The State Space Method: Generalizations and Applications [1st ed.] 9783764373702, 3764373709

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Operator Theory: Advances and Applications Vol. 161 Editor: I. Gohberg

Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel Editorial Board: D. Alpay (Beer-Sheva) J. Arazy (Haifa) A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) A. Ben-Artzi (Tel Aviv) H. Bercovici (Bloomington) A. Böttcher (Chemnitz) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) K. R. Davidson (Waterloo, Ontario) R. G. Douglas (College Station) A. Dijksma (Groningen) H. Dym (Rehovot) P. A. Fuhrmann (Beer Sheva) B. Gramsch (Mainz) J. A. Helton (La Jolla) M. A. Kaashoek (Amsterdam) H. G. Kaper (Argonne) S. T. Kuroda (Tokyo)

Subseries Linear Operators and Linear Systems Subseries editors: Daniel Alpay Department of Mathematics Ben Gurion University of the Negev Beer Sheva 84105 Israel

P. Lancaster (Calgary) L. E. Lerer (Haifa) B. Mityagin (Columbus) V. V. Peller (Manhattan, Kansas) L. Rodman (Williamsburg) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) I. M. Spitkovsky (Williamsburg) S. Treil (Providence) H. Upmeier (Marburg) S. M. Verduyn Lunel (Leiden) D. Voiculescu (Berkeley) H. Widom (Santa Cruz) D. Xia (Nashville) D. Yafaev (Rennes) Honorary and Advisory Editorial Board: C. Foias (Bloomington) P. R. Halmos (Santa Clara) T. Kailath (Stanford) P. D. Lax (New York) M. S. Livsic (Beer Sheva)

Joseph A. Ball Department of Mathematics Virginia Tech Blacksburg, VA 24061 USA André M.C. Ran Division of Mathematics and Computer Science Faculty of Sciences Vrije Universiteit NL-1081 HV Amsterdam The Netherlands

The State Space Method Generalizations and Applications

Daniel Alpay Israel Gohberg Editors

Birkhäuser Verlag Basel . Boston . Berlin

Editors: Daniel Alpay Department of Mathematics Ben-Gurion University of the Negev P.O. Box 653 Beer Sheva 84105 Israel e-mail: [email protected]

Israel Gohberg School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University Ramat Aviv 69978 Israel e-mail: [email protected]

2000 Mathematics Subject Classification 47Axx, 93Bxx

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .

ISBN 3-7643-7370-9 Birkhäuser Verlag, Basel – Boston – Berlin 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. © 2006 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Cover design: Heinz Hiltbrunner, Basel Printed in Germany ISBN-10: 3-7643-7370-9 e-ISBN: 3-7643-7431-4 ISBN-13: 978-3-7643-7370-2 987654321

www.birkhauser.ch

Contents Editorial Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

D. Alpay and I. Gohberg Discrete Analogs of Canonical Systems with Pseudo-exponential Potential. Definitions and Formulas for the Spectral Matrix Functions . . . . . . . . . . 1 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Review of the continuous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The asymptotic equivalence matrix function . . . . . . . . . . . . . . . . . . . . 2.2 The other characteristic spectral functions . . . . . . . . . . . . . . . . . . . . . . 2.3 The continuous orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The discrete case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 First-order discrete system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The asymptotic equivalence matrix function . . . . . . . . . . . . . . . . . . . . 3.3 The reflection coefficient function and the Schur algorithm . . . . . . 3.4 The scattering function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The Weyl function and the spectral function . . . . . . . . . . . . . . . . . . . . 3.6 The orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 The spectral function and isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Two-sided systems and an example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Two-sided discrete first-order systems . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 An illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 4 4 8 14 16 19 19 22 27 29 31 33 37 39 39 41 44

D. Alpay and D.S. Kalyuzhny˘-Verbovetzki˘ ˘ Matrix-J-unitary Non-commutative Rational Formal Power Series . . .

49

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 More on observability, controllability, and minimality in the non-commutative setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Matrix-J-unitary formal power series: A multivariable non-commutative analogue of the line case . . . . . . . . . . . 4.1 Minimal Givone–Roesser realizations and the Lyapunov equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 54 60 67 68

vi

Contents

5

6

7

8

4.2 The associated structured Hermitian matrix . . . . . . . . . . . . . . . . . . . . 4.3 Minimal matrix-J-unitary factorizations . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Matrix-unitary rational formal power series . . . . . . . . . . . . . . . . . . . . . Matrix-J-unitary formal power series: A multivariable non-commutative analogue of the circle case . . . . . . . . . 5.1 Minimal Givone–Roesser realizations and the Stein equation . . . . 5.2 The associated structured Hermitian matrix . . . . . . . . . . . . . . . . . . . . 5.3 Minimal matrix-J-unitary factorizations . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Matrix-unitary rational formal power series . . . . . . . . . . . . . . . . . . . . . Matrix-J-inner rational formal power series . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 A multivariable non-commutative analogue of the half-plane case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 A multivariable non-commutative analogue of the disk case . . . . . Matrix-selfadjoint rational formal power series . . . . . . . . . . . . . . . . . . . . . . . 7.1 A multivariable non-commutative analogue of the line case . . . . . . 7.2 A multivariable non-commutative analogue of the circle case . . . . Finite-dimensional de Branges–Rovnyak spaces and backward shift realizations: The multivariable non-commutative setting . . . . . . . . 8.1 Non-commutative formal reproducing kernel Pontryagin spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Minimal realizations in non-commutative de Branges–Rovnyak spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72 74 75 77 77 83 84 85 87 87 91 96 96 100 102 102 106 110 111

D.Z. Arov and O.J. Staffans State/Signal Linear Time-Invariant Systems Theory, Part I: Discrete Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 1 2 3 4 5 6 7 8 9 10

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . State/signal nodes and trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The driving variable representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The output nulling representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The input/state/output representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal behaviors, external equivalence, and similarity . . . . . . . . . . . . . . . . Dilations of state/signal systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowlegment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

116 120 123 128 132 138 146 153 167 176 176 176

Contents

vii

J.A. Ball, G. Groenewald and T. Malakorn Conservative Structured Noncommutative Multidimensional Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Structured noncommutative multidimensional linear systems: basic definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Adjoint systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Dissipative and conservative structured multidimensional linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conservative SNMLS-realization of formal power series in the class SAG (U, Y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Gohberg, I. Haimovici, M.A. Kaashoek and L. Lerer The Bezout Integral Operator: Main Property and Underlying Abstract Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Spectral theory of entire matrix functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 A review of the spectral data of an analytic matrix function . . . . 2.2 Eigenvalues and Jordan chains in terms of realizations . . . . . . . . . . 2.3 Common eigenvalues and common Jordan chains in terms of realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Common spectral data of entire matrix functions . . . . . . . . . . . . . . . 3 The null space of the Bezout integral operator . . . . . . . . . . . . . . . . . . . . . . . 3.1 Preliminaries on convolution integral operators . . . . . . . . . . . . . . . . . 3.2 Co-realizations for the functions A, B, C, D . . . . . . . . . . . . . . . . . . . . . 3.3 Quasi commutativity in operator form . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Intertwining properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Proof of the first main theorem on the Bezout integral operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 A general scheme for defining Bezout operators . . . . . . . . . . . . . . . . . . . . . . 4.1 A preliminary proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Definition of an abstract Bezout operator . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Haimovici-Lerer scheme for defining an abstract Bezout operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Bezout integral operator revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 The null space of the Bezout integral operator . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

180 183 191 193 199 220

225 226 228 229 232 234 237 241 242 244 248 251 254 256 257 260 262 264 266 268

Editorial Introduction This volume of the Operator Theory: Advances and Applications series (OTAA) is the first volume of a new subseries. This subseries is dedicated to connections between the theory of linear operators and the mathematical theory of linear systems and is named Linear Operators and Linear Systems (LOLS). As the existing subseries Advances in Partial Differential Equations (ADPE), the new subseries will continue the traditions of the OTAA series and keep the high quality of the volumes. The editors of the new subseries are: Daniel Alpay (Beer–Sheva, Israel), Joseph Ball (Blacksburg, Virginia, USA) and Andr´ ´e Ran (Amsterdam, The Netherlands). In the last 25–30 years, Mathematical System Theory developed in an essential way. A large part of this development was connected with the use of the state space method. Let us mention for instance the “theory of H∞ control”. The state space method allowed to introduce in system theory the modern tools of matrix and operator theory. On the other hand the state space approach had an important impact on Algebra, Analysis and Operator Theory. In particular it allowed to solve explicitly some problems from interpolation theory, theory of convolution equations, inverse problems for canonical differential equations and their discrete analogs. All these directions are planned to be present in the subseries LOLS. The editors and the publisher are inviting authors to submit their manuscripts for publication in this subseries. This volume contains five essays. The essay of D. Arov and O. Staffans, State/signal linear time-invariant systems theory, part I: discrete time systems, contains new results in classical system theory. The essays of D. Alpay and D.S. Kalyuzhny˘ ˘ı-Verbovetzki˘ı, Matrix-J-unitary non-commutative rational formal power series, and of J.A. Ball, G. Groenewald and T. Malakorn, Conservative structured noncommutative multidimensional linear systems are dedicated to a new branch in Mathematical system theory where discrete time is replaced by the free semigroup with N generators. The essay of I. Gohberg, I. Haimovici, M.A. Kaashoek and L. Lerer, The Bezout integral operator: main property and underlying abstract scheme contains new applications of the state space method to the theory of Bezoutiants and convolution equations. The essay of D. Alpay and I. Gohberg Discrete analogs of canonical systems with pseudo-exponential potential. Definitions and formulas for the spectral matrix functions is concerned with new results and formulas for the discrete analogs of canonical systems. Daniel Alpay, Israel Gohberg

Operator Theory: Advances and Applications, Vol. 161, 1–47 c 2005 Birkhauser
¨ Verlag Basel/Switzerland

Discrete Analogs of Canonical Systems with Pseudo-exponential Potential. Definitions and Formulas for the Spectral Matrix Functions Daniel Alpay and Israel Gohberg Abstract. We first review the theory of canonical differential expressions in the rational case. Then, we define and study the discrete analogue of canonical differential expressions. We focus on the rational case. Two kinds of discrete systems are to be distinguished: one-sided and two-sided. In both cases the analogue of the potential is a sequence of numbers in the open unit disk (Schur coefficients). We define the characteristic spectral functions of the discrete systems and provide exact realization formulas for them when the Schur coefficients are of a special form called strictly pseudo-exponential. Mathematics Subject Classification (2000). 34L25, 81U40, 47A56.

