∈ [E, H− ]; (iii) F+ is an orthogonal projection in H+ and H; (iv) the set ρ(D, F+ , K) of all points z ∈ C where (D − zF+ )−1 exists on H + Ran(K) and (−, ·)-continuous is open.

12.4. Deﬁnitions of NCI-systems and NCL-systems

471

˙ A ∈ [H+ , H− ] be a (∗)-extension of T , K be a bounded Let T ∈ Λ(A), linear operator from a ﬁnite-dimensional Hilbert space E into H− , K ∗ ∈ [H+ , E], J = J ∗ = J −1 ∈ [E, E], and Im A = KJK ∗ . Let also F+ be an orthogonal projection in H+ and H. Consider the system of equations ⎧ dχ ⎨ iF+ dt + Aχ(t) = KJψ− (t), (12.52) ⎩ χ(0) = x ∈ H+ , ∗ ψ+ = ψ− − 2iK χ(t). Let L2[0,τ0 ] (E) be the Hilbert space of E-valued functions equipped with an inner product (12.13). We have the following lemma. Lemma 12.4.2. If for a given continuous ψ− (t) ∈ L2[0,τ0 ] (E) we have that χ(t) ∈ H+ and ψ+ (t) ∈ L2[0,τ0 ] (E) satisfy (12.52), then a system of the form (12.52) satisﬁes the metric conservation law τ τ 2F+ χ(τ )2 − 2F+ χ(0)2 = (Jψ− , ψ− )E dt − (Jψ+ , ψ+ )E dt, (12.53) 0 0 τ ∈ [0, τ0 ]. The proof of Lemma 12.4.2 follows from Lemmas 6.3.3 and 12.1.3. Given an input vector ψ− = ϕ− eizt ∈ E, we seek solutions to the system (12.52) as an output vector ψ+ = ϕ+ eizt ∈ E and a state-space vector χ(t) = xeizt ∈ H+ . Substituting the expressions for ψ± (t) and χ(t) allows us to cancel exponential terms and convert the system (12.52) to the stationary form

(A − zF+ )x = KJϕ− , z ∈ ρ(A, F+ , K), (12.54) ϕ+ = ϕ− − 2iK ∗ x, where ρ(A, F+ , K) is deﬁned below. Following the canonical case we can re-write (12.54) as an array and have the following deﬁnition. Deﬁnition 12.4.3. Let A˙ be a closed symmetric operator in a Hilbert space H and ˙ The array let H+ ⊂ H ⊂ H− be the rigged Hilbert space associated with A. A F+ K J Θ = ΘF+ = , (12.55) H+ ⊂ H ⊂ H− E where E is a ﬁnite-dimensional Hilbert space is called a non-canonical L-system or NCL-system if: ˙ (i) A ∈ [H+ , H− ] is a (∗)-extension of T ∈ Λ(A); (ii) J = J ∗ = J −1 : E → E; (iii) A − A∗ = 2iKJK ∗, where K ∈ [E, H− ]; (iv) F+ is an orthogonal projection in H+ and H;

472

Chapter 12. Non-canonical Systems

(v) the set ρ(A, F+ , K) of all points z ∈ C, where (A − zF+ )−1 exists on H + Ran(K) and (−, ·)-continuous, is open; (vi) the set ρ(Re A, F+ , K) of all points z ∈ C, where (Re A − zF+ )−1 exists on H + Ran(K) and (−, ·)-continuous, and the set ρ(A, F+ , K) ∩ ρ(Re A, F+ , K) are both open; (vii) if z ∈ ρ(A, F+ , K), then z¯ ∈ ρ(A∗ , F+ , K); if z ∈ ρ(Re A, F+ , K), then z¯ ∈ ρ(Re A, F+ , K). We say that ΘF+ is a scattering NCL-system if J = I. In this case the state-space operator A in (12.55) is dissipative: Im A ≥ 0. It is easy to see that Theorem 4.3.10 implies that any L-system can be considered as an NCL-system with F+ = I. To each NCL-system in Deﬁnition 12.4.3 one can associate a transfer function, via WΘF+ (z) = I − 2iK ∗ (A − zF+ )−1 KJ,

z ∈ ρ(A, F+ , K).

(12.56)

Lemma 12.4.4. Let ΘF+ be an NCL-system of the form (12.55). Then for all z, w ∈ ρ(A, F+ , K), WΘF+ (z)JWΘ∗ F+ (w) − J = 2i(w ¯ − z)K ∗ (A − zF+ )−1 F+ (A∗ − wF ¯ + )−1 K, WΘ∗ F+ (w)JWΘF+ (z) − J = 2i(w ¯ − z)JK ∗ (A∗ − wF ¯ + )−1 F+ (A − zF+ )−1 KJ. Proof. By the properties (iii) and (vi) in Deﬁnition 12.4.3 one has, for all z, w ∈ ρ(A, F+ , K), (A−zF+ )−1 − (A∗ − wF ¯ + )−1 =(A − zF+ )−1 [(A∗ − wF ¯ + ) − (A − zF+ )](A∗ − wF ¯ + )−1 =(z − w)(A ¯ − zF+ )−1 F+ (A∗ − wF ¯ + )−1 − 2i(A − zF+ )−1 KJK ∗ (A∗ − wF ¯ + )−1 . This identity together with (12.56) implies that WΘF+ (z)JWΘ∗ F (w) − J +

= [I − 2iK ∗ (A − zF+ )−1 KJ ]J[I + 2iJK ∗ (A∗ − wF ¯ + )−1 K] − J = 2i(w ¯ − z)K ∗ (A − zF+ )−1 F+ (A∗ − wF ¯ + )−1 K. This proves the ﬁrst equality. Likewise one proves the second identity by using ∗ (A − zF+ )−1 − (A∗ − wF ¯ + )−1 = (z − w)(A ¯ − wF ¯ + )−1 F+ (A∗ − zF+ )−1

− 2i(A∗ − wF ¯ + )−1 KJK ∗ (A − zF+ )−1 . This completes the proof.

12.4. Deﬁnitions of NCI-systems and NCL-systems

473

Lemma 12.4.4 shows that the transfer function WΘF+ (z) in (12.56) associated to an NCL-system of the form (12.55) is J-unitary on the real axis, Jexpansive in the upper half-plane, and J-contractive in the lower half-plane with z ∈ ρ(A, F+ , K). There is another function that one can associate to each NCL-system ΘF+ of the form (12.55). This is the impedance function of the form VΘF+ (z) = K ∗ (Re A − zF+ )−1 K,

z ∈ ρ(Re A, F+ , K),

(12.57)

where ρ(Re A, F+ , K) is deﬁned above. Clearly, ρ(Re A, F+ , K) is symmetric with respect to the real axis. Theorem 12.4.5. Let ΘF+ be an NCL-system of the form (12.55) and let WΘF+ (z) and VΘF+ (z) be deﬁned by (12.56) and (12.57), respectively. Then for all z, w ∈ ρ(Re A, F+ , K), the impedance function VΘF+ (z) satisﬁes VΘF+ (z) − VΘF+ (w)∗ = (z − w)K ¯ ∗ (Re A − zF+ )−1 F+ (Re A − wF ¯ + )−1 K, (12.58) VΘF+ (z) is a Herglotz-Nevanlinna function, and for each z ∈ ρ(Re A, F+ , K) ∩ ρ(A, F+ , K) the operators I + iVΘF+ (z)J and I + WΘF+ (z) are invertible. Moreover,

and

VΘF+ (z) = i[WΘF+ (z) + I]−1 [WΘF+ (z) − I]J

(12.59)

WΘF+ (z) = [I + iVΘF+ (z)J]−1 [I − iVΘF+ (z)J].

(12.60)

Proof. For each z, w ∈ ρ(Re A, F+ , K) one has (Re A − zF+ )−1 − (Re A − wF ¯ + )−1 = (z − w)(Re ¯ A − zF+ )−1 F+ (Re A − wF ¯ + )−1 . In view of (12.57) this relation implies (12.58). Clearly, VΘF+ (z)∗ = VΘF+ (¯ z ). Moreover, it follows from (12.58) and the Deﬁnition 12.4.3 that VΘF+ (z) is an operator-valued Herglotz-Nevanlinna function. The following identity for z ∈ ρ(A, F+ , K) ∩ ρ(Re A, F+ , K), (Re A − zF+ )−1 − (A − zF+ )−1 = i(A − zF+ )−1 Im A(Re A − zF+ )−1 , leads to K ∗ (Re A − zF+ )−1 K−K ∗ (A − zF+ )−1 K = iK ∗ (A − zF+ )−1 KJK ∗ (Re A − zF+ )−1 K.

474

Chapter 12. Non-canonical Systems

Now in view of (12.56) and (12.57), we have 2VΘF+ (z) + i(I − WΘF+ (z))J = (I − WΘF+ (z))VΘF+ (z), or equivalently, that [I + WΘF+ (z)][I + iVΘF+ (z)J] = 2I.

(12.61)

Similarly, the identity (Re A − zF+ )−1 − (A − zF+ )−1 = i(Re A − zF+ )−1 Im A(A − zF+ )−1 , with z ∈ ρ(A, F+ , K) ∩ ρ(Re A, F+ , K) leads to [I + iVΘF+ (z)J][I + WΘF+ (z)] = 2I.

(12.62)

The equalities (12.61) and (12.62) show that the operators are invertible and consequently one obtains (12.59) and (12.60).

12.5 NCI realizations of Herglotz-Nevanlinna functions The realization of Herglotz-Nevanlinna functions has been obtained for various subclasses. In this section earlier realizations are combined to present a general realization of an arbitrary Herglotz-Nevanlinna function by an NCI-system. Lemma 12.5.1. Let Q be a self-adjoint operator in a ﬁnite-dimensional Hilbert space E. Then V (z) ≡ Q admits a representation of the form V (z) = K ∗ (D − zF+ )−1 K,

z ∈ ρ(D, F ),

where K is an invertible mapping from E into a Hilbert space H, D is a bounded self-adjoint operator in H, and F+ is an orthogonal projection in H whose kernel ker(F+ ) is ﬁnite-dimensional. Proof. First assume that Q is invertible. Let H = E, let K be any invertible mapping from E onto H, and let D = KQ−1 K ∗ . Then D is a bounded self-adjoint operator in H. Clearly, V (z) = K ∗ (D − zF )−1 K with F = 0, an orthogonal projection in H. In the general case, Q can be written as the sum of two invertible self-adjoint operators Q = Q(1) + Q(2) (for example, Q(1) = Q − εI and Q(2) = εI, where ε is a real number), so that Q(1) = K (1)∗ (D(1) − zF (1) )−1 K (1) ,

Q(2) = K (2)∗ (D(2) − zF (2) )−1 K (2) ,

where K (i) is an operator from E into a Hilbert space H(i) = E, D(i) is a bounded self-adjoint operator in H(i) , and F (i) = 0 is an orthogonal projection in H(i) ,

12.5. NCI realizations of Herglotz-Nevanlinna functions

475

i = 1, 2. (Note that since K (i) is an arbitrary operator from E into H(i) = E it may as well be chosen as K (i) = I). Deﬁne (1) (1) (1) K D 0 F 0 (1) (2) H=H ⊕H , K = ,D= , F+ = . K (2) 0 D (2) 0 F (2) Then K is an operator from E into the Hilbert space H, D is a bounded self-adjoint operator, and F+ = 0 is an orthogonal projection in H. Moreover, ∗

∗

Q = Q(1) + Q(2) = K (1) (D(1) − zF (1) )−1 K (1) + K (2) (D(2) − zF (2) )−1 K (2) = K ∗ (D − zF+ )−1 K,

which proves the lemma.

Herglotz-Nevanlinna functions of the form (12.6) which belong to the class N (R) can be realized using Theorem 6.5.4. By means of Lemma 12.5.1 these realizations can be extended to Herglotz-Nevanlinna functions of the form (12.6) with L = 0. Theorem 12.5.2. Let V (z) be a Herglotz-Nevanlinna function in a ﬁnite-dimensional Hilbert space E, with the integral representation 1 t V (z) = Q + − dG(t), (12.63) t−z 1 + t2 R where Q = Q∗ and G(t) is a non-decreasing operator-valued function on R. Then V (z) admits a realization of the form V (z) = K ∗ (D − zF+ )−1 K,

z ∈ C \ R ⊂ ρ(D, F+ , K),

(12.64)

˙ where D ∈ [H+ , H− ] is a t-self-adjoint bi-extension of a symmetric operator A, H+ ⊂ H ⊂ H− is a rigged Hilbert space, F+ is an orthogonal projection in H+ and H, K is an operator from E into H− , K ∗ ∈ [H+ , E]. Moreover, the operators D and F+ can be selected such that the following commutativity condition holds: F− D = DF+ ,

F− = R−1 F+ R ∈ [H− , H− ],

(12.65)

where R is the Riesz-Berezansky operator deﬁned in (2.1). Proof. According to Theorem 6.5.4 each operator-valued Herglotz-Nevanlinna function of the form (12.63) admits a realization as the impedance function of an L-system of the form (6.36), i.e., V (z) = K ∗ (Re A − zI)−1 K = i[WΘ (z) + I]−1 [WΘ (z) − I], where WΘ (z) is the transfer function (6.44), if and only if the following condition holds: t Qf = dG(t)f, (12.66) 1 + t2 R

476

Chapter 12. Non-canonical Systems

for every vector f ∈ E, such that (dG(t)f, f )E < ∞.

(12.67)

R

To prove the existence of the representation (12.64) for Herglotz-Nevanlinna functions V (z) which do not satisfy the condition (12.66), the realization result in Lemma 12.5.1 will be used. Denote by E1 the subspace of vectors f ∈ E with the property (12.67) and let E2 = E E1 , so that E = E1 ⊕ E2 . Rewrite Q in the block matrix form Q11 Q12 Q= , Qij = PEi Q Ej , j = 1, 2, Q21 Q22 and let G(t) = (Gij (t))2i,j=1 be decomposed accordingly. Observe, that by (12.66), (12.67) the integrals t t S11 := dG (t), S := dG12 (t), 11 12 2 1 + t 1 + t2 R R are convergent in the strong topology. Let the self-adjoint operator S be deﬁned by S11 S12 S= , (12.68) ∗ S12 C where C = C ∗ is arbitrary. Now rewrite V (z) = V1 (z) + V2 (z) with 1 t V1 (z) = Q − S, V2 (z) = S + − dG(t). t−z 1 + t2 R Clearly, for every f ∈ E1 the equality Sf = R

(12.69)

t dG(t)f, 1 + t2

holds. Consequently, V2 (z) ∈ N (R) and thus by Theorem 6.5.2 admits the representation V2 (z) = K2∗ (Re A(2) − zI)−1 K2 , where K2 : E → H−2 , K2∗ : H+2 → E with H+2 ⊂ H2 ⊂ H−2 a rigged Hilbert space, and where Re A(2) is a t-self-adjoint bi-extension of a symmetric operator A˙ 2 . The operator K2 has the properties Ran(K2 ) ⊂ Ran(A(2) − zI), (A(2) − zI)−1 K2 ∈ [E, H+ ],

Ran(K2 ) ⊂ Ran(Re A(2) − zI), (Re A(2) − zI)−1 K2 ∈ [E, H+ ];

(12.70)

for further details, see the proof of Theorems 6.5.1–6.5.2. Now, by Lemma 12.5.1 the function V1 (z) admits the representation V1 (z) = K1∗ (D1 − zF+,1 )−1 K1 ,

12.5. NCI realizations of Herglotz-Nevanlinna functions

477

where D1 = D1∗ and F+,1 = 0 are acting on a ﬁnite-dimensional Hilbert space H1 = E ⊕ E. Recall from Lemma 12.5.1 that (1) (1) D1 0 K1 D1 = K1 = (2) , (2) , 0 D1 K1 (i)

(1)

(2)

where K1 : E → E, i = 1, 2, and D1 , D1 are deﬁned by means of the decomposition of Q − S into the sum of two invertible self-adjoint operators Q − S = (Q(1) − S (1) ) + (Q(2) − S (2) ). Then

(i)

(i)∗

D1 = K1

(i)

(Q(i) − S (i) )−1 K1 ,

i = 1, 2.

To obtain the realization (12.64) for V (z) in (12.63), introduce the triplet of Hilbert spaces (1)

(1)

H+ := E ⊕ E ⊕ H+2 ⊂ E ⊕ E ⊕ H2 ⊂ E ⊕ E ⊕ H−2 := H− ,

(12.71)

i.e., a rigged Hilbert space corresponding to the block representation of symmetric operator D1 ⊕ A˙ 2 in H(1) := H1 ⊕ H2 (where H1 = E ⊕ E). Also introduce the operators ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ (1) (1) D1 0 0 0 0 0 K1 ⎜ ⎟ ⎜ ⎟ (2) D=⎝ 0 D1 0 ⎠ , F+ = ⎝0 0 0⎠ , K = ⎝K1(2) ⎠ . (12.72) 0 0 I K2 0 0 Re A(2) It is straightforward to check that (1)∗

V (z) = V1 (z) + V2 (z) = K1 (2)∗

+ K1

(2)

(1)

(1)

(D1 − zF+,1 )−1 K1 (2)

(D1 − zF+,1 )−1 K1 + K2∗ (Re A(2) − zI)−1 K2

(12.73)

= K ∗ (D − zF+ )−1 K. By the construction, A˙ 2 ⊂ Aˆ2 = Aˆ∗2 ⊂ Re A(2) , where Aˆ2 is the quasi-kernel of Re A(2) and A˙ 2 is a symmetric operator in H2 . Moreover, D as an operator in (1) (1) [H+ , H− ] is self-adjoint, i.e., D = D∗ , and since D1 0 D1 0 ˆ D= ⊂ = D, 0 Re A(2) 0 Aˆ2 ˆ the operator D is a t-self-adjoint bi-extension of the symand A˙ = D1 ⊕ A˙ 2 ⊂ D, ˙ metric operator A in H1 ⊕ H2 . It is easy to see that with operators in (12.72) one obtains the representation (12.64) for V (z) in (12.63) and the system constructed with these operators satisﬁes the Deﬁnition 12.4.1 of a NCI-system. Finally, from (12.72) one obtains F− D = DF+ , where F+ and F− are connected as in (12.65). This completes the proof of the theorem.

478

Chapter 12. Non-canonical Systems

The general impedance realization result for Herglotz-Nevanlinna functions of the form (12.7) is going to be built on Theorem 12.5.2 and the following representation for the linear term in integral representation (12.6). Lemma 12.5.3. Let L be a non-negative operator in a ﬁnite-dimensional Hilbert space E. Then it admits a realization of the form ˆ ∗P K ˆ = K ∗ (D3 − zF3 )−1 K3 , zL = z K 3

(12.74)

where D3 is a self-adjoint operator in a Hilbert space H3 , P is the orthogonal projection onto Ran(L), and K3 is an invertible operator from E into H3 . Proof. Since L ≥ 0, there is a unique non-negative square root L1/2 ≥ 0 of L with ker(L1/2 ) = ker(L),

Ran(L1/2 ) = Ran(L).

ˆ in E by Deﬁne the operator K

ˆ = Ku

u, L1/2 u,

u ∈ ker(L), u ∈ Ran(L).

(12.75)

ˆ is invertible and L1/2 = P K, ˆ where P denotes the orthogonal projection Then K onto Ran(L). Deﬁne ˆ PK 0 iI 0 0 H3 = E ⊕ E, K3 = , D = , F = . (12.76) 3 +,3 ˆ −iI 0 0 I K Then K3 is an invertible operator from E into H3 , D3 is a bounded self-adjoint operator, and F+,3 is an orthogonal projection in H3 . Moreover, ˆ ∗P K ˆ = K3∗ (D3 − zF+,3 )−1 K3 . V3 (z) = zL = z K

This completes the proof.

The general realization result for Herglotz-Nevanlinna functions of the form (12.7) is now being obtained by combining the earlier realizations. Theorem 12.5.4. Let V (z) be an operator-valued Herglotz-Nevanlinna function in a ﬁnite-dimensional Hilbert space E with the integral representation 1 t V (z) = Q + zL + − 2 dG(t), (12.77) t−z t +1 R where Q = Q∗ , L ≥ 0, and G(t) is a non-decreasing, non-negative operator-valued function on R. Then V (z) admits a realization of the form V (z) = K ∗ (D − zF+ )−1 K

(12.78)

where D ∈ [H+ , H− ] is a t-self-adjoint bi-extension in a rigged Hilbert space H+ ⊂ H ⊂ H− , F+ is an orthogonal projection in H+ and H, and K ∈ [E, H− ].

12.5. NCI realizations of Herglotz-Nevanlinna functions Proof. Let us deﬁne the functions 1 t V1 (z) = Q + − dG(t), t−z 1 + t2 R

479

V2 (z) = zL.

According to Theorem 12.5.2 the function V1 (z) has a representation V1 (z) = K1∗ (D1 − zF+,1 )−1 K1 , where D1 , K1 and F+,1 are given by the formula (12.72). We recall that D1 is a (1) (1) t-self-adjoint bi-extension in a rigged Hilbert space H− ⊂ H(1) ⊂ H+ given by (1) (12.71), F+,1 is an orthogonal projection in H+ , and K1 is an invertible mapping (1) from E into H− . According to Lemma 12.5.3 the functions V2 (z) has a realization of the form (12.74) with components H3 , D3 , K3 and F+,3 described by (12.76). Now the ﬁnal result follows by introducing the rigged Hilbert space H3 ⊕ (1) (1) H+ ⊂ H3 ⊕ H(1) ⊂ H3 ⊕ H− and the operators D=

D3 0

0 D1

∈

(1) [H3 ⊕ H+ , H3

(1) ⊕ H− ],

F+,3 F+ = 0

0 F+,1

,

K=

K3 . K1

It is straightforward to check that with these operators one obtains the representation (12.78) for V (z) in (12.77) and the system constructed with these operators satisﬁes the Deﬁnition 12.4.1 of a NCI-system. For the sake of clarity an extended version for the NCI-realization in the proof of Theorem 12.5.4 is provided. The rigged Hilbert space used is E ⊕ E ⊕ E ⊕ E ⊕ H+2 ⊂ E ⊕ E ⊕ E ⊕ E ⊕ H2 ⊂ E ⊕ E ⊕ E ⊕ E ⊕ H−2 , (12.79) and the operators are given by ⎛

0 ⎜−iI ⎜ D=⎜ ⎜ 0 ⎝ 0 0 ⎛ 0 0 ⎜0 I ⎜ F+ = ⎜ ⎜0 0 ⎝0 0 0 0

iI 0 0 0 0 0 0 0 0 0

0 0 (1) D1 0 0 0 0 0 0 0

⎞ 0 0 ⎟ ⎟ 0 ⎟ ⎟, 0 ⎠ Re A(2) ⎛ ⎞ ˆ PK ⎜ K ⎟ ⎜ ˆ ⎟ ⎜ (1) ⎟ K = ⎜ K1 ⎟ . ⎜ (2) ⎟ ⎝ K1 ⎠ K2

0 0 0 (2) D1 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟, 0⎠ I

All the operators in (12.80) are deﬁned above.