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Review of the continuous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 The asymptotic equivalence matrix function . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The other characteristic spectral functions . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 The continuous orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 The discrete case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 First-order discrete system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 The asymptotic equivalence matrix function . . . . . . . . . . . . . . . . . . . . . . 22 3.3 The reflection coefficient function and the Schur algorithm . . . . . . . . 27 3.4 The scattering function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.5 The Weyl function and the spectral function . . . . . . . . . . . . . . . . . . . . . . 31 3.6 The orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.7 The spectral function and isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4 Two-sided systems and an example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1 Two-sided discrete first-order systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 An illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2

D. Alpay and I. Gohberg

1. Introduction Canonical differential expressions are differential equations of the form −iJ

∂Θ (x, λ) = λΘ(x, λ) + v(x)Θ(x, λ), ∂x

where v(x) =




0 k(x)∗

k(x) 0



,

J=




x ≥ 0, λ ∈ C, In 0

0 −IIn



(1.1)

,

and where k ∈ Ln×n (R+ ) is called the potential. Such systems were introduced by 1 M.G. Kre˘ ˘ın (see, e.g., [37], [38]). Associated to (1.1) are a number of functions of λ, which we called in [10] the characteristic spectral functions of the canonical system. These are: 1. 2. 3. 4. 5.

The The The The The

asymptotic equivalence matrix function V (λ). scattering function S(λ). spectral function W (λ). Weyl function N (λ). reflection coefficient function R(λ).

Direct problems consist in computing these functions from the potential k(x) while inverse problems consist in recovering the potential from one of these functions. In the present paper we study discrete counterparts of canonical differential expressions. To present our approach, we first review various facts on the telegraphers’ equations. By the term telegraphers’ equations, one means a system of differential equations connecting the voltage and the current in a transmission line. The case of lossy lines can be found for instance in [45] and [18]. We here consider the case of lossless lines and follow the arguments and notations in [16, Section 2], [19, p. 110–111] and [46]. The telegraphers’ equations which describe the evolution of the voltage v(x, t) and current i(x, t) in a lossless transmission line can be given as: ∂v ∂i (x, t) + Z(x) (x, t) = 0 ∂x ∂t ∂i ∂v (x, t) + Z(x)−1 (x, t) = 0. ∂x ∂t

(1.2)

In these equations, Z(x) represents the local impedance at the point x. A priori there may be points where Z(x) is not continuous, but it is important to bear in mind that voltage and current will be continuous at these points. Let us assume that Z(x) > 0 and is continuously differentiable on an interval (a, b), and introduce the new variables V (x, t) = Z(x)−1/2 v(x, t), I(x, t) = Z(x)1/2 i(x, t),

Analogs of Canonical Systems with Pseudo-exponential Potential

3

and V (x, t) + I(x, t) , 2 V (x, t) − I(x, t) WL (x, t) = . 2

WR (x, t) =

Then the function W (x, t) =



1 Z(x)−1/2 WR (x, t) = WL (x, t) 2 Z(x)−1/2

Z(x)1/2 −Z(x)1/2


 v(x, t) i(x, t)

(1.3)

satisfies the differential equation, also called symmetric two components wave equation (see [16, equation (2.6) p. 362], [46, p. 256], [19, equation (3.3) p. 111])
 ∂W (x, t) ∂W (x, t) 0 −κ(x) = −J + W (x, t), −κ(x) 0 ∂x ∂t where
 Z ′ (x) 1 0 J= and κ(x) = . (1.4) 0 −1 2Z(x) We distinguish two cases: (a) The case where Z(x) > 0 and is continuously differentiable on R+ . Taking   f(λ) = R eiλt f (t)dt on both sides we get the (inverse) Fourier transform f → to a canonical differential expressions (also called Dirac type system), with  (x, λ). The theory of canonical differential k(x) = iκ(x) and Θ(x, λ) = W expressions is reviewed in the next section. (b) The case where Z(x) is constant on intervals [nh, (n + 1)h) for some preassigned h > 0. We are then lead to discrete systems. The paper consists of three sections besides the introduction. In Section 2 we review the main features of the continuous case. The third section presents the discrete systems to be studied. These are of two kinds, one-sided and two-sided. Section 3 also contains a study of one-sided systems and of their associated characteristic spectral functions. In Section 4 we focus on two-sided systems and we also present an illustrative example. In the parallel between the continuous and discrete cases a number of problems remains to be considered to obtain a complete picture. In the sequel to the present paper we study inverse problems associated to these first-order systems. To conclude this introduction we set some definitions and notation. The open unit disk will be denoted by D, the unit circle by T, and the open upper half-plane by C+ . The open lower half-plane is denoted by C− and its closure by C− . We will make use of the Wiener algebras of the real line and of the unit circle. These are defined as follows. The Wiener algebra of the real line W n×n (R) = W n×n consists of the functions of the form  ∞ f (λ) = D + eiλt u(t)dt (1.5) −∞

4

D. Alpay and I. Gohberg

where D ∈ Cn×n and where u ∈ Ln×n (R). Usually we will not stress the depen1 n×n n×n dence on R. The sub-algebra W+ (resp. W− ) consists of the functions of the form (1.5) for which the support of u is in R+ (resp. in R− ). The Wiener algebra W(T) (we will usually write W rather than W(T)) of the unit circle consists of complex-valued functions f (z) of the form  f (z) = fℓ z ℓ Z

for which

def.

f W =

 Z

|ffℓ | < ∞.

2. Review of the continuous case 2.1. The asymptotic equivalence matrix function We first review the continuous case, and in particular the definitions and main properties of the characteristic spectral functions. See, e.g., [7], [11], [10] for more information. We restrict ourselves to the case where the potential is of the form   −1 ∗ k(x) = −2ceixa Ip + Ω Y − e−2ixa Y e2ixa (b + iΩc∗ ) , (2.1)

where (a, b, c) ∈ Cp×p × Cp×n × Cn×p is a triple of matrices with the properties that ℓ ℓ p and ∪m ∩m ℓ=0 ker ca = {0} ℓ=0 Im a b = C for m large enough. In system theory, see for instance [30], the first condition means that the pair (c, a) is observable while the second means that the pair (a, b) is controllable. When both conditions are in force, the triple is called minimal. See also [14] for more information on these notions. We assume moreover that the spectra of a and of a× = a − bc are in the open upper half-plane. Furthermore Ω and Y in (2.1) belong to Cp×p and are the unique solutions of the Lyapunov equations i(Ωa×∗ − a× Ω) = −i(Y a − a∗ Y ) =

bb∗ , c∗ c.

(2.2) (2.3)

This class of potentials was introduced in [7] and called in [26] strictly pseudoexponential potentials. Note that both Ω and Y are strictly positive since the triple (a, b, c) is minimal, and that Ip + ΩY and Ip + Y Ω are invertible since √ √ det(IIp + ΩY ) = det(IIp + Y Ω) = det(IIp + Y Ω Y ). Note also that asymptotically, k(x) ∼ −2ceixa (IIp + ΩY )−1 (b + iΩc∗ )

(2.4)

as x → +∞. Potentials of the form (2.1) can also be represented in a different form; see (2.22).

Analogs of Canonical Systems with Pseudo-exponential Potential

5

We first define the asymptotic equivalence matrix function. To that purpose (and here we follow closely our paper [12]) let F, G and T be the matrices given by


 −c 0 ia 0 0 f1∗ F =i , T = , G = , (2.5) 0 f1 0 −ia∗ c∗ 0 where f1 = (b∗ − icΩ)(IIp + Y Ω)−1 .

Theorem 2.1. Let Q(x, y) be defined by Q(x, y) = F exT (II2p − exT ZexT )−1 eyT G

where (F, G, T ) are defined by (2.5) and where Z is the unique solution of the matrix equation T Z + ZT = −GF. Then the matrix function  ∞ U (x, λ) = eiλJx + Q(x, u)eiλJu du x

is the unique solution of (1.1) with the potential as in (2.1), subject to the condition
−ixλ  e In 0 lim U (x, λ) = I2n , λ ∈ R. (2.6) x→∞ 0 eixλ In

Furthermore, the Cn×n -valued blocks in the decomposition of the matrix function U (0, λ) = (U Uij (0, λ)) are given by U11 (0, λ) U21 (0, λ) U12 (0, λ) U22 (0, λ)

= In + icΩ(λIIp − a∗ )−1 c∗ ,

= (−b∗ + icΩ)(λIIp − a∗ )−1 c∗ ,

= −c(IIp + ΩY )(λIIp − a)−1 (IIp + ΩY )−1 (b + iΩc∗ ),

= In − (ib∗ Y + cΩY )(λIIp − a)−1 (IIp + ΩY )−1 (b + iΩc∗ ).

See [9, Theorem 2.1]. Definition 2.2. The function V (λ) = U (0, λ) is called the asymptotic equivalence matrix function. The terminology asymptotic equivalence matrix function is explained in the following theorem: Theorem 2.3. The asymptotic equivalence matrix function has the following property: let x ∈ R and ξ0 and ξ1 in C2n . Let f0 (x, λ) = eiλxJ ξ0 be the C2n -valued solution to (1.1) corresponding to k(x) = 0 and f0 (0, λ) = ξ0 and let f1 (x, λ) corresponding to an arbitrary potential k of the form (2.1), with f1 (0, λ) = ξ1 . The two solutions are asymptotic in the sense that lim f1 (x, λ) − f0 (x, λ) = 0

x→∞

if and only if ξ1 = U (0, λ)ξ0 . For a proof, see [10, Section 2.2].

6

D. Alpay and I. Gohberg

The asymptotic equivalence matrix function takes J-unitary values on the real line: V (λ)JV (λ)∗ = J, λ ∈ R.

We recall the following: if R be a C2n×2n -valued rational functions analytic at infinity, it can be written as R(λ) = D + C(λIIm − A)−1 B, where A, B, C and D are matrices of appropriate sizes. Such a representation of R is called a realization. The realization is said to be minimal if the size of A is minimal (equivalently, the triple (A, B, C) is minimal, in the sense recalled above). The McMillan degree of R is the size of the matrix A in any minimal realization. Minimal realizations of rational matrix-valued functions taking J-unitary values on the real line were characterized in [5, Theorem 2.8 p. 192]: R takes J-unitary values on the real line if and only if there exists an Hermitian invertible matrix H ∈ Cm×m solution of the system of equations i(A∗ H − HA) = C =

C ∗ JC iJB ∗ H.

(2.7) (2.8)

The matrix H is uniquely defined by the minimal realization of R and is called the associated Hermitian matrix to the minimal realization matrix function. The matrix function R is moreover J-inner, that is J-contractive in the open upper half-plane: R(λ)JR(λ) ≤ J

for all points of analyticity in the open upper half-plane,

if and only if H > 0. The asymptotic equivalence matrix function V (λ) has no pole on the real line, but an arbitrary rational function which takes J-unitary values on the real line may have poles on the real line. See [5] and [4] for examples. The next theorem presents a minimal realization of the asymptotic equivalence matrix function and its associated Hermitian matrix. Theorem 2.4. Let k(x) be given in the form (2.1). Then, a minimal realization of the asymptotic equivalence matrix function associated to the corresponding canonical differential system is given by V (λ) = I2n + C(λII2p − A)−1 B, where
∗ 
∗  a 0 c 0 A= , B= 0 a 0 (IIp + ΩY )−1 (b + iΩc∗ ) and C=




icΩ −b∗ + icΩ

−c(IIp + ΩY ) −ib∗ Y − cΩY



,

and the associated Hermitian matrix is given by 
Ω i(IIp + ΩY ) H= . −i(IIp + Y Ω) (IIp + Y Ω)Y We now prove a factorization result for V (λ). We first recall the following: let as above R be a rational matrix-valued function analytic at infinity. The factorization

Analogs of Canonical Systems with Pseudo-exponential Potential

7

R = R1 R2 of R into two other rational matrix-valued functions analytic at infinity (all the functions are assumed to have the same size) is said to be minimal if deg R = deg R1 + deg R2 . Minimal factorizations of rational matrix-valued functions have been characterized in [14, Theorem 1.1 p. 7]. Assume now that R takes J-unitary values on the real line. Minimal factorizations of R into two factors which are J-unitary on the real line were characterized in [5]. Such factorizations are called J-unitary factorizations. To recall the result (see [5, Theorem 2.6 p. 187]), we introduce first some more notations and definitions: let H ∈ Cp×p be an invertible Hermitian matrix. The formula [x, y]H = y ∗ Hx,

x, y ∈ Cp

defines a non-degenerate and in general indefinite inner product. Two vectors are orthogonal with respect to this inner product if [x, y]H = 0. The orthogonal complement of a subspace M ⊂ Cp is: M[⊥] = {x ∈ Cp ; [x, m]H = 0 ∀m ∈ M} . We refer to [29] for more information on finite-dimensional indefinite inner product spaces. Theorem 2.5. Let R be a rational matrix-valued function analytic at infinity and J-unitary on the real line, and let R(λ) = D + C(zIIp − A)−1 B be a minimal realization of R, with associated matrix H. Let M be a A-invariant subspace of Cp non-degenerate with respect to the inner product [·, ·]H . Let π denote the orthogonal (with respect to [·, ·]H ) projection such that ker π = M,

Im π = M[⊥]

and let D = D1 D2 be a factorization of D into two J-unitary constants. Then R = R1 R2 with R1 (z) = D1 + C(zIIp − A)−1 (IIp − π)BD2−1

R2 (z) = D2 + D1−1 Cπ(zIIp − A)−1 BD2

is a minimal J-unitary factorization of R. Conversely, every J-unitary factorization of R is obtained in such a way. As a consequence we have: Theorem 2.6. Let V (λ) be the asymptotic equivalence matrix function of a canonical differential expression (1.1) with potential of the form (2.1). Then it admits a minimal factorization V (λ) = V1 (λ)V V2 (λ)−1 where V1 and V2 are J-inner and of same degree.

8

D. Alpay and I. Gohberg

To prove this result we consider the realization of V (λ) given in Theorem 2.4
p and note that the space C0 is A-invariant and H-non-degenerate (in fact, Hpositive). The factorization follows from Theorem 2.5. The fact that V2 is inner follows from


∗ Ip 0 Ω 0 Ip 0 H= . −i(IIp + Y Ω)Ω−1 Ip 0 −Ω−1 − Y −i(IIp + Y Ω)Ω−1 Ip To prove this last formula we have used the formula for Schur complements: 



 A11 I 0 0 A11 A12 I A−1 11 A12 = A21 A22 A21 A−1 I 0 A22 − A21 A−1 0 I 11 11 A12

for matrices of appropriate sizes and A11 being invertible. See [20, formula (0.3) p. 3].




One could have started with the space C0p , which is also A-invariant and Hpositive. In particular, the above factorization is not unique. 2.2. The other characteristic spectral functions In this section we review the definitions and main properties of the characteristic spectral functions associated to a canonical differential expression. It follows from Theorem 2.4 that U (0, λ) has no pole on the real line and that, furthermore: U11 (0, λ)U11 (0, λ)∗ − U12 (0, λ)U12 (0, λ)∗ = In U22 (0, λ)U U22 (0, λ)∗ − U21 (0, λ)U U21 (0, λ)∗ = In

and

U11 (0, λ)∗ U12 (0, λ) = U21 (0, λ)∗ U22 (0, λ) for real λ. In particular, U11 (0, λ) is invertible on the real line and U21 (0, λ)U11 (0, λ)−1 is well defined and takes contractive values on the real line. Definition 2.7. The function R(λ) = (U U21 (0, λ)U11 (0, λ)−1 )∗ = U12 (0, λ)U U22 (0, λ)−1 , is called the reflection coefficient function.

λ ∈ R,

To present an equivalent definition of the reflection coefficient function, we need some notation: if
 A B Θ= ∈ C(p+q)×(p+q) , A ∈ Cp×p , and X ∈ Cp×q C D

we set

TΘ (X) = (AX + B)(CX + D)−1 . Note that TΘ1 Θ2 (X) = TΘ1 (T TΘ2 (X)) when all expressions are well defined.

(2.9)

Analogs of Canonical Systems with Pseudo-exponential Potential

9

Theorem 2.8. Let Θ(x, λ) = U (x, λ)U (0, λ)−1 . Then, Θ(x, λ) is also a solution of (1.1). It is an entire function of λ. It is J-expansive in C+ ,

λ∈R ∗ = 0, J − Θ(x, λ)JΘ(x, λ) ≤ 0, λ ∈ C+ , and satisfies the initial condition Θ(0, λ) = I2n . Moreover R(λ) = lim TΘ(x,λ)−1 (0), x→∞

λ ∈ R.

(2.10)

The matrix function Θ(x, λ) is called the matrizant, or fundamental solution of the canonical differential expression. Its properties may be found in [22, p. 150]. For real λ the matrix function U (0, λ) is J-unitary. Hence we have: Θ(x, λ)−1 = U (0, λ)U (x, λ)−1 . The result follows using (2.9) and the asymptotic property (2.6). In fact, the function R is analytic and takes contractive values in the closed lower half-plane. For a proof and references, see [10] and [13, Theorem 3.1 p 6]. Theorem 2.9. A minimal realization of R(λ) is given by R(λ) = −c(λIIp − (a + iΩc∗ c))−1 (b + iΩc∗ ).

(2.11) ∗

See [10]. It follows in particular that the spectrum of the matrix a + iΩc c is in the open upper half-plane. Note that Ω is not arbitrary but is related to a, b and c via the Lyapunov equation (2.2). A direct proof that R is analytic and contractive in C− can be given using the results in [33], as we now explain. Definition 2.10. A Cn×n -valued rational function R is called a proper contraction if it takes contractive values on the real line and if moreover it is analytic at infinity and such that R(∞)R(∞)∗ < In . The following results are respectively [33, Theorem 3.2 p. 231, Theorem 3.4 p. 235]. Theorem 2.11. Let R be a Cn×n -valued rational function analytic at infinity and let R(z) = D + C(zI − A)−1 B be a minimal realization of W . Let

 α β A + BD∗ (IIn − DD∗ )−1 C B(IIn − D∗ D)−1 B ∗ A= = . γ α∗ C ∗ (IIn − DD∗ )−1 C A∗ + C ∗ (IIn − DD∗ )−1 DB ∗ Then the 1) The 2) The 3) The

following are equivalent: matrix function R is a proper contraction. real eigenvalues of A have even partial multiplicities. Riccati equation XγX − iXα∗ + iαX + β = 0.

has an Hermitian solution.