(12.80)

480

Chapter 12. Non-canonical Systems

12.6 Realization by NCL-systems In the NCI-realization results of Theorem 12.5.2 and Theorem 12.5.4, the realizations are presented in terms of the operators in (12.64) and (12.78), respectively. It remains to identify the Herglotz-Nevanlinna functions as impedance functions of appropriate NCL-systems. Theorem 12.6.1. Let V (z) be a Herglotz-Nevanlinna function acting on a ﬁnitedimensional Hilbert space E with the integral representation 1 t V (z) = Q + − dG(t), t−z 1 + t2 R where Q = Q∗ and G(t) is a non-decreasing, non-negative operator-valued function on R. Then the function V (z) can be realized as the impedance function of a scattering NCL-system ΘF+ deﬁned in (12.55), that is V (z) = i[WΘF+ (z) + I]−1 [WΘF+ (z) − I],

(12.81)

where WΘF+ (z) is the transfer function of ΘF+ . Proof. By Theorem 12.5.2 the function V (z) can be represented in the form V (z) = K ∗ (D − zF+ )−1 K, where K, D, and F+ are as in (12.72) corresponding to the decomposition V (z) = V1 (z) + V2 (z),

where V1 (z) = Q − G,

V2 (z) = G +

R

1 t − t − z 1 + t2

dG(t),

with a self-adjoint operator G of the form (12.68). With the notation used in the proof of Theorem 12.5.2 one may rewrite V1 (z) and V2 (z) as in (12.73) with (1)

D1 = (Q − G − εI)−1 ,

(2)

D1 = (εI)−1 ,

(1)

K1

= λIE ,

(2)

K1

= IE ,

(12.82)

Re A(2) ∈ [H+2 , H−2 ], A(2) is a (∗)-extension of an operator T2 ∈ Λ(A˙ 2 ) for which (−i) ∈ ρ(T2 ), cf. Section 6.5. The remaining operators are deﬁned in (12.72) and ε, λ > 0. Recall that K2 and the resolvents (A(2) − zI)−1 , (Re A(2) − zI)−1 satisfy the properties (12.70). To construct an NCL-system of the form (12.55) we introduce an operator A by A = D + iKK ∗ ∈ [H+ , H− ], where K, D, and F+ are deﬁned in (12.72). Then the block-matrix form of A is ⎛ ⎞ (1) D1 + iλ2 I iλI iλK2∗ ⎜ ⎟ (2) A=⎝ iλI D1 + iI iK2∗ ⎠ . iλK2 iK2 A(2)

12.6. Realization by NCL-systems

Let ΘF+ =

481

A H+ ⊂ H ⊂ H−

F+

K

I , E

where the rigged Hilbert triplet H+ ⊂ H ⊂ H− is deﬁned in (12.71), i.e., E ⊕ E ⊕ H+2 ⊂ E ⊕ E ⊕ H2 ⊂ E ⊕ E ⊕ H−2 . It remains to show that all the properties in Deﬁnition 12.4.3 are satisﬁed. For this purpose, consider the equation (A − zF+ )x = (D + iKK ∗ )x − zF+ x = Kg, ⎛

or

(1)

D + iλ2 I ⎜ 1 ⎝ iλI iλK2

iλI (2) D1 + iI iK2

g ∈ E,

⎞⎛ ⎞ ⎛ ⎞ iλK2∗ x1 λg ⎟ iK2∗ ⎠ ⎝x2 ⎠ = ⎝ g ⎠ . (2) x3 K2 g A − zI

Using the decomposition of the operators and taking into account that A(2) = Re A(2) + iK2 K2∗ , we can re-write this equation in form of the system ⎧ (1) ⎪ ⎨ D1 x1 + iλ2 Ix1 + iλIx2 + iλK2∗ x3 = λg, (2) (12.83) D1 x2 + iλIx1 + iIx2 + iK2∗ x3 = g, ⎪ ⎩ (A(2) − zI)x + iλK x + iK x = K g, 3 2 1 2 2 2 ⎧ ⎪ ⎨

or

⎪ ⎩

1 (1) D x1 + iλIx1 + iIx2 + iK2∗ x3 = g, λ 1 (2) D1 x2 + iλIx1 + iIx2 + iK2∗ x3 = g, (A(2) − zI)x3 + iλK2 x1 + iK2 x2 = K2 g.

In a neighborhood of (−i) the resolvent (A(2) − zI)−1 is well deﬁned so that by (12.70) the third equation in (12.83) can be solved for x3 : x3 = (A(2) − zI)−1 K2 g − i(A(2) − zI)−1 K2 (λx1 + x2 ).

(12.84)

Substituting (12.84) into the ﬁrst line of the system yields 1 (1) D x1 + iI(λx1 + x2 ) + K2∗ (A(2) − zI)−1 K2 (λx1 + x2 ) λ 1 = g − iK2∗ (A(2) − zI)−1 K2 g, Denoting the right-hand side by C and using (12.56) we get C = g − iK2∗ (A(2) − zI)−1 K2 g =

1 [I + WΘ2 (z)] g. 2

Then 1 (1) D x1 + iI(λx1 + x2 ) + K2∗ (A(2) − zI)−1 K2 (λx1 + x2 ) = C. λ 1

482

Chapter 12. Non-canonical Systems

Multiplying both sides by 2i and using (12.56) one more time yields 2i (1) D x1 − [I + WΘ2 (z)] (λx1 + x2 ) = 2iC. λ 1 Writing for further convenience B = [I + WΘ2 (z)] we obtain 2i (1) D x1 − λBx1 − Bx2 = 2iC λ 1 or

2i (1) D x1 − λBx1 − 2iC = Bx2 . (12.85) λ 1 Now we subtract the second equation of the system from the ﬁrst and obtain (1) (2) D1 x1 = λD1 x2 , or (1) (2) λ(D1 )−1 D1 x2 = x1 . (12.86) Applying (12.86) to (12.85) we get (2)

(1)

(2)

2iD1 x2 − Bλ2 (D1 )−1 D1 x2 − Bx2 = 2iC, and using (12.82)

or

2i 1 Ix2 − B[λ2 (Q − G − εI) + I]x2 = 2iC, ε ε 2iI − [I + WΘ2 (z)] [λ2 (Q − G) + ε(1 − λ2 )I] x2 = 2iεC.

(12.87)

Choosing λ and ε suﬃciently small the operator on the left-hand side of (12.87) can be made invertible for z = −i. Using an invertibility criteria from [89] we deduce that (12.87) is also invertible in a neighborhood of (−i). Consequently, the system (12.83) has a unique solution and (A − zF+ )−1 K is well deﬁned in a neighborhood of (−i). In order to show that the remaining properties in Deﬁnition 12.4.3 are satis˙ such that A is a (∗)-extension of T . ﬁed we need to present an operator T ∈ Λ(A) To construct T we note ﬁrst that (A − zF+ )H+ ⊃ H for some z in a neighborhood of (−i). This can be conﬁrmed by considering the equation (A − zF+)x = g,

x ∈ H+ ,

(12.88)

and showing that it has a unique solution for every g ∈ H. The procedure then is reduced to solving the system (12.83) with an arbitrary right-hand side g ∈ H. Following the steps for solving (12.83) we conclude that the system (12.88) has a unique solution. Similarly one shows that (A∗ − zF+ )H+ ⊃ H. Using the technique developed in Section 4.5 we can conclude that operators (A + iF+ )−1 and (A∗ − iF+ )−1 are (−, ·)-continuous. Deﬁne T = A, Dom(T ) = (A + iF+ )−1 H, T1 = A∗ , Dom(T1 ) = (A∗ − iF+ )−1 H.

(12.89)

12.6. Realization by NCL-systems

483

One can see that both Dom(T ) and Dom(T1 ) are dense in H while operator T is closed in H. Indeed, assuming that there is a vector φ ∈ H that is (·)-orthogonal to Dom(T ) and representing φ = (A∗ − iF+ )ψ, we can immediately get φ = 0. It is also easy to see that T1 = T ∗ . Thus, operator T deﬁned by (12.89) ﬁts the deﬁnition of a (∗)-extension for operator A. Property (vi) of Deﬁnition 12.4.3 follows from Theorem 12.5.4 and the fact that Re A = D. Consequently all the properties for an NCL-system Θ in Deﬁnition 12.4.3 are fulﬁlled with the operators and spaces deﬁned above. Now we present the principal result of this section. Theorem 12.6.2. Let V (z) be an operator-valued Herglotz-Nevanlinna function in a ﬁnite-dimensional Hilbert space E with the integral representation 1 t V (z) = Q + zL + − 2 dG(t), (12.90) t−z t +1 R where Q = Q∗ , L ≥ 0 is an invertible operator, and G(t) is a non-decreasing, nonnegative operator-valued function on R. Then V (z) can be realized as the impedance function of a scattering NCL-system ΘF+ of the form (12.55). Proof. Decompose the function V (z) as follows: 1 t V1 (z) = Q + − 2 dG(t) t−z t +1 R

and

V2 (z) = zL,

and use the earlier realizations for each of these functions. By Theorem 12.6.1 the function V1 (z) can be represented by V1 (z) = i[WΘF1,+ (z) + I]−1 [WΘF1,+ (z) − I], where WΘF1,+ (z) is an operator-valued transfer function of some scattering NCLsystem, WΘF1,+ (z) = I − 2iK1∗ (A1 − zF1,+ )−1 K1 , (12.91) A1 = D1 + iK1 K1∗ maps H+1 continuously into H−1 , D1 is a self-adjoint biextension, and D1 ∈ [H+1 , H−1 ], K1 ∈ [E, H−1 ]. Following the proof of Theorem 12.5.4, the function V2 (z) can be represented in the form V2 (z) = K2∗ (D2 − zF2,+ )−1 K2 , where

D2 =

0 −iI

iI , 0

F2,+

0 = 0

0 , I

K2 =

: PK : K

,

(12.92)

: are as in (12.75), so that K2 is an operator from E into H2 = E ⊕ E. and P and K Introduce the triplet H+1 ⊕ H2 ⊂ H1 ⊕ H2 ⊂ H−1 ⊕ H2 , and consider the operator A = D + iKK ∗,

(12.93)

484

Chapter 12. Non-canonical Systems

from H+1 ⊕ H2 into H−1 ⊕ H2 given by the block form D1 0 K1 ∗ K1 K2∗ A= +i 0 D2 K2 A1 iK1 K2∗ = . iK2 K1∗ A2

(12.94)

Here A2 = D2 + iK2 K2∗ . It will be shown that the equation (A − zF+ )x = Kh,

with F+ =

F1,+ 0

0 F2,+

,

h ∈ E,

(12.95)

K1 K= , K2

has always a unique solution x ∈ H+1 ⊕ H2 and (A − zF+ )−1 K ∈ [E, H+1 ⊕ H2 ]. Taking into account (12.94), the equation (12.95) can be written as the system

(A1 − zF1,+ )x1 + iK1 K2∗ x2 = K1 h, (12.96) (A2 − zF2,+ )x2 + iK2 K1∗ x1 = K2 h, where

A1 = D1 + iK1 K1∗ ,

A2 = D2 + iK2 K2∗ .

By Theorem 12.6.1 it follows that (A1 − zF1,+ )−1 K1 ∈ [E, H+1 ]. Therefore, the ﬁrst equation in (12.96) gives x1 = (A1 − zF1,+ )−1 K1 h − i(A1 − zF1,+ )−1 K1 K2∗ x2 .

(12.97)

Now substituting x1 in the second equation in (12.96) yields (A2 − zF2,+)x2 + K2 K1∗ (A1 − zF1,+ )−1 K1 K2∗ x2 = K2 h − iK2 K1∗ (A1 − zF1,+ )−1 K1 h.

(12.98)

Taking into account (12.91), (12.92), and (12.94) the identity (12.98) leads to (2) 2iI − (D2 − zF+ )−1 K2 [I + WΘF1,+ (z)]K2∗ x2 (2) (1) = 2i(D2 − zF+ )−1 K2 h − iK2 K1∗ (A1 − zF+ )−1 K1 h . It will be shown that the operator-function on the left-hand side, in front of x2 , is invertible. First by straightforward calculations one obtains zI iI (2) (D2 − zF+ )−1 = ∈ [E ⊕ E, E ⊕ E]. −iI 0

12.6. Realization by NCL-systems

485

The operator-function M (z) deﬁned by M (z) = I + WΘF1,+ (z) ∈ [E, E] is invertible according to Theorem 12.4.5. It follows from (12.92) that 1/2 : L M (z) PK K2 M (z) = M (z) = ∈ [E ⊕ E, E ⊕ E], : (z) : KM K and that K2 M (z)K2∗

=

L1/2 M (z)L1/2 : (z)L1/2 KM

For any 2 × 2 block-matrix Z=

: L1/2 M (z)K : : KM (z)K a c

∈ [E ⊕ E, E ⊕ E].

b d

with entries in E, deﬁne the operator-function a b 2 + zai − c (2) N (z) = 2iI − (D2 − zF+ )−1 =i c d a

zbi − d . b+2

ˆ = L1/2 . Now Since the operator L > 0 is invertible, therefore ker(L) = {0} and K choose 1/2 L M (z)L1/2 L1/2 M (z)L1/2 A0 A0 Z= = , A0 A0 L1/2 M (z)L1/2 L1/2 M (z)L1/2 where A0 = A0 (z) = L1/2 M (z)L1/2 . Note that the operator-function A0 is invert−1/2 ible and that A−1 M (z)−1 L−1/2 . With this choice of Z one obtains 0 =L 2I + ziA0 − A0 ziA0 − A0 N = N (z) = i . A0 A0 + 2I To investigate the invertibility of N consider the system 2I + ziA0 − A0 ziA0 − A0 x1 0 = , A0 A0 + 2I x2 0 or

(2I + ziA0 − A0 )x1 + (ziA0 − A0 )x2 = 0, A0 x1 + (A0 + 2)x2 = 0.

Solving the second equation for x1 yields

2x1 + ziA0 x1 − A0 x1 + ziA0 x2 − A0 x2 = 0, x1 = −x2 − 2A−1 0 x2 .

486

Chapter 12. Non-canonical Systems

Substituting x1 into the ﬁrst equation gives (2A−1 0 + zi)x2 = 0, or equivalently, A0 x2 =

2i x2 . z

(12.99)

Recall that A0 = A0 (z) = L1/2 M (z)L1/2 = L1/2 [I + WΘ1 (z)]L1/2 . For every z in the lower half-plane WΘ1 (z) is a contraction (see (6.46)) and thus A0 (z) ≤ 2L. This means that for every z (Im z < 0) the norm of the lefthand side of (12.99) is bounded while the norm of the right-hand side can be made unboundedly large by letting z → 0 along the imaginary axis. This leads to a conclusion that x2 = 0 and then also x1 = 0. Hence, N = N (z) is invertible. Consequently, (2) 2iI − (D2 − zF+ )−1 K2 [I + WΘ1 (z)]K2∗ is invertible and x2 depends continuously on h ∈ E in (12.98), while (12.97) shows that x1 depends continuously on h ∈ E. Now we will follow the steps taken in the proof of Theorem 12.6.1 to show that the remaining properties in Deﬁnition 12.4.3 are satisﬁed. We introduce an ˙ such that A is a (∗)-extension of T . To construct T we note operator T ∈ Λ(A) ﬁrst that (A − zF+ )H+ ⊃ H for some z in a neighborhood of (−i). This can be conﬁrmed by considering the equation (A − zF+)x = g,

x ∈ H+ ,

(12.100)

and showing that it has a unique solution for every g ∈ H. The procedure then is reduced to solving the system (12.96) with an arbitrary right-hand side g ∈ H. Inspecting the steps of solving (12.96) we conclude that the system (12.100) has a unique solution. Similarly one shows that (A∗ − zF+)H+ ⊃ H. Once again relying on Section 4.5 we can conclude that operators (A + iF+ )−1 and (A∗ − iF+ )−1 are (−, ·)-continuous and deﬁne T = A, Dom(T ) = (A + iF+ )−1 H, T1 = A∗ , Dom(T1 ) = (A∗ − iF+ )−1 H.

(12.101)

Using arguments similar to the proof of Theorem 12.6.1, we note that both Dom(T ) and Dom(T1 ) are dense in H while operator T is closed in H. It is also easy to see that T1 = T ∗ . Thus, operator T deﬁned by (12.101) ﬁts the construction of (∗)-extension A. Property (vi) of Deﬁnition 12.4.3 follows from Theorem 12.5.4 and the fact that Re A = D. Therefore, the array A K F+ I ΘF+ = H+1 ⊕ H2 ⊂ H1 ⊕ H2 ⊂ H−1 ⊕ H2 E is an NCL-system and V (z) admits the realizations V (z) = K ∗ (D − zF+ )−1 K = i[WΘF+ (z) + I]−1 [WΘF+ (z) − I]. This completes the proof.

12.7. Minimal NCL-realization

487

12.7 Minimal NCL-realization Recall from Section 6.6 that a symmetric operator A˙ in a Hilbert space H is called a prime operator if there exists no reducing non-trivial invariant subspace on which it induces a self-adjoint operator. An NCL-system of the form (12.55) is called F+ 1 1 minimal if there are no nontrivial reducing invariant subspaces H1 = H+ , (H+ is a (+)-subspace of Ran(F+ )) of H where the symmetric operator A˙ induces a self-adjoint operator. Here the closure is taken with respect to the (·)-metric. In the case that F+ = I this deﬁnition coincides with the deﬁnition of minimality for L-systems given in Section 6.5. Theorem 12.7.1. Let the operator-valued Herglotz-Nevanlinna function V (z) be realized as the impedance function of a scattering NCL-system ΘF+ of the form (12.55) Then this NCL-system can be reduced to an F+ -minimal NCL-system that also realizes V (z). Proof. Let the operator-valued Herglotz-Nevanlinna function V (z) be realized in the form (12.81) by an NCL-system of the type (12.55). Assume that its symmetric 1 , (H1 is a (+)-subspace operator A˙ has a reducing invariant subspace H1 = H+ + of Ran(F+ )) on which it generates a self-adjoint operator A1 . Then there is the (·, ·)-orthogonal decomposition H = H0 ⊕ H 1 ,

A˙ = A˙ 0 ⊕ A1 ,

(12.102)

where A˙ 0 is an operator induced by A˙ on H0 . The identity (12.102) shows that the adjoint of A˙ in H admits the orthogonal decomposition A˙ ∗ = A˙ ∗0 ⊕ A1 . Now consider operators T ⊃ A˙ and T ∗ ⊃ A˙ as in the deﬁnition of the system ΘF+ . It is easy to see that both T and T ∗ admit the (·, ·)-orthogonal decompositions T = T0 ⊕ A1 ,

and

T ∗ = T0∗ ⊕ A1 ,

˙ the identity where T0 ⊃ A˙ 0 and T0∗ ⊃ A˙ 0 . Since T ∈ Λ(A), A˙ 0 ⊕ A1 = T ∩ T ∗ = (T0 ∩ T0∗ ) ⊕ A1 , holds and (−i) is a regular point of T = T0 ⊕ A1 or, equivalently, (−i) is a regular point of T0 . (Here we use the notation T ∩ T ∗ to denote the maximal symmetric part of the operators T and T ∗ .) The above relation shows that T0 ∈ Λ(A˙ 0 ). Clearly, 0 1 H+ = H+ ⊕ H+ = Dom(A∗0 ) ⊕ Dom(A1 ). This decomposition remains valid in the sense of (+)-orthogonality. Indeed, if 0 1 f 0 ∈ H+ and f1 ∈ H+ = Dom(A1 ), then by considering the adjoint of A˙ : H0 (= ˙ dom A) → H as a mapping from H into H0 one obtains (f0 , f1 )+ = (f0 , f1 ) + (A˙ ∗ f0 , A˙ ∗ f1 ) = (f0 , f1 ) + (A˙ ∗0 f0 , A1 f1 ) = 0 + 0 = 0.

488

Chapter 12. Non-canonical Systems

Consequently, the inclusions H+ ⊂ H ⊂ H− can be rewritten in the decomposed forms 0 1 0 1 H+ ⊕ H+ ⊂ H0 ⊕ H1 ⊂ H− ⊕ H− 0 0 1 = H+ ⊕ Dom(A1 ) ⊂ H0 ⊕ H1 ⊂ H− ⊕ H− .

Now let A ∈ [H+ , H− ] be the (∗)-extension of A˙ in the deﬁnition of the system ΘF+ . Then A admits the decomposition A = A0 ⊕ A1 and A∗ = A∗0 ⊕ A1 . Since A1 is self-adjoint in H1 , A0 is a (∗)-extension of T0 . Moreover, A − A∗ (A0 ⊕ A1 ) − (A∗0 ⊕ A1 ) A0 − A∗0 A1 − A1 = = ⊕ 2i 2i 2i 2i A0 − A∗0 = ⊕ O, 2i

(12.103)

where O stands for the zero operator. Then (12.103) implies that KJK ∗ = K0 JK0∗ ⊕ O. 0 Let P+0 be the orthogonal projection operator of H+ onto H+ and set K = K0 ⊕O. ∗ ∗ 0 Then K = K0 P+ , since for all f ∈ E, g ∈ H+ one has

(Kf, g) = (K0 f, g) = (K0 f, g0 + g1 ) = (K0 f, g0 ) + (K0 f, g1 ) = (K0 f, g0 ) = (f, K0∗ g0 ) = (f, K0∗ P+0 g). 1 Since H+ is a closed subspace of Ran(F+ ), P+0 commutes with F+ and there0 0 fore F+ := F+ P+0 deﬁnes an orthogonal projection in H+ . Now, let h ∈ E, 0 1 z ∈ ρ(A, F+ , K), and φ = φ0 + φ1 ∈ H+ = H+ ⊕ H+ be such that

(A − zF+)φ = Kh. Since K = K0 ⊕ O, the previous identity is equivalent to (A0 ⊕ A1 − zF+ )(φ0 + φ1 ) = (K0 ⊕ O)h. Since F+ φ1 = φ1 and P+0 commutes with F+ , this yields (A0 − zF+0 )φ0 = K0 h,

(A1 − zI)φ1 = 0.

It follows from the previous equations that z ∈ ρ(A1 ) because z ∈ ρ(A, F+ , K). Thus, ρ(A, F+ , K) ⊂ ρ(A0 , F+0 , K0 ) and hence φ0 = (A0 − zF+0 )−1 K0 h. On the other hand, φ0 = φ = (A − zF+ )−1 Kh and therefore for all h ∈ E one obtains (A − zF+ )−1 Kh = (A0 − zF+0 )−1 K0 h and K ∗ (A − zF+ )−1 Kh = K0∗ (A0 − zF+0 )−1 K0 h.

12.7. Minimal NCL-realization

489

This means that the transfer functions of the system ΘF+ in (12.55) and of the system A0 F+0 K0 J 0 ΘF+ = 0 0 H+ ⊂ H 0 ⊂ H− E coincide. Therefore, the system ΘF+ in (12.55) can be reduced to an F+ -minimal system of the same form such that the corresponding transfer and, consequently, impedance functions coincide. This completes the proof of the theorem. The deﬁnition of minimality can be extended to NCI-systems in the same manner. Moreover, an NCL-system of the form (12.3)

(A − zF+ )x = KJϕ− , ϕ+ = ϕ− − 2iK ∗ x, and a NCI-system of the form (12.1)

(Re A − zF+ )x = Kϕ− , ϕ+ = K ∗ x, where Re A is the real part of A, are minimal (or non-minimal) simultaneously. For the NCI-systems constructed in Section 12.4 the minimality can be characterized as follows. Theorem 12.7.2. The realization of the operator-valued Herglotz-Nevanlinna function V (z) constructed in Theorem 12.5.4 is minimal if and only if the symmetric part A˙ 2 of Re A(2) deﬁned by (12.80) is prime. Proof. Assume that the system constructed in Theorem 12.5.4 is not minimal. Let 1 H1 (with H+ ⊂ Ran(F+ )) be a reducing invariant subspace from Theorem 12.7.1 on which A˙ generates a self-adjoint operator A1 . Then D = D0 ⊕ A1 and it follows from the block representations of D and F+ in (12.80) that H1 is necessarily a 1 subspace of H2 in (12.79) while H+ is a subspace of H+2 . To see this let us describe Ran(F+ ) ﬁrst. According to (12.79) H+ ⊂ H ⊂ H− = E 4 ⊕ H+2 ⊂ E 4 ⊕ H2 ⊂ E 4 ⊕ H−2 , where E 4 = E ⊕ E ⊕ E ⊕ E. Hence every vector x ∈ H+ can be written as x1 x= By (12.80),

x2 x3 x4 x5

F+ x =

, where x1 , x2 , x3 , x4 ∈ E, x5 ∈ H+2 .