(2.12)

10

D. Alpay and I. Gohberg

The matrix A is called the state characteristic matrix of W and the Riccati equation (2.12) is called its state characteristic equation. Theorem 2.12. Let R be a Cn×n -valued proper contraction, with minimal realization R(z) = D + C(zI − A)−1 B and let (2.12) be its state characteristic equation. Then, any Hermitian solution of (2.12) is invertible and the number of negative eigenvalues of X is equal to the number of poles of R in C− . Consider now the minimal realization (2.11). The corresponding state characteristic equation is Xc∗ cX − iX(a∗ − icc∗ Ω) + i(a + iΩcc∗ )X + (b + iΩc∗ )(b∗ − icΩ) = 0. To show that X = Ω is a solution of this equation is equivalent to prove that Ω solves the Lyapunov equation (2.3). Indeed, 0 = Ωc∗ cΩ − iΩ(a∗ − icc∗ Ω) + i(a + iΩcc∗ )Ω + (b + iΩc∗ )(b∗ − icΩ) ⇐⇒

0 = −iΩa∗ + iaΩ + bb∗ − iΩ(a − c∗ b∗ ) + i(a − bc)Ω + bb∗ ⇐⇒

0 = i(a× Ω − Ωa×∗ ) + bb∗ , which is (2.3). The scattering matrix function is defined as follows: Theorem 2.13. The differential equation (1.1) has a uniquely defined C2n×n -valued solution such that for λ ∈ R,  In −IIn X(0, λ) = 0,  lim 0 eixλ In X(x, λ) = In . x→∞

The limit

 lim e−ixλ In

x→∞

0 X(x, λ) = S(λ)

exists for all real λ and is called the scattering matrix function of the canonical system. The scattering matrix function takes unitary values on the real line, belongs to the Wiener algebra W and admits a factorization S = S+ S− where S+ and its inverse are analytic in the closed upper half-plane while S− and its inverse are analytic in the closed lower half-plane. We note that the general factorization of a function in the Wiener algebra and unitary on the real line involves in general a diagonal term taking into account quantities called partial indices; see [31], [32], [34], [17]. We also note that conversely, functions with the properties as in the theorem are scattering matrix functions of a more general class of differential equations; see [41] and the discussion in [7, Appendix].

Analogs of Canonical Systems with Pseudo-exponential Potential

11

Theorem 2.14. The scattering matrix function of a canonical system (1.1) with potential (2.1) is given by: = (IIn + b∗ (λIIp − a∗ )−1 c∗ )−1

S(λ)

×(IIn − (ib∗ Y − c)(λIIp − a)−1 (IIp + ΩY )−1 (b + iΩc∗ )).

A minimal realization of the scattering matrix function is given by S(λ) = In + C(λII2p − A)−1 B, where
 a b(icΩ − b∗ ) , A= 0 a×∗
 b B= , (IIp + Y Ω)−1 (c∗ + iY b) C = (c

Set G= Then it holds that




icΩ − b∗ ).

−Ω −iIIp

iIIp −Y (IIp + ΩY )−1

i(AG − GA∗ ) = CG =



.

−BB ∗ , iB ∗ ,

and thus S takes unitary values on the real line. For a proof, see [8, p. 7]. The last statement follows from [5, Theorem 2.1 p. 179], that is from equations (2.7) and (2.8) with H = X −1 and J = Ip . Since


∗ Ip 0 −Ω 0 Ip 0 X= iΩ−1 Ip 0 (Ω + ΩY Ω)−1 iΩ−1 Ip the space leads to:




Cp 0



is A invariant and H-negative. Thus Theorem 2.5 on factorizations

Theorem 2.15. The scattering matrix function of a canonical system (1.1) with potential (2.1) admits a minimal factorization of the form S(z) = U1 (z)−1 U2 (z) where both U1 and U2 are inner (that is, are contractive in C+ and take unitary values on the real line). The fact that U2 is inner (and not merely unitary) stems from the fact that the Schur complement of −Ω in H is equal to −Y (IIp + ΩY )−1 − iIIp (−Ω)−1 (−iIIp ) = (Ω + ΩY Ω)−1

and in particular is strictly positive. Such a factorization result was also proved in [12, Theorem 7.1] using different methods. It is a particular case of a factorization result of M.G. Kre˘n ˘ and H. Langer for functions having a finite number of negative squares; see [39].

12

D. Alpay and I. Gohberg

We now turn to the spectral function. We first recall that the operator df (x) − v(x)f (x) dx restricted to the space of C2n -valued absolutely continuous functions with entries in L2 and such that (IIn − In )f (0) = 0 Hf (x) = −iJ

is self-adjoint.

Definition 2.16. A positive function W : R → Cn×n is called a spectral function if there is a unitary map U from Ln2 onto Ln2 (W ) mapping H onto the operator of multiplication by the variable in Ln2 (W ). Theorem 2.17. The function W (λ) = (V V22 (λ) − V12 (λ))−∗ (V V22 (λ) − V12 (λ))−1 is a spectral function, the map U being given by  ∞  1 In In Θ(x, λ)∗ f (x)dx. F (λ) = √ 2π 0

(2.13)

A direct proof in the rational case can be found in [26]. When k(x) ≡ 0, we have that W (λ) = In dλ, and the unitary map (2.13) is readily identified with the Fourier transform. Definition 2.18. The Weyl coefficient function N (λ) is defined in the open upper half plane; it is the unique Cn×n -valued function such that 


 ∞  In In In In −iN (λ) ∗ ∗ iN (λ) In dx Θ(x, λ) Θ(x, λ) In −IIn In −IIn In 0 is finite for −i(λ − λ∗ ) > 0.

In the setting of differential expressions (1.1), the function N was introduced in [27]. The motivation comes from the theory of the Sturm-Liouville equation. The Weyl coefficient function is analytic in the open upper half-plane and has a nonnegative imaginary part there. Such functions are called Nevanlinna functions. Theorem 2.19. The Weyl coefficient function is given by the formula N (λ) = i(U12 (0, λ) + U22 (0, λ))(U12 (0, λ) − U22 (0, λ))−1 = i(IIn − 2c(λIIp − a× )−1 (b + iΩc∗ )).

(2.14)

Proof. We first look for a Cn×2n -valued function P (λ) such that x → P (λ)Θ(x, λ)∗ has square summable entries for λ ∈ C+ . Let U (λ, x) be the solution of the differential system (1.1) subject to the asymptotic condition (2.6). Then, U (x, λ) = Θ(x, λ)U (0, λ). We thus require the entries of the function x → P (λ)U (0, λ)−∗ U (x, λ)

(2.15)

Analogs of Canonical Systems with Pseudo-exponential Potential

13

to be square summable. By definition of U , it is necessary for P (λ)U (0, λ)−∗ to be of the form (0, p(λ)) where p(λ) is Cn×n -valued. It follows from the definition of U (0, λ) that one can take   P (λ) = 0 In U (0, λ)∗ = U12 (0, λ)∗ U22 (0, λ)∗

and hence the necessity condition. Conversely, we have to show that the function (2.15) has indeed summable entries. But this is just doing the above argument backwards. The realization formula follows then from the realization formulas for the block entries of the asymptotic equivalence matrix function.
Any of the functions in the spectral domain determines all the others, as follows from the next theorem:

Theorem 2.20. Assume given a differential system of the form (1.1) with potential k(x) of the form (2.1). Assume W (λ), V (λ), R(λ), S(λ) and N (λ) are the characteristic spectral functions of (1.1), and let S = S− S+ be the spectral factorization of the scattering matrix function S, where S− and its inverse are invertible in the closed lower half-plane and S+ and its inverse are invertible in the closed upper half-plane. Then, the connections between these functions are: W (λ) W (λ) S(λ) R(λ) N (λ) V (λ)

= S− (λ)−1 S− (λ)−∗ = S+ (λ)S+ (λ)∗ , = Im N (λ), = S− (λ)S+ (λ),

= (iN (λ)∗ − In )(iN (λ)∗ + In )−1 ,

= i(IIn + R(λ)∗ )(IIn − R(λ)∗ )−1 ,
 1 (iN (λ)∗ + In )S− (λ)∗ (−iN (λ) − In )S+ (λ)−∗ = (iN (λ)∗ − In )S− (λ)∗ (−iN (λ) + In )S+ (λ)−∗ 2

for λ ∈ R. See [10, Theorem 3.1]. We note that R∗ = TV (0). We now wish to relate V to a unitary completion of the reflection coefficient function. It is easier to look at

 0 In 0 In

V (λ) = V (λ) . In 0 In 0 We set

P =

I2n + J = 2




In 0

0 0



and

Q=

I2n − J = 2




0 0

0 In



.

Theorem 2.21. Let Θ ∈ C2n×2n be such that det(P +QΘ) = 0. Then det(P −ΘQ) = 0 and def def. Θ× = (P Θ + Q)(P + QΘ)−1 = (P − ΘQ)−1 (ΘP − Q) (2.16)

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D. Alpay and I. Gohberg

Finally ∗

=

I2n − Θ× Θ×

=

I2n − Θ× Θ× ∗




A C

B C



(P − ΘQ)−1 (J − ΘJΘ∗ ) (P − ΘQ)−∗

(P + QΘ)−∗ (J − Θ∗ JΘ) (P + QΘ)−1 .

where A ∈ Cn×n . We have:

In 0 In P + QΘ = , P − ΘQ = C D 0

Proof. We set Θ =

−B −D



(2.17) (2.18)

.

Thus either of these matrices is invertible if and only if D is invertible. Thus both equalities in (2.16) make sense. To prove that they define the same object is equivalent to prove that (P − ΘQ)(P Θ + Q) = (ΘP − Q)(P + QΘ), i.e., since P Q = QP = 0, P Θ − ΘQ = ΘP − QΘ. This in turn clearly holds since P + Q = I2n . We now prove (2.17). The proof of (2.18) is similar and will be omitted. We have I2n − Θ× Θ×

∗

= = =

I2n − (P − ΘQ)−1 (ΘP − Q)(ΘP − Q)∗ (P − ΘQ)−∗

(P − ΘQ)−1{(P − ΘQ)(P − ΘQ)∗−(ΘP − Q)(ΘP − Q)∗ } ×(P − ΘQ)−∗ (P − ΘQ)−1 {P − Q + ΘQΘ∗ − ΘP Θ∗ } (P − ΘQ)−∗

and hence the result since J = P − Q.