0 x2 0 0 x5

,

and

D(F+ x) =

ix2 0 0 0 Re A(2) x5

.

490

Chapter 12. Non-canonical Systems

1 This means that x ∈ H+ ⊂ Ran(F+ ) only if x2 = 0. Therefore the only possibility for a reducing invariant subspace H1 to exist, is to be a subspace of H2 while 1 1 H+ is a subspace of H+2 . This proves the claim H+ ⊂ H+2 . Consequently, H1 is a reducing invariant subspace for the symmetric operator A2 , in which case the operator A2 is not prime. Conversely, if the symmetric operator A2 is not prime, then a reducing invariant subspace on which A2 generates a self-adjoint operator is automatically a reducing invariant subspace for the operator A˙ which belongs to Ran(F+ ). This completes the proof.

Finally, Theorem 12.7.2 implies that a realization of an arbitrary operatorvalued Herglotz-Nevanlinna function in Theorem 12.5.4 can be provided by a minimal NCI-system.

12.8 Examples and non-canonical system interpolation Example. Consider the Herglotz-Nevanlinna function i V (z) = 1 + z − i tanh zl , z ∈ C \ R, 2

(12.104)

where l > 0. We are going to construct an NCL-system ΘF+ whose impedance function coincides with V (z). Let the diﬀerential operator T2 in H2 = L2[0,l] be given by 1 dx , i dt Dom(T2 ) = { x(t) ∈ H2 : x(t) − abs. cont., x (t) ∈ H2 , x(0) = 0 } , T2 x =

with the adjoint 1 dx , i dt ∗ Dom(T2 ) = { x(t) ∈ H2 : x(t) − abs. cont., x (t) ∈ H2 , x(l) = 0 } . T2∗ x =

Let A˙ 2 be the symmetric operator deﬁned by 1 dx A˙ 2 x = , i dt Dom(A˙ 2 ) = { x(t) ∈ H2 : x(t) − abs. cont., x (t) ∈ H2 , x(0) = x(l) = 0 } , (12.105) with the adjoint 1 dx A˙ ∗2 x = , i dt

Dom(A˙ ∗2 ) = { x(t) ∈ H2 : x(t) − abs. cont., x (t) ∈ H2 } .

12.8. Examples and non-canonical system interpolation

491

Then H+ = Dom(A˙ ∗2 ) = W21 is a Sobolev space with the scalar product (x, y)+ =

l

x(t)y(t) dt + 0

l

x (t)y (t) dt.

0

Now consider the rigged Hilbert space W21 ⊂ L2[0,l] ⊂ (W21 )− and the operators A2 x =

1 dx 1 dx + ix(0) [δ(x − l) − δ(x)] , A∗2 x = + ix(l) [δ(x − l) − δ(x)] , i dt i dt

where x(t) ∈ W21 and δ(x), δ(x − l) are delta-functions in (W21 )− . Deﬁne the operator K2 by 1 K2 c = c · √ [δ(x − l) − δ(x)], c ∈ C1 , 2 so that 1 1 K2∗ x = x, √ [δ(x − l) − δ(x)] = √ [x(l) − x(0)], x(t) ∈ W21 . 2 2 ∗ Let D1 = K1 Q−1 1 K1 = 1, where Q = 1, and K1 = 1, K1 : C → C. Following (12.76) deﬁne 1 0 iI 0 0 H3 = C ⊕ C, K3 = , D3 = , F+,3 = . 1 −iI 0 0 I

Now the corresponding NCL-system can be constructed. According to (12.80) one has ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 i 0 0 0 0 0 0 1 ⎜−i 0 0 ⎟ ⎜0 1 0 0 ⎟ ⎜1 ⎟ 0 ⎟ , F+ = ⎜ ⎟ ⎜ ⎟ D=⎜ ⎝0 0 1 ⎝0 0 0 0 ⎠ , K = ⎝ 1 ⎠ , 0 ⎠ 0 0 0 Re A2 0 0 0 I K2 and it follows from (12.93)-(12.94) that ⎛ i ⎜ 0 A = D + iKK ∗ = ⎜ ⎝ i iK2

2i i i iK2

i i 1+i iK2

⎞ iK2∗ iK2∗ ⎟ ⎟. iK2∗ ⎠ A2

Consequently, the corresponding NCL-system is given by A K F+ ΘF+ = C3 ⊕ W21 ⊂ C3 ⊕ L2[0,l] ⊂ C3 ⊕ (W21 )−

I , C

(12.106)

where C3 = C ⊕ C ⊕ C and all the operators are described above. Since the symmetric operator A˙ 2 deﬁned in (12.105) is prime, then according to Theorem

492

Chapter 12. Non-canonical Systems

12.7.2, the system (12.106) is prime. Thus, V (z) in (12.104) are the impedance functions of the F+ -minimal NCL-system (12.106). The transfer function of this system is 1 − i − zi − tanh 2i zl 2 − i(1 + eizl )(z + 1) . WΘF+ (z) = izl = 2e + i(1 + eizl )(z + 1) 1 + i + zi + tanh 2i zl Example. Consider the Herglotz-Nevanlinna function −i tanh (πiz) 1 0 1 0 V (z) = +z + 0 0 1 0 1

0

1−z z2 −z−1

.

(12.107)

An explicit NCL-system ΘF+ will be constructed so that V (z) ≡ VΘF+ (z). Let T21 be a diﬀerential operator H2 = L2[0,2π] given by 1 dx , i dt Dom(T21 ) = { x(t) ∈ H2 : x(t) − abs. cont., x (t) ∈ H2 , x(0) = 0 } , T21x =

with the adjoint 1 dx , i dt ∗ Dom(T21 ) = { x(t) ∈ H2 : x(t) − abs. cont., x (t) ∈ H2 , x(2π) = 0 } . ∗ T21 x=

Let A˙ 21 be the symmetric operator deﬁned by 1 dx A˙ 21 x = , i dt Dom(A˙ 21 ) = { x(t) ∈ H2 : x(t) − abs. cont., x (t) ∈ H2 , x(0) = x(2π) = 0 } , (12.108) with the adjoint 1 dx A˙ ∗21 x = , i dt

Dom(A˙ ∗21 ) = { x(t) ∈ H2 : x(t) − abs. cont., x (t) ∈ H2 } .

Then H+ = Dom(A˙ ∗21 ) = W21 is a Sobolev space with the scalar product 2π 2π (x, y)+ = x(t)y(t) dt + x (t)y (t) dt. 0

0

Consider the rigged Hilbert space W21 ⊂ L2[0,2π] ⊂ (W21 )− and the operators 1 dx + ix(0) [δ(x − 2π) − δ(x)] , i dt 1 dx A∗21 x = + ix(2π) [δ(x − 2π) − δ(x)] , i dt A21 x =

12.8. Examples and non-canonical system interpolation

493

where x(t) ∈ W21 and δ(x), δ(x − 2π) are delta-functions in (W21 )− . Deﬁne the operator K21 by 1 K21 c = c · √ [δ(x − 2π) − δ(x)], 2

so that ∗ K21 x

=

1 x, √ [δ(x − 2π) − δ(x)] 2

for x(t) ∈ W21 . Deﬁne

T22 =

and set A2 =

A21 0

c ∈ C1 ,

1 = √ [x(2π) − x(0)], 2

i i 1 and K22 = , −i 1 0 0 T22

and K2 =

Now let D1 = K1 Q−1 K1∗ = I2 , where Following (12.76) deﬁne ⎛ ⎞ ⎛ 1 0 ⎜1⎟ ⎜0 4 ⎟ ⎜ H3 = C , K3 = ⎜ ⎝1⎠ , D3 = ⎝−i 1 0

K21 0

(12.109)

0 . K22

(12.110)

Q = I2 , and K1 = I2 , K1 : C2 → C2 . 0 i 0 0 0 0 −i 0

⎞ 0 i⎟ ⎟, 0⎠ 0

F+,3

⎛ 0 ⎜0 =⎜ ⎝0 0

0 0 0 0

0 0 1 0

⎞ 0 0⎟ ⎟. 0⎠ 1

Now the corresponding NCL-system will be constructed. According to (12.80) one has ⎛ ⎞ ⎛ ⎞ .. .. ⎛ ⎞ D . 0 F . 0 K3 ⎜ 3 ⎟ ⎜ +,3 ⎟ ⎜· · · · · · ⎟ ⎜ · · · · · · · · ·⎟ ⎜· · ·⎟ · · · ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ 0 ⎟ 0 0⎟ D=⎜ ⎜ 0 D1 ⎟ , F+ = ⎜ 0 ⎟ , K = ⎜K1 ⎟ , (12.111) ⎜· · · · · · ⎟ ⎜ · · · · · · · · ·⎟ ⎝· · ·⎠ · · · ⎝ ⎠ ⎝ ⎠ .. .. K2 0 . Re A2 0 . I and it follows from (12.93)-(12.94) that A = D + iKK ∗. Consequently, the corresponding NCL-system is given by A K F+ ΘF+ = C6 ⊕ W21 ⊂ C6 ⊕ L2[0,2π] ⊂ C6 ⊕ (W21 )−

(12.112) I , C2

(12.113)

where all the operators are described above. The transfer function of this system is given by 1−i−zi−tanh(πiz) 0 1+i+zi+tanh(πiz) WΘF+ (z) = . z 3 +iz 2 −(3+i)z−i 0 −z 3 +iz 2 +(3−i)z−i

494

Chapter 12. Non-canonical Systems

It is easy to see that the maximal symmetric part of the operator T22 in (12.109) is a non-densely-deﬁned operator

4 0 i 0 A˙ 22 = , Dom(A˙ 22 ) = : c∈C . (12.114) −i 1 c Consequently, the symmetric operator A˙ 2 deﬁned by A2 in (12.110), D in (12.111), and A in (12.112) is given by ⎛ 1 dx ⎞ 0 0 i dt A˙ 2 = ⎝ 0 0 i⎠ , 0 −i 1 ⎧⎛ ⎫ ⎞ ⎨ x(t) ⎬ Dom(A˙ 2 ) = ⎝ 0 ⎠ : x(t), x (t) ∈ H2 , x(0) = x(2π) = 0, c ∈ C . ⎩ ⎭ c Hence, this operator A˙ 2 does not have nontrivial reducing invariant subspaces on which it induces self-adjoint operators. Thus, the NCL-system in (12.113) is an F+ -minimal realization of the function V (z) in (12.107). Example. Let us consider the interpolation data z1 = i,

z2 = 2i,

v1 = 1 + i,

i v2 = 1 + . 2

(12.115)