The function defined by (2.16) is called the Potapov–Ginzburg transform of Θ. We have
 A − BD−1 C BD−1 × Θ = . (2.19) −D−1 C D−1 Theorem 2.22. The Potapov–Ginzburg transform of V is a unitary completion of the reflection coefficient function.

Indeed, from (2.19) the 22 block of the Potapov–Ginzburg transform of V is exactly R. It is not a minimal completion (in particular it has n poles in C− ). See [20] for more information on this transform. Minimal unitary completions of a proper contraction are studied in [33, Theorem 4.1 p. 236]. 2.3. The continuous orthogonal polynomials As already mentioned, for every x ≥ 0 the function λ → Θ(x,λ) = U (x,λ)U (0,λ)−1 is entire. Albeit their name, the continuous orthogonal polynomials are entire functions, first introduced by M.G. Kre˘ın (see [37]) and in terms of which one can

Analogs of Canonical Systems with Pseudo-exponential Potential

15

compute the matrix function Θ(x, λ). To define these functions we start with a function W of the form  W (λ) = In − eitλ ω(t)dt, λ ∈ R, (2.20) R

with ω ∈ Ln×n (R) and such that W (λ) > 0 for all λ ∈ R. This last condition 1 insures that the integral equation  T ΓT (t, s) − ω(t − u)ΓT (u, s)du = ω(t − s), t, s ∈ [0, T ] 0

has a unique solution for every T > 0.

Definition 2.23. The continuous orthogonal polynomial is given by:
  2t P (t, λ) = eitλ In + Γ2t (u, 0)e−iλu du . 0

Theorem 2.24. It holds that   In In Θ(x, λ) = P (t, −λ) R(t, λ)   2t where R(t, λ) = eitλ In + 0 Γ2t (2t − u, 2t)e−iλu du .

In view of Theorem 2.20, note that every rational function analytic at infinity, such that W (∞) = In , with no poles and strictly positive on the real line, is the spectral function of a canonical differential expression of the form (1.1) with potential of the form (2.1). Furthermore, let W (λ) = In + C(λIIp − A)−1 B be a minimal realization of W . Then, W is of the form (2.20) with

iCe−iuA (IIp − P )B, u > 0, ω(u) = −iCe−iuA P B, u < 0,

where P is the Riesz projection of A in C+ . We recall that  P = (ζIIp − A)−1 dζ γ

where γ is a positively oriented contour which encloses only the eigenvalues of A in C+ .

Theorem 2.25. Let W be a rational Cn×n -valued function analytic and invertible on R and at infinity. Assume moreover that W (λ) > 0 for real λ and that W (∞) = In . Let W (λ) = In + C(λIIp − A)−1 B be a minimal realization of W . Let P (resp. P × ) denote the Riesz projection corresponding to the eigenvalues of A (resp. of A× = A − BC) in C+ . Then, the continuous orthogonal polynomials P (t, λ) are given by the formula   × P (t, λ) = eiλt In + C(λIIp + A× )−1 (IIp − e−2iλt e−2itA )π2t B where

×

πt = (IIp − P + P e−itA )−1 (IIp − P ).

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D. Alpay and I. Gohberg

Furthermore, lim e−itλ P (t, λ) = S− (−λ)∗ .

t→∞

(2.21)

See [7, Theorem 3.3 p 10]. The computations in [7] use exact formulas for the function ΓT (t, s) in terms of the realization of W which have been developed in [15]. We note that the potential k(x) can be written as −1  × k(x) = 2C P e−2ixA |Im P PB (2.22) in terms of the realization of the spectral function W .

2.4. Perturbations In this subsection we address the following question: assume that k(x) is a strictly pseudo-exponential potential. Is −k(x) also such a potential? This is not quite clear from formulas (2.1) or (2.22). One could attack this problem using the results in [11], where we studied a trace formula for a pair of self-adjoint operators corresponding to the potentials k(x) and −k(x). Here we present a direct argument in the rational case. More precisely, if N is a Nevanlinna function so are the three functions λ λ λ

→ −N −1 (λ),

→ −N −1 (−λ∗ )∗ , → N (−λ∗ )∗ ,

and we have three associated weight functions W− (λ) W1 (λ)

= =

W2 (λ)

=

Im − N (λ)−1 , Im − N (−λ∗ )−∗ , Im N (−λ∗ )∗ .

The relationships between these three weight functions and the original weight function W and the associated potential have been reviewed in the thesis [36] and we recall the results in form of a table: The potential The weight function
 0 k(x) v(x) = W (λ) = Im N (λ) k(x)∗ 0
 0 k(x) −v(x) = − W− (λ) = Im − N (λ)−1 k(x)∗ 0
 0 k(x)∗ − W1 (λ) = Im N (−λ∗ )∗ k(x) 0 
0 k(x)∗ W2 (λ) = Im − N (−λ∗ )−∗ k(x) 0

Analogs of Canonical Systems with Pseudo-exponential Potential

17

Let N (λ) = i(I + c(λI − a)−1 b)

be a minimal realization of N . Then,

W (λ) = I + C(λI − A)−1 B is a minimal realization of the weight function W , where

 1 a 0 b c A= , B= , C= 0 a∗ c∗ 2

b∗ ,

(2.23)

and the Riesz projection corresponding to the spectrum of A in the open upper half-plane C+ is
 I 0 P = . (2.24) 0 0 Furthermore, the potential associated to the weight function W is given by (2.22) where A, B, C and P are given by (2.23) and (2.24), and
 ∗ a − bc − bb2 2 ∗ A× = A − BC = . ∗ − c2c (a − bc 2) Consider now the weight function W− . A minimal realization of −N (λ)−1 is given by −N (λ)−1 = i(I − c(λI − a× )−1 b), a× = a − bc, and a minimal realization of W− is given by

W− (λ) = I + C− (λI − A− )−1 B− , where A− =




a× 0

0 a

×∗



,

B− = B =






b c∗

,

C− = −C = −

1 c 2

b∗ ,

and the Riesz projection corresponding to the spectrum of A− in the open upper half-plane C+ is P− = P given by (2.24).

The potential associated to the weight function W− is given by  −1 × k− (x) = −2C P e−2itA− |Im P P B, where

A× − = A− − B− C− = Setting D= we have




a − bc 2 0

0 ∗ (a − bc 2)

A× = D − Z






bc 2 c c 2

a− ∗

,

(a

Z=

bb∗ 2 ∗ − bc 2)




0

∗

cc 2

and A× − = D + Z.



.

b∗ b 2

0



,

18

D. Alpay and I. Gohberg

We are now in a position to prove the following result: Theorem 2.26. Let k(x) be a strictly pseudo-exponential potential with associated Weyl function N (λ). The potential associated to Im − N −1 is equal to k− (x) = −k(x). Proof. To prove that k− (x) = −k(x), it is enough to prove that ×

P e−itA |Im

P

= P e−it(A− −B− C− ) |Im

P.

To prove this equality, it is enough in turn to prove that for all positive integers ℓ, it holds that P A×ℓ |Im P = P (A− − B− C− )ℓ |Im P , i.e., that



 I 0 I 0 I 0 I 0 = (D − Z)ℓ (D + Z)ℓ 0 0 0 0 0 0 0 0

for all positive integers ℓ. Let ǫ = ±1. The expression (D + ǫZ)ℓ consists of a sum of terms of the form Dα1 (ǫZ)β1 Dα2 (ǫZ)β2 · · · ,  where the αi and the βi are equal to 1 or 0 and i (αi + βi ) = ℓ. Each factor Dαi Z βi for which βi = 0 is anti block diagonal. We consider two cases, namely   β being odd or even. When β is odd, we have the product of an odd i i i i number of anti block diagonal matrices, and the result is
antiblock diagonal, and so, premultiplying and postmultiplying this product by I0 00 we obtain the zero  matrix. When i βi is even, the product is an even function of ǫ and have the same value at ǫ = 1 and at ǫ = −1. The case of the other two weight functions is treated in much the same way. We focus on W1 (λ) = Im N (−λ∗ )∗ . A minimal realization of N (−λ∗ )∗ is given by N (−λ∗ )∗ = i(I − b∗ (λI + a∗ )−1 c∗ ),

and a minimal realization of the weight function W1 is therefore given by W1 (λ) = I + C1 (λI − A1 )−1 B1 ,

where




−a∗ 0 and the Riesz projection half-plane C+ is P1 = P function W1 is given by A1 =


∗  1 0 c , B1 = , C1 = − b∗ c , −a b 2 corresponding to the spectrum of A1 in the open upper given by (2.24). The potential associated to the weight

 × k1 (x) = 2C1 P1 e−2itA1 |Im

P1

We claim that k1 (x) = −k(x)∗ . Indeed,  ∗× k1 (x)∗ = 2B1∗ P1∗ P1 e2itA1 |Im

−1

P1

P1 B1 .

−1

P1∗ C1∗ .

Analogs of Canonical Systems with Pseudo-exponential Potential But we have that B1∗ P1∗

 = 2CP = c

which allows to conclude.



0 ,

P1 C1∗

1 = −P B = − 2




b 0



,

19

× A∗× 1 = −A ,




3. The discrete case 3.1. First-order discrete system In our previous work [6] we studied inverse problems for difference operators associated to Jacobi matrices. Such operators are the discrete counterparts of Sturm– Liouville differential operators, and one can associate to them a number of functions analytic in the open unit disk similar to the characteristic spectral functions of a canonical differential expression. In the present paper we chose a different avenue to define discrete systems, which has more analogy to the continuous case and is more natural. The analogies between the two cases are gathered in form of two tables at the end of the paper. We note that another type of discrete systems has been considered by L. Sakhnovich in [42, Section 2 p. 389]. Our starting point is the telegraphers’ equations (1.2). We now assume that the local impedance function Z(x) defined in (1.2) is equal to a constant, say Zn , on the interval [nh, (n + 1)h) for n = 0, 1, . . . In particular, Z(x) may have discontinuities at the points nh. On the open interval (nh, (n + 1)h), we have k(x) = 0 and equation (1.3) becomes
∂  ∂ ( ∂x + ∂t ) 0 W (x, t) = 0. ∂ ∂ 0 ( ∂x − ∂t ) Hence one can write


 v1n (x − t) v2n (x + t) on the interval (nh, (n + 1)h). Voltage and current are continuous at the points nh. Let us set α(n, t) = lim W (x, t). W (x, t) =

x→nh x>nh

Taking into account (1.3) one gets to:  1 Zn−1/2 α(n, t) = 2 Zn−1/2  1 Zn−1/2 −1 lim W (x, t) = x→nh 2 Zn−1/2 −1 x 0 and the function √1t H0 (z) is J-unitary on the unit circle, with minimal realization 1 1 1 √ H0 (z) = √ D + C(zI − A)−1 √ B. t t t

26

D. Alpay and I. Gohberg

The associated Hermitian matrix to this realization is given by
 −Ω −IIp X= . −IIp −a∆a∗ We now recall the analogue of Theorem 2.5 for minimal J-unitary factorizations on the unit circle (see [5, Theorem 3.7 p. 205]): Theorem 3.7. Let R be a rational function J-unitary on the unit circle and analytic and invertible at ∞. Let R(z) = D + C(zI − A)−1 B be a minimal realization of R, with associated Hermitian matrix H. Let M be a A-invariant subspace nondegenerate in the metric [·, ·]H induced by H. Finally, let π denote the orthogonal projection defined by ker π = M, Im π = M[⊥] . Then R = R1 R2 with

R1 (z) = (I + C(zI − A)−1 (I − π)BD−1 )D1−1 R2 (z) = D2 (I + D−1 Cπ(zI − A)−1 B)D with D1 = I + C1 H1−1 (I − αA∗1 )−1 C1∗ J,

D2 = DD1−1

where |α| = 1 and

C1 = C|M ,

A1 = A|M ,

H1 = πH|M

is a minimal J-unitary factorization of R, and every minimal J-unitary factorization of R is obtained in such a way. Using this result we obtain: Theorem 3.8. The matrix function H0 admits a minimal J-unitary factorization H0 (z) = U1 (z)−1 U2 (z) where U1 and U2 are J-inner. The asymptotic equivalence matrix function admits a minimal J-unitary factorization 1 V (z) = V1 (z)−1 V2 (z) det H0 (z) where V1 and V2 are J-inner.


p



Indeed, the space C0 is A invariant and H-negative. Furthermore,



∗ −Ω −IIp Ip 0 −Ω 0 Ip 0 = , −IIp −a∆a∗ Ω−1 Ip 0 Ω−1 − a∆a∗ Ω−1 Ip and by (3.6) and (3.4), Ω−1 − a∆a∗ > 0. This insures that U2 is J-inner.

To prove the second claim, we remark that the function


set V1 (z) = U2 (z)

1 0

0 z −1






and V2 (z) = U1 (z)

1 0

0 z −1



.




1 0

0 z −1



is J-inner and


Analogs of Canonical Systems with Pseudo-exponential Potential

27

3.3. The reflection coefficient function and the Schur algorithm We now associate to a one-sided first-order discrete system a function analytic and contractive in the open unit disk. We first set
 1 −ρ C(ρ) = −ρ∗ 1 and Mn (z) = C(ρ0 )


z 0


0 z C(ρ1 ) 1 0


0 z · · · C(ρn ) 1 0

 0 . 1

(3.21)

Theorem 3.9. Let ρn , n = 1, 2, . . . be a strictly pseudo-exponential sequence and let Mn (z) be defined by (3.21). The limit R(z) = lim TMn (z) (0)

(3.22)

n→∞

exists and is equal to β0 (1/z). α0 It is a function analytic and contractive in the open unit disk, called the reflection coefficient function. It takes strictly contractive values on the unit circle. R(z) =

Proof. From (3.15) we have that:  n 
n+1  z 2 Mn (z) = (1 − |ρℓ | ) H0 (z ∗ )∗ 0 ℓ=0

 0 Hn+1 (z ∗ )∗ . 1

The result follows then from the definition of the linear fractional transformation and from the equality (see (3.16)) γ0 (z ∗ )∗ β0 = (1/z). δ0 (z ∗ )∗ α0

For every n the matrix function

n

ℓ=0

√1

1−|ρℓ |2

Mn is J-inner and thus the function

TMn (z) (0) is analytic and contractive in the open unit disk. It follows that R(z) is analytic and contractive in the open unit disk. The fact that R(z) is strictly contractive on T is proved as follows. One first notes that α0 and β0 have no pole on the unit circle. From the J-unitarity on the unit circle of √ 1 H0 (z) (recall det H0 (z)

that det H0 (z) is a strictly positive constant; see (3.17)) stems the equality 1 , 2 ℓ=0 (1 − |ρℓ | )

|α0 (z)|2 − |β0 (z)|2 = det H0 (z) = ∞

and hence | αβ00 (z)| < 1 for z ∈ T.

z ∈ T,


We note the complete analogy between the characterizations (2.10) and (3.22) of the reflection coefficient functions for the continuous and discrete cases respectively.

28

D. Alpay and I. Gohberg

We now present a realization for R: Theorem 3.10. Let ρn , n = 0, 1, . . . be a strictly pseudo-exponential sequence of the form (3.3). The reflection coefficient function of the associated discrete system (3.2) is given by the formula: R(z) = c {(I − ∆a∗ Ωa) − z(I − ∆Ω)a}

−1

b.

(3.23)

In particular R(0) = c(I − ∆a∗ Ωa)−1 b = −ρ0 . Proof. We first compute α0 (z)−1 using the formula (1 + AB)−1 = 1 − A(I + BA)−1 B

with A = cz(zI − a)−1 and B = (I − ∆Ω)−1 ∆c∗ . We obtain

α0 (z)−1 = 1 − cz(zI − a)−1 (I + (I − ∆Ω)−1 ∆c∗ cz(zI − a)−1 )−1 (I − ∆Ω)−1 ∆c∗ −1

= 1 − cz {(I − ∆Ω)(zI − a) + ∆c∗ cz}

Therefore

∆c∗ .

 α0 (z)−1 β0 (z) = 1 − cz {(I − ∆Ω)(zI − a) + ∆c∗ cz}−1 ∆c∗ × (cz(zI − a)−1 (I − ∆Ω)−1 b)

= cz(zI − a)−1 (I − ∆Ω)−1 b

−1

− cz {(I − ∆Ω)(zI − a) + ∆c∗ cz} × ∆c∗ cz(zI − a)−1 (I − ∆Ω)−1 b.

Writing ∆c∗ cz = (I − ∆Ω)(zI − a) + ∆c∗ cz − (I − ∆Ω)(zI − a),

we have that

−1

α0 (z)−1 β0 (z) = cz {(I − ∆Ω)(zI − a) + ∆c∗ cz} × (zI − a)−1 (I − ∆Ω)−1 b,

(I − ∆Ω)(zI − a)

and hence the result since (I − ∆Ω)(zI − a) + ∆c∗ cz = z(I − ∆a∗ Ωa) − (I − ∆Ω)a.




The Schur algorithm starts from a function R(z) analytic and contractive in the open unit disk (a Schur function), and associates to it recursively a sequence of functions Rn with R0 (z) = R(z) and, for n ≥ 1: Rn+1 (z) =

Rn (z) − Rn (0) . z(1 − Rn (0)∗ Rn (z))

The recursion continues as long as |Rn (0)| < 1. By the maximum modulus principle, all the functions in the (finite or infinite) sequence are Schur functions; see [43], [23].

Analogs of Canonical Systems with Pseudo-exponential Potential

29

The numbers ρn = Rn (0) bear various names: Schur coefficients, reflection coefficients,. . . . They give a complete characterization of Schur functions. In various places (see, e.g., [44]), they are also called Verblunsky coefficients. Theorem 3.11. Let ρn be a strictly pseudo-exponential sequence. The functions  −1 βn Rn (z) = (1/z) = can (I − ∆a∗(n+1) Ωan+1 ) − z(I − ∆a∗n Ωan )a b αn are Schur functions. Furthermore, the Schur coefficients of Rn are −ρm , m ≥ n.

Proof. The first claim follows from the previous theorem, replacing c by can and Ω by a∗n Ωan . To prove the second fact, we rewrite (3.18) (with m instead of n) as: αm+1 (z) = βm+1 (z) = zγm+1 (z) = δm+1 (z) =

αm (z) + ρ∗m βm (z),

(3.24)

z(ρm αm (z) + βm (z)), γm (z) + ρ∗m δm (z),

(3.25)

δm (z) + ρm γm (z)

Dividing (3.25) by (3.24) side by side we obtain: βm (z) + ρm βm+1 (z) = z αm αm+1 1 + ρ∗m αβm (z) m




and hence the result. Corollary 3.12. For every n ≥ 0 there exists a Schur function Sn such that R = TMn (Sn ).

(3.26)

3.4. The scattering function We now turn to the scattering function. We first look for the C2 -valued solution of the system (3.2), with the boundary conditions  1 −1 Y0 (z) = 0,  0 1 Yn (z) = 1 + o(n).