We construct Pick matrices P and Q of the form (11.38) and get 1 12 1 1 P= , Q = . 1 1 1 1 2 4 Clearly det(P ) = 0 and P is not invertible. Thus we can not apply Theorem 11.6.1 to deduce the existence of a canonical interpolation system that is a solution to the Nevanlinna-Pick interpolation problem for the data (12.115). Moreover, such a canonical system does not exist. To show that we assume the contrary, i.e., that there is a system of the form (11.66) with state-space H, dim H = 2, and the statespace operator T having one-dimensional imaginary part Im T = (·, g)g. Without loss of generality we can assume that T is prime. Then, by Lemma 11.1.3, Re T has a cyclic vector g. Consequently, we can apply Theorem 11.7.1 that gives us invertibility of P. Thus, we have arrived at a contradiction. v¯1 1−i 2 Let ϕ be a vector in C of the form (11.40), i.e., ϕ = = . v¯2 1 − 2i Consider the function V (z) = ((Q − zP)−1 ϕ, ϕ). (12.116) It was shown in the proof of Theorem 11.6.1 that V (zk ) = vk , (k = 1, 2). Taking into account that 1 − z 1 − z2 Q − zP = , 1 − z2 1 − z4

12.8. Examples and non-canonical system interpolation

495

we compute V (z) using (12.116) to obtain 1 V (z) = ((Q − zP)−1 ϕ, ϕ) = 1 − . z According to Theorem 12.3.6 the function V (z) can be realized as the impedance function of a non-canonical scattering F-system ΘF of the form (12.17). The algorithm of reconstruction of ΘF is described in the proof of Theorem 12.3.6 while the elements of ΘF are given by (12.50). This yields 1 1 0 0 0 ∗ K= , K = (1, 1), D= , F = , 1 0 1 0 1 1+i i ∗ and M = D + iKK = . Finally, i i M F K 1 ΘF = , (12.117) C2 C and

z − iz + i . z + iz − i Therefore we have shown that even though the canonical system interpolation solution to the Nevanlinna-Pick problem for the data (12.115) does not exist, there is a non-canonical system solution provided by the system ΘF in (12.117) and its impedance function VΘF (z) = 1 − 1/z. Example. Consider the interpolation data WΘF (z) = I − 2iK ∗ (M − zF )−1 K =

z1 = i,

z2 = 2i,

v1 = 2i,

v2 =

5i , 2

and construct Pick matrices P and Q of the form (11.38) 2 32 0 −i P= , Q= . 3 5 i 0 2 4

(12.118)

(12.119)

Unlike Example 12.8, det(P ) = 0 and P is invertible. Hence we can apply The of the form (11.47) orem 11.6.1 which provides us with a canonical system Θ with the state space C2P of dimension 2 that is a solution to the Nevanlinna-Pick interpolation problem for the data (12.118). Applying (11.40) we get v¯1 −2i 5 −6 5i −1 ϕ= = , g = P ϕ = ϕ = . (12.120) v¯2 −6 8 −8i − 5i 2 of Θ are deﬁned by (11.41) and The state-space and channel operators T and K (11.42) using P, Q, and g from (12.119)–(12.120) as T = P−1 Q + i(· , g)C2P g ,

= c · g, Kc

c ∈ C.

496

Chapter 12. Non-canonical Systems

Furthermore, 10z , (12.121) −4 is the impedance function of the canonical interpolation system solution I T K = Θ . (12.122) C2P C −1 VΘ ϕ, ϕ) = − (z) = ((Q − zP)

z2

On the other hand, it is easy to see that our interpolation data (12.118) also satisﬁes another function 1 V (z) = z − . (12.123) z Clearly, the function V (z) does not belong to the class N (R) and hence can not be realized as an impedance function of any canonical L-system. However, V (z) can be realized as an impedance function of a non-canonical scattering F-system ΘF of the form (12.17). We follow the algorithm of reconstruction described in the proof of Theorem 12.3.6 and obtain the elements of ΘF via (12.50). This yields ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 i 0 0 0 0 K = ⎝1⎠ , K ∗ = (1, 1, 1), D = ⎝−i 0 0⎠ , F = ⎝0 1 0⎠ , 1 0 0 0 0 0 1 ⎛ ⎞ i 2i i and M = D + iKK ∗ = ⎝ 0 i i ⎠ . Finally, i i i ΘF = and

M C3

F

K

1 , C

(12.124)

z 2 + iz − 1 , VΘF (z) = z − 1/z. −z 2 + iz − 1 Therefore we have shown that there are two system solutions to the Nevanlinna-Pick problem for the data (12.118): one canonical and one non-canonical. of the form (12.122) and has the The canonical one is given by the system Θ 2 impedance function VΘ (z) = −10z/(z − 4) in (12.121). The non-canonical system solution is provided by the system ΘF in (12.124) and its impedance function VΘF (z) = z − 1/z. WΘF (z) =

Notes and Comments Chapter 1 Theorem 1.1.2 for the case of a densely-deﬁned symmetric operator belongs to von Neumann and for a non-densely-deﬁned operator to Krasnoselski˘i [169]. Theorems from Sections 1.1–1.4 were stated and proved by von Neumann. Deﬁnition of the aperture of two linear manifolds in Hilbert space due to Sz.-Nagy [242] and Kre˘in– Krasnoselski˘i [170] (see also [3]). Formula (1.21) was used by Kre˘in–Krasnoselski˘i– Milman [171] to determine the aperture of two linear manifolds in Banach space. Semi-deﬁciency subspaces and semi-deﬁciency indices have been introduced by Krasnoselski˘i [169]. Theorem 1.5.3 belongs to Krasnoselski˘i [169]. Symmetric and self-adjoint extensions of non-densely-deﬁned symmetric operators were considered for the ﬁrst time in [169]. Admissible isometric extensions were introduced and used by Krasnoselski˘i [169] to obtain the analogues of the ﬁrst and second von Neumann’s formulas in order to get parametrization of symmetric and self-adjoint extensions. Theorems 1.7.4 and 1.7.5 belong to M. Naimark [206] and Krasnoselski˘i [169]. The results of Section 1.7 were published in [169].

Chapter 2 Rigged Hilbert spaces were ﬁrst studied by Leray [186] and Lax [184], but even earlier Krein [172] had studied the case when H+ is a Banach space and H− is its dual. The theory of rigged Hilbert spaces (triplets of Hilbert spaces) used in the text was developed by Berezansky in [81]. Another approach to the rigged Hilbert spaces via so-called complimentary Hilbert spaces belongs to de Branges [85]. ˘ Theorem 2.1.1 is taken from the paper by Tsekanovski˘i and Smuljan [264]. The construction of an operator generated rigging was ﬁrst carried out independently ˘ by Tsekanovski˘i [247] and Smuljan [264]. Theorem 2.1.2 is due to Douglas [112]. The analogues of von Neumann formulas (2.14) and (2.15) as well as Theorem 2.3.1 are contained in the papers by Arlinski˘ı and Tsekanovski˘i [44], [45], and ˘ of Smuljan [233]. Theorem 2.3.3 is due to Arlinski˘ı and Tsekanovski˘i [44]. The decomposition of the class of all closed symmetric operators into the classes of regular and singular operators was found by Krasnoselskii [169]. Theorem 2.4.1 Y. Arlinskii et al., Conservative Realizations of Herglotz–Nevanlinna Functions, Operator Theory: Advances and Applications, DOI 10.1007/978-3-7643-9996-2, © Springer Basel AG 2011

497

498

Notes and Comments

is taken from [264]. The concept of an O-operator was introduced and studied ˘ independently by Arlinski˘ı and Tsekanovski˘i [45] and by Smuljan [233]. Theorem 2.4.3 was ﬁrst published in [264]. The notion of the minimal angle between two subspaces in a Hilbert space, investigations of its properties and results very close to Lemma 2.5.2 can be found in [171], [179], and [139]. Theorems 2.5.1–2.5.4 are ˘ due to Smuljan [233], [264], while Theorems 2.5.5–2.5.6 are due to Arlinski˘ı, ˘ Tsekanovski˘i, and Smuljan [44], [264]. The deﬁnition of quasi-kernel belongs to ˘ Tsekanovskii [254], [264].

Chapter 3 The concept of bi-extensions, including self-adjoint bi-extensions was introduced by Tsekanovski˘i [251]. Most of the contents of Sections 3.1–3.2 belong to Tseka˘ novski˘i and Smuljan, and can be found in [264] and [265]. The content of Section 3.3 is due to Arlinski˘ı and Tsekanovski˘i [44], [45]. The results of Section 3.4 were obtained for the ﬁrst time by Tsekanovski˘i [251], [252], [253] for denselydeﬁned symmetric operators with ﬁnite and equal defect indices, and for a denselydeﬁned symmetric operator with arbitrary and equal defect indices by Okunski˘i and Tsekanovski˘i, and published in [265]. Theorem 3.4.11 and the description of self-adjoint extensions of a symmetric densely-deﬁned operator with ﬁnite equal deﬁciency indices in terms of abstract boundary conditions (3.40) belong to Tsekanovski˘i [251]. Analogues to Theorem 3.4.11 result for quasi-self-adjoint extensions and the corresponding abstract boundary conditions can be found in [253]. The t-self-adjoint bi-extensions of a densely-deﬁned symmetric operator have been used in the abstract problem of singular perturbation of a self-adjoint operator by Arlinski˘ı and Tsekanovski˘i in [52].

Chapter 4 ˘ The results of this chapter are due to Arlinski˘ı, Tsekanovski˘i, and Smuljan [264], [265]. Lemma 4.1.4 and Theorems 4.1.9, 4.1.11 in Section 4.1 belong to Arlinski˘i and Tsekanovski˘i and can be found in [264]. Theorems 4.1.10, 4.1.12 belong to ˘ Arlinski˘i, while Theorems 4.1.5 and 4.1.13 are due to Tsekanovski˘i and Smuljan [264], [265]. In Section 4.2, Theorems 4.2.5 and 4.2.8 for symmetric operators with dense domain and ﬁnite and equal defect indices and formulas (4.23) were obtained for the ﬁrst time by Tsekanovski˘i [254], [263] while Theorem 4.2.9 belongs to Arlinski˘i [20]. In Section 4.3 the notion of (∗)-extension belongs to Tsekanovski˘i and ˘ Smuljan [264], [265], [263]. Theorems 4.3.2, 4.3.3, 4.3.10, and 4.3.11 are due to Arlinski˘ı [20]. In the case of a densely-deﬁned symmetric operator with equal defect indices, Theorems 4.3.2, 4.3.3, 4.3.5, and 4.3.7 via formulas (4.23) were obtained by Tsekanovski˘i [254], [253]. Uniqueness Theorem 4.3.9 in the densely-deﬁned case is due to Tsekanovski˘i [254], [253], while the presented proof is due to Arlinski˘i. In

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˘ Section 4.4 Theorem 4.4.2 belongs to Arlinski˘i, Tsekanovski˘i and Smuljan [264], ˘ [265], and Theorem 4.4.3 is due to Arlinskii [20]. The Uniqueness Theorem 4.4.6 for a densely-deﬁned symmetric operator with equal defect indices was obtained by Tsekanovski˘i [254], [253]; the presented proof belongs to Arlinski˘i. The results of ˘ Section 4.5 belong to Arlinski˘i, Tsekanovski˘i, and Smuljan [254], [264], [265]. The˘ ˘ ˘ orems 4.5.1–4.5.6 belong to Arlinskii, Tsekanovskii and independently to Smuljan ˘ [17], [231], [264], [265]. Theorem 4.5.11 was obtained by Smuljan [231], [233]. Theorems 4.5.12–4.5.18 were originally set up for the densely-deﬁned case with ﬁnite and equal defect indices by Tsekanovski˘i [254], [264], [265].