The first condition implies that the solution is of

n n−1  1 0 z Yn (z) = (1 − |ρℓ |2 ) Hn (z)−1 0 z 0 ℓ=0

the form 
0 1 H0 (z) 1 0


 x(z) z −1 x(z) 0

where x(z) is to be determined via the second boundary condition. We compute
n 
 n−1    x(z) z 0 0 1 Yn (z) = (1 − |ρℓ |2 ) 0 z Hn (z)−1 H0 (z) x(z) . 0 1 z ℓ=0

Taking into account that limn→∞ Hn (z) = I2 we get that ∞    lim 0 1 Yn (z) = (1 − |ρℓ |2 ) 0

n→∞

ℓ=0


 x(z) z H0 (z) x(z) z

30

D. Alpay and I. Gohberg

∞ and hence 1 = ( ℓ=0 (1 − |ρℓ |2 ))(zγ0 (z) + δ0 (z))x(z), that is Furthermore,  lim 1

n→∞

1 x(z) = ∞ . 2 ( ℓ=0 (1 − |ρℓ | ))zγ0 (z) + δ0 (z)




  1 0 x(z) 1 0 0 Yn (z)z −n = 1 0 H0 (z) 0 z −1 x(z) 0 z ∞      α0 (z) + β0 (z) 2 z = (1 − |ρℓ | ) 1 0 x(z) γ0 (z) + δ0z(z) ℓ=0 α0 (z) + β0z(z) . zγ0 (z) + δ0 (z)

= Definition 3.13. The function

S(z) =

α0 (z) + β0z(z) zγ0 (z) + δ0 (z)

is called the scattering function associated to the discrete system (3.2). Theorem 3.14. The scattering function admits the factorizations S(z) = S+ (z)S− (z) =

B1 (z) B2 (z)

where S+ and its inverse are invertible in the closed unit disk, S− and its inverse are invertible in the outside of the open unit disk, and where B1 and B2 are two finite Blaschke products. Proof. Using (3.16) we see that β0 (z) z and so S takes unitary values on the unit circle. It follows from Theorem 3.9 and from [24, Theorem 3.1, p. 918] that (zγ0 ) (1/z ∗ )∗ =

zγ0 (z) + δ0 (z) = δ0 (z)(1 + zR(z ∗ )∗ ) is analytic and invertible in |z| < 1. This gives the first factorization with 1 , zγ0 (z) + δ0 (z) 1 β0 (z) S− (z) = = α0 (z) + . S+ (1/z ∗ )∗ z S+ (z) =

The second factorization is a direct consequence of the fact that S is rational and takes unitary values on T.


Analogs of Canonical Systems with Pseudo-exponential Potential

31

3.5. The Weyl function and the spectral function To introduce the Weyl coefficient function we consider the matrix function 1 Un (z) = √ 2


1 1


 ℓ=n−1 
1 1 −ρ∗ℓ −1 ℓ=0

−ρℓ 1


z 0

 
0 1 1 1 √ . 1 2 1 −1

Definition 3.15. The Weyl coefficient function N (z) is defined for z ∈ D by the
iN (z ∗ )∗ following property: The sequence n → Un (z) belongs to ℓ22 . 1 A similar definition appears in [40, Theorem 1, p. 231]. Theorem 3.16. It holds that 1 − zR(z) . (3.27) 1 + zR(z) n−1 Proof. Indeed, by (3.15) and with cn−1 = ℓ=0 (1 − |ρℓ |2 ), we have that:

 
cn−1 1 1 iN (z ∗ )∗ 1 0 Un (z) = Hn (z)−1 1 1 −1 0 z 2
n 

 z 0 1 0 1 + iN (z ∗ )∗ × H0 (z) 0 1 0 z −1 −1 + iN (z ∗ )∗ 

n 
cn−1 1 1 1 0 z 0 = Hn (z)−1 1 −1 0 z 0 1 2
 β0 (z) ∗ ∗ α0 (z)(1 + iN (z ) − z (1 − iN (z ∗ )∗ ) × , zγ0 (z)(1 + iN (z ∗ )∗ ) − δ0 (z)(1 − iN (z ∗ )∗ )
 iN (z ∗ )∗ and so the sequence n → Un (z) belongs to ℓ22 if and only if it holds 1 that N (z) = i

zγ0 (z)(1 + iN (z ∗ )∗ ) = δ0 (z)(1 − iN (z ∗ )∗ ).

(3.28)

This equation in turns is equivalent to iN (z) =

zγ0 (z ∗ )∗ − δ0 (z ∗ )∗ zβ0 (1/z) − α0 (1/z) zR(z) − 1 = = . zγ0 (z ∗ )∗ + δ0 (z ∗ )∗ zβ0 (1/z) + α0 (1/z) zR(z) + 1

where we took into account (3.16).

(3.29)


For similar results, see [44, Theorem 5.2 p. 520]. Theorem 3.17. The Weyl coefficient function associated to a one-sided first-order discrete system with strictly pseudo-exponential sequence is given by:  −1 N (z) = i 1 + 2zc {I − ∆a∗ Ωa + zbc − z(I − ∆Ω)a} b . (3.30)

32

D. Alpay and I. Gohberg  1 − 2(1 + zR(z))−1 . On the other hand,  −1 −1 = 1 + zc {(I − ∆a∗ Ωa) − z(I − ∆Ω)a} b

Proof. We have N (z) = (1 + zR(z))−1

1 zR(z)−1 i zR(z)+1

=

1 i

−1

= 1 − zc {(I − ∆a∗ Ωa) − z(I − ∆Ω)a} −1  −1 × 1 + zbc {(I − ∆a∗ Ωa) − z(I − ∆Ω)a} b −1

and hence the result.

= 1 + zc {I − ∆a∗ Ωa + zbc − z(I − ∆Ω)a}

b,




Remark 3.18. Let N be the Weyl function associated to the sequence ρn , n = 0, 1, 2, . . .. Then −N −1 is the Weyl function associated to the sequence −ρn , n = 0, 1, 2, . . .. The spectral function W (z) =

c , |α0 (1/z) + zβ0 (1/z)|2

1 , (1 − |ρℓ |2 ) ℓ=0

c = ∞

will play an important role in the sequel.

|z| = 1.

(3.31)

Theorem 3.19. The Weyl coefficient function N (z) is such that Im N (z) = W (z) on the unit circle. Proof. From (3.16) we have that |α0 (z)|2 − |β0 (z)|2 is a constant for |z| = 1. Therefore:   1 1 zR(z) − 1 1 z ∗ R(z)∗ − 1 Im N (z) = + 2i i zR(z) + 1 i z ∗ R(z)∗ + 1 2 1 − |R(z)| = |1 + zR(z)|2 |α0 (1/z)|2 − |β0 (1/z)|2 = = W (z). |α0 (1/z) + zβ0 (1/z)|2
Theorem 3.20. The characteristic spectral functions of a one-sided first-order discrete system are related by the formulas c 1 W (z) = , z ∈ T, c = ∞ , |S− (1/z)|2 (1 − |ρℓ |2 ) ℓ=0 W (z) = Im N (z), z ∈ T, 1 − zR(z) , 1 + zR(z) 1 1 + iN (z) R(z) = , z 1 − iN (z)
1 (1 + iN (z ∗ )∗ )S+ (z)−1 V (z) = 2 −(1 − iN (z ∗ )∗ )S+ (z)−1

N (z) = i

 −(1 + iN (1/z))S−(1/z) . (1 − iN (1/z))S−(1/z)

Analogs of Canonical Systems with Pseudo-exponential Potential

33

We will prove only the last identity. From (3.19) and (3.28) we have that 1 + iN (z ∗ )∗ δ0 (z) = 2 zγ0 (z) + δ0 (z)

1 + iN (z ∗ )∗ zγ0 (z) = . 2 zγ0 (z) + δ0 (z)

and

Thus, 1 + iN (z ∗ )∗ S+ (z)−1 2 Similarly, from (3.29) we obtain δ0 (z) =

and zγ0 (z) =

1 + iN (z) zβ0 (1/z) = 2 zβ0 (1/z) + α0 (1/z)

1 − iN (z ∗ )∗ S+ (z)−1 . 2

1 − iN (z) α0 (1/z) = , 2 zβ0 (1/z) + α0 (1/z)

and

and hence the result. 3.6. The orthogonal polynomials The solution Mn (given by (3.21)) to the system (3.2) with the initial condition M0 (z) = I2 is polynomial. It can be expressed in terms of the orthogonal polynomials associated to the weights Im N (z) and Im − N −1 (z) (where |z| = 1), and we recall now  the definition of the  orthogonal polynomials. We start with a function W (eit ) = Z wℓ eiℓt such that Z |wℓ | < ∞ (that is, W belongs to the Wiener algebra of the unit circle). We assume moreover that W (eit ) > 0 for all real t. Set ⎛ ⎞ ∗ w1∗ ··· wm w0 ∗ ⎜ w1 ⎟ w0 . . . wm−1 ⎜ ⎟ Tm = ⎜ . (3.32) ⎟. . . .. .. ⎝ .. ⎠ w0 wm wm−1 · · · Then Tm is invertible, and we define: ⎛ (m) (m) γ00 γ01 ⎜ (m) (m) ⎜ γ10 γ11 ⎜ . T−1 = .. m ⎜ . ⎝ . . (m) (m) γm0 γm1 Definition 3.21. The family

1

⎛

γ0m (m) γ1m .. .