Chapter 5 Canonical open systems (5.6), (5.4) with a bounded main (state-space) operator and its transfer functions (5.17) were introduced by Livˇsic [191]. The metric conservation law (5.2) belongs to Livˇsic and can be found in books by LivˇsicYantsevich [193] and by Zolotarev [272]. In Operator Theory [89], [193] the array (5.6) is called the operator colligation and the expression (5.17) is called the characteristic function of this colligation. The notion of the characteristic function of a bounded linear operator on a separable Hilbert space with nuclear imaginary part and its analytical properties without colligation approach was discovered and studied for the ﬁrst time by Livˇsic [190]. Theorems 5.1.2 and 5.6 belong to Brodski˘i [89], and Theorem 5.4 is due to Livˇsic [190]; the presented proof of this theorem is due to Brodski˘i [89]. The geometrical theory of operator colligations was developed by Brodski˘i [89] and we follow this approach and terminology (principal and excess subspaces, system projections, etc.) in our considerations. Coupling of two linear bounded operators originally belongs to Livˇsic [190], [91]. On its basis Livˇsic and Potapov set up for the ﬁrst time the multiplication theorem [192] of the characteristic matrix-valued functions. The fact that the characteristic function of a bounded linear operator with nuclear imaginary part is a unitary invariant of this operator was discovered by Livˇsic [190]. The new approach to inverse problems in the theory of non-self-adjoint operators belongs to Livˇsic [190], [193], [188]. He described the class of matrix-valued functions that can be realized as characteristic functions of some bounded linear operator and set up connections with the spectrum of this operator. Theorems 5.4.1, 5.4.3, 5.5.3–5.5.8 are a colligation version of the Livˇsic approach and belong to M.Brodski˘i [89]. Theorem 5.5.1 belongs to the authors and is a colligation version of a result implicitly contained in [91]. Triangular models of linear operators with nuclear imaginary part in Hilbert spaces were constructed by Livˇsic using factorizations of characteristic (transfer) functions and are presented in [188], [189], [190], [193], [91], [272]. Theorem 5.6.4 is a special case of a more general result established by Livˇsic. Functional models for contractions were obtained by B.Sz.-Nagy and Foias [243], by de Branges and Rovnyak [86], [87].

500

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Chapter 6 ˘ Theorems 6.1.2–6.2.8 belong to Smuljan [231], [232], [233], Theorem 6.2.10 is ˘ due to Arlinskii [18]. The deﬁnition of a rigged canonical system in the case of a densely-deﬁned symmetric part of a given non-self-adjoint state-space operator with ﬁnite and equal deﬁciency indices was introduced by Tsekanovski˘i [254]. The deﬁnition of an L-system, an impedance system associated with an L-system, and simply an impedance system, belongs to the authors. The metric conservation law (6.34) for L-systems with unbounded operators is presented for the ﬁrst time and its proof is based on bi-extension and (∗)-extension theory developed in Chapters 3 and 4. Theorems 6.3.7 and 6.3.8 also belong to the authors and are being published for the ﬁrst time. Class N (R) of Herglotz-Nevanlinna functions were introduced by Belyi and Tsekanovski˘i [74], [75], but in the implicit form this class was considered earlier by Arlinski˘i [18], [19]. Matrix-valued functions of the class N (R) were also treated by Alpay and Gohberg in [7]. The conservative state-space realizations of analytic functions that maps the right half-plane into itself were considered by Ball and Staﬀans [68], [69]. The realization problems with passive systems were treated by Arov and Nudelman [56], [57], [60]. Theorems 6.4.3–6.6.1 belong to Arlinski˘i [18], Belyi and Tsekanovski˘i [74], [75]. The proof of these theorems is based on the approach considered by Tsekanovski˘i [254] and its further general development by Arlinski˘i in [18], [19] regarding inverse problems of the theory of characteristic functions of unbounded operators. A description of the Hilbert space L2G (Cn ) with matrix-valued function G(t) has been given by Kac [158] (see also [61], [207]). In the case of dim E = ∞ and operator-valued function G(t) the characterization of L2G (E) and the corresponding functional model for the symmetric operator were obtained by Gesztesy, Kalton, Makarov, and Tsekanovski˘i [135] for the case of the dense domain and by S. Malamud and M. Malamud in [199]. The functional model for a symmetric operator and its primeness was obtained by Gesztesy and Tsekanovski˘i in [137] for the case of ﬁnite and equal deﬁciency indices and in the general situation by S. Malamud and M. Malamud in [199]. Our proof of the primeness for a symmetric operator in Theorem 6.6.7 uses the approach of [199] to a ﬁnite-dimensional space E. Lemmas 6.6.4–6.6.9 belong to Tsekanovski˘i [258], [260]. Theorem 6.6.10 and Corollary 6.6.12 are due to Arlinski˘i and Tsekanovski˘i [53]. Operator colligations (open systems) and their characteristic functions (transfer functions) related to unbounded quasi-self-adjoint state space operators, were deﬁned and studied using diﬀerent methods by Strauss [241], A. Kuzhel [181], Gubreev [144], [145], and Gubreev-Kovalenko [147]. Some connections between A. Kuzhel’s approach and the (∗)-extensions are established in [146]. In Zolotarev’s papers [273] and [274] Kuzhel’s approach is applied to the study of open systems with commuting unbounded operators. Triangular models for unbounded operators were constructed by A. Kuzhel [181], Arlinski˘ı [21], and Tsekanovski˘ı [259]. Functional models for unbounded dissipative and non-dissipative operators have

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been applied by Pavlov (see [214] and references therein), by Naboko [203], [204] for spectral analysis, and by Tikhonov [244].

Chapter 7 Subclasses N0 (R), N1 (R), and N01 (R) were introduced by Belyi and Tsekanovski˘i [77]. Theorems 7.1.4–7.1.11 are due to Belyi and Tsekanovski˘i [77]. Classes Ω(R, J), Ω0 (R, J), Ω1 (R, J), Ω01 (R, J) are deﬁned by Belyi and Tsekanovski˘i [76]. In implicit form Ω0 (R, J) was considered in the inverse problem of characteristic functions of unbounded operators by Tsekanovski˘i [254] and other classes by Arlinski˘i [18]. Theorem 7.2.4 and Corollary 7.2.5 are due to Belyi and Tsekanovski˘i [76]. Theorems 7.3.2–7.3.3, 7.3.5–7.3.8 belong to Belyi and Tsekanovski˘i [76]. Coupling of two rigged canonical systems with transfer functions from the class Ω0 (R, J) was considered for the ﬁrst time by Arlinski˘i and Tsekanovski˘i [46]. Theorem 7.3.4 is due to Arlinski˘i and Tsekanovski˘i [46]. Proposition 7.4.4, Theorem 7.4.5 belongs to Arlinsk˘i and Kaplan [43]; Theorems 7.5.2 and 7.5.3 were established by Arlinski˘i. The deﬁnition of the Q-function is due to Krein and Langer [174], [175], [176]. The deﬁnition of the Weyl-Titchmarsh function via boundary triplet (boundary value spaces) belong to Derkach and Malamud [100]. In [99], [102], [104] these authors established connections between Strauss characteristic functions [241] of quasi-self-adjoint (proper) extensions of a symmetric operator and the Weyl (Weyl-Titchmarsh) functions of boundary triplets.

Chapter 8 Lemma 8.1.1, Theorem 8.1.2 are due to Arlinski˘i and Tsekanovski˘i [46], [51]. An auxiliary system of the form (8.15) was introduced in [46] and [51]. Theorems 8.2.1– 8.2.2 on the constant J-unitary factor belong to Arlinski˘i and Tsekanovski˘i [46], [51]. Theorem 8.2.3 is due to Arlinski˘i, Belyi and Tsekanovski˘i and is presented for the ﬁrst time. Transform (8.31) was introduced in the scalar case by Donoghue [111]. Theorem 8.2.4 is due to Tsekanovski˘i [255], [260]. Theorems 8.3.1 and 8.3.2 were discovered in the matrix-valued case for normalized Weyl-Titchmarsh functions by Makarov and Tsekanovski˘i. The version of the theorems presented here belongs to Makarov and the authors. The Donoghue transform for normalized Herglotz-Nevanlinna matrix-functions was studied by Gesztesy and Tsekanovski˘i [137]; the operator-valued version of the Donoghue transform has been considered and used in Krein’s resolvent formula by Gesztesy, Makarov and Tsekanovski˘i [134] and Gesztesy, Kalton, Makarov and Tsekanovski˘i [135]. The results of Section 8.4 belong to Tsekanovski˘i [247], [248], [249]. Deﬁnition of normalized canonical sysil tems belongs to Tsekanovski˘i [247], [248], [249]. Realization of e λ as transfer function belongs to Livˇsic, while realization of eilλ is due to Tsekanovski˘i [247], [248], [249].

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Notes and Comments

Chapter 9 The statements in Lemma 9.1.1 are due to Ando [13] and Krein–Ovcharenko [177]. Operator HΩ presented in formula (9.8) was introduced by Krein [172], [173], the name shorted operator for HΩ was ﬁrst used by Anderson [10]. Properties of shorted operators were studied by Anderson [10], Anderson and Trapp [11], Anderson and Duﬃn [12], Fillmore and Williams[123], Kre˘in and Ovcharenko [177], Nishio and Ando [210], Pekarev [215], Shmul’yan [230]. Theorem 9.1.3 belongs to Crandall [94]. Theorems 9.1.4–9.1.5 belong to Arsene and Geondea [61], Davis, ˘ Kahan and Weinberger [96], Smuljan and Yanovskaya [234]; diﬀerent proofs are given by Foias and Frazho [125], Dritchel and Rovnyak [114], Kolmanovich and Malamud [165], Arlinski˘i [37], Arlinski˘i, Hassi, de Snoo [40]. The extremal scextensions Aμ and AM , called rigid and soft contractive self-adjoint extensions of symmetric (Hermitian) contraction, were discovered by Krein [172], [173]; criterion (9.55) for uniqueness of sc-extensions, Proposition 9.2.6, and Theorem 9.2.4 belong to Krein [172]. Kre˘in’s approach is presented in Akhiezer and Glazman [3] and also in A. Kuzhel and S. Kuzhel [182]. The matrix form of the rigid and soft extensions Aμ and AM were obtained by Ando in [14] (see also [65], [165], [150]). Deﬁnition of a sectorial operator belongs to Kato [163]; α-cosectorial contraction, class CH (α) as well as qsc-extension were introduced by Arlinski˘i and Tsekanovski˘i [47], [49], [50], [257], [260], [25], [27]. Theorem 9.2.7 on parametrization of all α-cosectorial contractive extensions of a given symmetric contraction as well as Corollaries 9.2.14 and 9.2.15 belong to Arlinski˘i and Tsekanovski˘i [47], [49]. More general problem: extensions of the class CH (β) of a non-densely-deﬁned CH (α) sub-operator A (α ≤ β < π/2) have been studied by Arlinski˘i in [28], [31], [34], [37] and Malamud in [196], [197], [198]. The deﬁnition of extremal qscextension of symmetric contraction as well as Proposition 9.2.9 are due to Arlinski˘i and Tsekanovski˘i [50]. Proposition 9.2.19 is a qsc-extended version of the corresponding proposition of Krein and Ovˇcarenko [177] obtained for sc-extensions of symmetric contraction. The condition (in terms of defect elements) for a nondensely-deﬁned symmetric operator to be prime was considered by Langer and Textorius [183]. Theorem 9.3.1 is due to Brodski˘i [89], Theorem 9.3.2, Propositions 9.3.3–9.3.6 are due to Arlinski˘i, Hassi, de Snoo [40]. Theorems 9.4.1–9.4.2 belong to Derkach and Tsekanovski˘i [108], [257], [258]. Proposition 9.5.2 and Theorem 9.5.4 were established by Krein [172], [173]. Theorem 9.5.7 belongs to Ando and Nishio [16]. Uniqueness of nonnegative self-adjoint extensions of the operator A0 (9.100) in Example 9.5 was established using diﬀerent methods by Gesztesy, Kalton, Makarov, and Tsekanovski˘i in [135] and by Adamyan in [1]. The set of equivalent conditions on accretive operators in Section 9.5 belongs to Phillips [217], [218], [219]. Lemma 9.5.12 and Theorem 9.5.13 are due to Tsekanovski˘i [255], [256]. The Phillips-Kato extension problem [217], [163] in the restricted sense consists of existence and description of maximal accretive and sectorial extensions T of a non-negative densely-deﬁned operator A˙ such that A˙ ⊂ T ⊂ A˙ ∗ . The solution of this problem was presented by Arlinski˘i (Theorem 9.5.14) and Tsekanovski˘i (Theo-

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rem 9.5.13) as well as in Theorem 9.2.7 [47], [49], [255], [256]. Theorems 9.6.1–9.6.3 and Lemma 9.6.4 were obtained by Tsekanovski˘i [255], [260], [257], [258]; Corollary 9.6.2 and Theorem 9.6.6 are due to Okunski˘i and Tsekanovski˘i [211]. The results of Section 9.7 belong to Arlinski˘i [23]. Theorem 9.8.2 was obtained by Derkach and Tsekanovski˘i [107]. Theorems 9.8.11–9.8.14 on realization of Stieltjes functions and their connections with the Krein-von Neumann extensions of non-negative operators are due to Dovzhenko and Tsekanovski˘i [113]. The L-system modiﬁcations are due to Belyi and Tsekanovski˘i and proofs are being published for the ﬁrst time. Deﬁnition of classes S −1 (R) and S0−1 (R) as well as Theorems 9.9.4 and 9.9.5 were introduced in [73]. Results of Section 9.9 on realization of inverse Stieltjes functions and their connections with the Friedrichs extensions of non-negative operators belong to Belyi and Tsekanovski˘i and are being published for the ﬁrst time. Quasi-self-adjoint m-sectorial extensions were studied by Derkach, Malamud, and Tsekanovski˘i [105], [106], and by Derkach and Malamud [104] via the characteristic function approach based on boundary triplets and corresponding Weyl functions. Unbounded m-sectorial operators and their characteristic functions were treated by Arlinski˘ı [36] by means of rigged Hilbert spaces.

Chapter 10 Theorem 10.1.1 belongs to Arlinski˘i [22] and Theorem 10.2.1 is due to Tsekanovski˘i [258], [260]. Theorem 10.2.2 on criterion for the rigged canonical system with Schr¨odinger operator to be an L-system belongs to the authors and is being published for the ﬁrst time. Formulas (10.30)–(10.31) were established by Arlinski˘i and Tsekanovski˘i [53]. Theorems 10.3.1–10.3.2 are due to Tsekanovski˘i [257], [258], [260]. Lemma 10.4.2 and Theorem 10.4.3 belong to Donoghue [111], for ﬁnite equal deﬁciency indices to Gesztesy and Tsekanovski˘i [137], and for equal inﬁnite deﬁciency indices to Gesztesy, Kalton, Makarov, and Tsekanovski˘i [135]. Theorem 10.5.1 is due to Tsekanovski˘i [260], Theorem 10.6.1 was obtained by Arlinski˘i and Tsekanovski˘i [53], Theorems 10.6.2–10.6.3 are due to Arlinski˘i, Belyi and Tsekanovski˘i, and their L-system version is being published for the ﬁrst time. Inequality (10.125) was discovered by Gesztesy, Kalton, Makarov, and Tsekanovski˘i in [135]; other inequalities of this nature were obtained by Arlinski˘ı and Tsekanovski˘i in [53]. An elementary proof of (10.125) was oﬀered by Kalton in private communications. Another proof of inequality (10.125) and other inequalities related to diﬀerential operators were established by Trigub in [246]. Theorems 10.7.1–10.7.5 belong to Arlinski˘i and Tsekanovski˘i [53], Theorem 10.7.8 is due to the authors and is presented for the ﬁrst time. Stieltjes-like functions were introduced by Belyi and Tsekanovski˘i [79]. Theorems 10.8.4–10.8.6 belong to Belyi and Tsekanovski˘i [79] as well as formulas and dynamics of restoration of the whole Lsystem with Schr¨ odinger operator based on its Stieltjes-like impedance function. Results of Section 10.9 belong to Belyi and Tsekanovski˘i [80] as well as formulas and dynamics of restoration of the whole L-system with Schr¨odinger operator

504

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based on its impedance inverse Stieltjes-like function.

Chapter 11 Theorems 11.1.1–11.1.2 belong to Stone [239]. Lemma 11.1.3 is a reformulation of the deﬁnition of a completely non-self-adjoint linear bounded operator [190], [193], [89], [91]. Theorems 11.1.4–11.2.6 on non-self-adjoint strengthening of Stone’s Theorem 11.1.1–11.1.2 were established by Arlinski˘i and Tsekanovski˘i [54]. Theb 1 orem 11.2.7 on realization of Herglotz-Nevanlinna function v(z) = a t−z dσ(t) as impedance function of some L-system with dissipative Jacobi matrix with onedimensional imaginary part as state-space operator of the system belong to Arlinski˘i and Tsekanovski˘i and is being published for the ﬁrst time. All the results on inverse spectral problems for non-self adjoint Jacobi matrices in Section 11.3 belong to Arlinski˘i and Tsekanovski˘i [54]. System interpolation was introduced and studied by Alpay and Tsekanovski˘i [8]. Theorems 11.5.2–11.6.10 are due to Alpay and Tsekanovski˘i [8]. The results of Section 11.7 on the Nevanlinna-Pick interpolation problem for a rational function with distinct poles and its connection with system interpolation and, in particular, with the Livˇsic system in the Pick form, belong to Arlinski˘i and Tsekanovski˘i and are presented for the ﬁrst time.

Chapter 12 Non-canonical F -systems were introduced by Hassi, de Snoo and Tsekanovski˘i in [151], [152], [153]. The metric conservation law of the forms (12.15), (12.53) for non-canonical systems is presented for the ﬁrst time and generalizes its canonical versions described in Chapters 5 and 6. The results in Sections 12.1–12.3 belong to Hassi, de Snoo and Tsekanovski˘i [151], [152], [153], [154]. The general realization Theorems 12.5.4 and 12.6.2 for Herglotz-Nevanlinna functions as well as other results in Sections 12.4–12.6 were established by Belyi, Hassi, de Snoo and Tsekanovski˘i [72], [73], [74], [75], [77]. Conservative state-space realizations of analytic functions that map the right half-plane into itself were treated by Ball and Staﬀans in [68], [69] and Staﬀans in [235], [237]. The systems with operator pencils as state-space operator have been studied by Rutkas [222], [223], [224]. Realizations of operator-valued Herglotz-Nevanlinna functions and pairs by means of the Weyl families of boundary relations have been obtained by Derkach and Malamud in [99], [100], [101], [103] and by Derkach, Hassi, Malamud, and de Snoo in [97], [98].

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Index (∗)-extension, 90 accumulative, 335 normalized at point zero, 254 α-co-sectorial contraction, 269 γ-ﬁeld, 224 m-function, 417 qsc-extension N-minimal, 280 Adjoint system, 123 Admissible extension, 15 operator, 71 Angle minimal between subspaces, 36 Aperture of two linear manifolds, 10 Bi-extension, 45 α-sectorial , 321 t-self-adjoint, 57 accretive, 311 q.s.-a., 81 quasi-self-adjoint, 81 twice-self-adjoint, 57 Boundary triplet, 224 Cayley transform, 9 Class CH (α), 269 N (R), 170 N0 (R), 206 N1 (R), 206 N01 (R), 206 ˙ 81 R(A),

S(R), 330 SL0 (R), 380 SL0 (R, K), 380 SL−1 0 (R), 397 SL−1 0 (R, F ), 397 SL−1 0,1 (R, F ), 397 SL01 (R, K), 380 S α1 ,α2 , 325 S α , 323 S0 (R), 330 S0K (R), 331 −1 S0,F (R), 338 Λ, 161 ˙ 161 Λ(A), Ω(R, J), 212 ˙ 69 Ω(A), Ω0 (R, J), 212 Ω1 (R, J), 212 ΩJ , 138 Ωm J , 139 Ω01 (R, J), 212 ˙ 312 Ξ(A), ΞAT , 318 N, 436 N0 , 419 ˙ 45 E(A), ˙ 69 M(A), N(G, [−1, 1]), 281 S(k) , 150 S −1 (R), 334 S0−1 (R), 335 Nr , 446 [H1 , H2 ], 11 525

526 Contraction prime, 278 Coupling of two canonical systems, 125 of two F-systems, 461 of two L-systems, 219 Cyclic vector, 415 Deﬁciency indices, 3 numbers, 3 subspace, 2 Direct sum of two operators, 17 Distribution function, 379 Dual pair, 268 Equal systems, 124 Excess part of a system, 124 subspace, 123 system, 123 Extension m-accretive extremal, 306 qsc-extension extremal, 272 of a dual pair, 268 contractive, 265 Friedrichs, 292 Kre˘ın-von Neumann, 294 qsc-extension, 269 regular symmetric, 36 rigid, 271 sc-extension, 269 self-adjoint disjoint, 65 relatively prime, 65 transversal, 65 soft, 271 Field of regularity, 2 Function Herglotz-Nevanlinna, 147, 167 impedance, 167

Index of an F-system, 458 of NCL-system, 473 transfer, 130 of an F-system, 458 of an L-system, 166 of NCL-system, 472 Weyl, 417 Weyl-Titchmarsh, 282, 346 via boundary triplet, 225 Inclusion of operator (into a system), 122 Inverse Stieltjes function, 334 Inverse Stieltjes-like function, 397 Isometry, 6 Jacobi matrices non-self-adjoint dissipative ﬁnite, 414 dissipative semi-inﬁnite, 414 self-adjoint, 414 Kre˘ın-Langer Q-function, 229 Left divisor of a system, 126 Linearly independent manifolds, 4 Maximal common symmetric part, 78 Moore-Penrose inverse, 262 Operator α-sectorial, 269 m-α-sectorial, 302 qsc-operator, 282 accretive, 301 α-sectorial, 302 adjoint, 25 bi-continuous, 111 channel, 120 closed , 1 defect, 262 directing, 120 dissipative, 74 exclusion, 15

Index isometric, 6 m-accretive, 301, 302 maximal accretive, 301, 302 maximal dissipative, 74 non-negative, 26, 292 O-operator, 35 prime, 122, 194 regular, 28 Regularizing, 158 Riesz-Berezansky, 24 Self-adjoint, 1 shorted, 265 singular, 34 state-space, 120 symmetric, 1 maximal, 7 unitary, 6 with simple spectrum, 415 Operator colligation, 121 Principal part of a system, 124, 195 Principal subspace, 123 Projection of a canonical system, 125 of an F-system, 462 Quasi-kernel, 26 Quasi-self-adjoint contractive extension, 269 extension, 69 mutually transversal, 104 regular, 69 relatively prime, 104 Range property (R), 81 Regular operator, 28 Regular point, 1 Regular type point, 1 Resolvent, 110, 455 canonical, 150 extended canonical, 150 set, 1, 455 Riesz-Berezansky operator, 24

527 Rigged canonical system, 163 Hilbert space, 24 Right divisor of a system, 126 Self-adjoint contractive extension, 269 extension with exit, 19 Semi-deﬁciency indices, 13 numbers, 13 subspace, 12 Space input-output , 120 state, 120 with negative norm, 24 with positive norm, 23 Spectral function canonical, 150 extended canonical, 153 Spectrum, 2 point spectrum, 2 spectrum of a system, 128 Stieltjes function, 322 Stieltjes-like functions, 379 Subspace channel, 121 excess, 123 principal, 123 System F -system, 454, 457 scattering, 457 accretive, 330 accumulative, 335 adjoint, 123 auxiliary rigged canonical, 244 bi-unitarily equivalent, 203 excess, 123 ﬁnite-dimensional, 139 in the Pick form, 438 interpolation, 431 L-system, 164 impedance, 167

528

Index minimal, 194 normalized at point zero, 254 Livsi˘c canonical, 120 in Jacobi form, 417 minimal, 123 NCI-system, 470 NCL-system, 471 F+ -minimal, 487 non-canonical impedance, 470 non-canonical L-system, 471 rigged canonical, 163 scattering, 165 unitary equivalent, 129

Transform Donoghue, 250 Potapov-Ginzburg, 212 von Neumann’s formula, 4