···

γmm

m 

⎝ pm (z) =  (m) j=0 γ00

⎞

(m)

··· ···

⎟ ⎟ ⎟. ⎟ ⎠

(m)

⎞

(m) γ0j z m−j ⎠

is called the family of orthonormal polynomials associated to the sequence wj . The term orthonormal is explained in the next theorem: Theorem 3.22. We have  2π 1 pk (eit )W (eit )pm (eit )∗ dt = δk,m . 2π 0

34

D. Alpay and I. Gohberg

We now consider a rational function W , analytic on T and at the origin. Then, W admits a minimal realization of the form W (z) = D + zC(IIp − zA)−1 B. The function W is in the Wiener algebra of the unit circle. Indeed, the matrix A has no spectrum on T and the Fourier coefficients of W are given by ⎧ ⎨ CAℓ−1 (I − P )B if ℓ = 1, 2, . . . wℓ = D − CP B if ℓ = 0 ⎩ −CAℓ−1 P B if ℓ = −1, −2, . . .

where P is the Riesz projection defined by  1 P =I− (ζI − A)−1 dζ. 2πi T Indeed, we have for |z| = 1:

W (z) = D + zC(I − zA)−1 B

= D + zC(I − zA)−1 (P + I − P )B

∞  = D + zC( z ℓ (A(I − P ))ℓ )B ℓ=0

− C(AP )−1 (I − z −1 (AP )−1 )−1 B,

and hence the result. Furthermore, for every m, the matrix Vm = (I − P + P A)−m (I − P + P A×m )

is invertible (with A× = A − BD−1 C). Moreover, a) for 0 ≤ j < i ≤ m. (m)

γij

−(m+1) = (D−1 C(A× )i Vm−1 (A× )m−j B − D−1 C(A× )i−j−1 BD−1 ). +1 P A

b) for 0 ≤ i ≤ j ≤ m (m)

γij

−(m+1) = δij D−1 + D−1 C(A× )i Vm−1 (A× )m−j BD−1 . +1 P A

These results are proved in [28, pp. 35–37] when D = I. They allow to prove: Theorem 3.23. Let W be a rational matrix-valued function analytic and invertible at the origin and infinity, and analytic on the unit circle. Let W (z) = D + zC(I − zA)−1 B be a minimal realization of W . Suppose that W (eit ) > 0, t ∈ [0, 2π]. Then, (1)

−(m+1) ×m pm (z) = (D−1 + D−1 CV Vm−1 A B)−1/2 +1 P A ⎧ ⎫ m ⎨ ⎬  −(m+1) × z m D−1 + D−1 CV Vm−1 ( A×(m−j) z m−j ) B. +1 P A ⎩ ⎭ j=0

Analogs of Canonical Systems with Pseudo-exponential Potential (2) For |z|
0, (m)

γ0j

(m)

× = w−j + K0j

= −CE × Ω×(j−1) P × B

+CE × Ω× (I − P × )V Vm−1 (I − Q)E × Ω×j P × B

−CE × Ω×m P × Vm−1 QE × Ω×(m−j) (I − P × )B.

Analogs of Canonical Systems with Pseudo-exponential Potential

37

Thus z m pm (1/z) = =

(m)

(m)

(m)

γ00 + zγ01 + · · · + z m γ0m In + CE × (I − P × )B m  −CE × Ω×−1 z j Ω×j P × B j=1

⎛

+CE × Ω× (I − P × )V Vm−1 (I − Q)E × ⎝ ⎛

−CE × Ω×m P × Vm−1 QE × Ω×m ⎝

m  j=0

from which the claim follows.

m  j=0

⎞

z j Ω×j ⎠ P × B ⎞

z j Ω×−j ⎠ (I − P × )B




One can also consider representations of the form W (z) = D + (1 − z)C(zG − A)−1 B. (m)

See [35]. One needs to develop formulas for the γij . Such formulas and the corresponding formulas for the orthogonal polynomials will be given elsewhere. 3.7. The spectral function and isometries Let

 1 1 1 0 1 U= √ and J1 = . 1 0 2 1 −1 We note that

J = U J1 U. Furthermore, let Θn (z) = U Mn (z)U where Mn (z) is given by (3.21). The matrix function Θn is J1 -inner. We denote by H(Θn ) the associated reproducing kernel (z)J1 Θn (w)∗ Hilbert space, with reproducing kernel J1 −Θn1−zw . We denote by L(N ) the ∗ reproducing kernel Hilbert space with reproducing kernel

Theorem 3.27. The map

N (z)−N (w)∗ i(1−zw ∗ ) .

 F → −iN (z) 1 F (z)

is an isometry from H(Θn ) into L(N ). Furthermore, elements of H(Θn ) are of the form
 f (z) F (z) = , i(pN ∗ f )(z) where f runs through the set of polynomials of degree less or equal to n and where p denotes the orthogonal projection from L2 onto H2 , and F 2H(Θn) = 2f 2L2 (Im

N ).

(3.37)

38

D. Alpay and I. Gohberg

Proof. Let us denote by H(R) the reproducing kernel Hilbert space with repro∗ R(z)R(w)∗ ducing kernel 1−zw1−zw . Then, by e.g., [2, Propositions 6.1 and 6.4] (but ∗ the result is well known and is related to the Carath´ ´eodory–Toeplitz extension problem), equation (3.26) implies that the map which to F associates the function  z → 1 −zR(z) F (z)

is an isometry from H(M Mn ) into H(R). Since

J1 − Θn (z)J J1 Θn (w)∗ J − Mn (z)JM Mn (w)∗ ∗ =M M , ∗ 1 − zw 1 − zw∗ N (z) − N (w)∗ 2 1 − zw∗ R(z)R(w)∗ 1 = , ∗ ∗ ∗ i(1 − zw ) 1 + zR(z) 1 − zw 1 + w R(w)∗

the maps

F → M F √ 2 f → f (1 + zR) are isometries from H(Θn ) onto H(M Mn ) and from H(R) onto L(N ). The first claim follows since √   2 −iN (z) 1 = 1 −zR(z) M. 1 + zR(z) The last claim can be obtained from [3, Section 7].
We note that a similar result for the continuous case was proved in [11]. The arguments are easier here because of the finite dimensionality. Using Theorem 3.27 we can relate the orthogonal polynomials and the entries of the matrix function Θn . Corollary 3.28. Let Θn be as in Theorem 3.27. Then for ℓ, k < n %

& 1 1 Θℓ , Θk = 2δℓ,k . 1 1 H(Θ ) n

In particular, for every n ≥ 0,

 pn (z) = 1


 1 0 Θn (z) . 1

Proof. Denote by H2,J the Kre˘ ˘ın space of C2 -valued functions with entries in the Hardy space H2 of the open unit disk, and with inner product: [F, G]H2,J = F, JGH22 . Then (see [4]), the space H(M Mn ) is isometrically included inside H2,J . Assume now that ℓ < k. The function
 k  z 0 −1 (Θℓ Θk )(z) = U C(ρi ) U 0 1 i=ℓ+1

Analogs of Canonical Systems with Pseudo-exponential Potential

39

belongs to H2,J and is such that (Θ−1 ℓ Θk )(0)



 1 0 = . 1 0

Thus, %

Θℓ



& 1 1 , Θk 1 1 H(Θ

= n)

%

& 1 1 , Θ−1 Θ k ℓ 1 1 H(Θ

=0 n)

The proof that the inner product is equal to 2 when ℓ = k is proved in the same way. The last claim follows from (3.37).


4. Two-sided systems and an example 4.1. Two-sided discrete first-order systems We now turn to the systems of the form (3.1), that is,

 1 −ρn z 0 Yn+1 (z) = Yn (z), −ρ∗n 1 0 z −1 and begin with the definition of the asymptotic equivalence matrix function. Theorem 4.1. Let ρn be a strictly pseudo-exponential sequence. Every solution of the system (3.1) is of the form

n 
 n−1  1 0 0 1 0 2 2 −1 z 2 (1 − |ρℓ | ) Hn (z ) H0 (z ) Y0 (z). Yn (z) = 0 z2 0 z −n 0 z12 ℓ=0

The solution such that


−n z lim n→∞ 0

 0 Yn (z) = I2 zn

corresponds to 1 Y0 (z) = ∞ (1 − |ρℓ |2 ) ℓ=0



1 0 2 −1 1 H0 (z ) 0 z2 0

0 z −2



,

while the solution with value I2 at n = 0 corresponds to Y0 (z) = I2 . Proof. Replacing z by z 2 in the recursion (3.18) we obtain:


 1 0 1 0 1 ρn 2 Hn+1 (z 2 ) = H (z ) . n 0 z12 0 z12 ρ∗n 1 Note that




1 −ρ∗n

−ρn 1




1 ρ∗n

ρn 1



= (1 − |ρn |2 )II2 .

(4.1)

40

D. Alpay and I. Gohberg

Thus, multiplying side by side (4.1) and (3.1) we obtain:


 1 0 1 0 z 0 2 2 2 Hn+1 (z ) Yn+1 (z) = (1 − |ρn | ) Hn (z ) Yn (z) 0 z12 0 z12 0 z −1

 z 0 1 0 = (1 − |ρn |2 ) Hn (z 2 ) Yn (z) 0 z −1 0 z12 from which we obtain:
 1 0 Hn+1 (z 2 ) Yn+1 (z) = 0 z12
n+1 z = 0

0 z −(n+1)




1 H0 (z ) 0

0

2

1 z2



Y0 (z)



n 

ℓ=0

2

1 − |ρℓ |




and hence the formula for Yn (z). Definition 4.2. The function 1 V (z) = n−1 2 ℓ=0 (1 − |ρℓ | )


1 0


0 2 −1 1 H (z ) 0 0 z2

0 z −2



is called the asymptotic equivalence matrix of the two-sided first-order discrete system (3.1). We note that it is related to the asymptotic equivalence matrix (3.19) of the discrete system (3.2) by the transformation z → z 2 . The proof of the following result is similar to the proof of Theorem 3.4. Theorem 4.3. Let c1 and c2 be in C2 , and let Y (1) and Y (2) be the C2 -valued solutions of (3.1), corresponding to the case of ρn ≡ 0 and to the strictly pseudo(1) exponential sequence ρn respectively and with initial conditions Y0 (z) = c1 and (2) Y0 (z) = c2 . Then, for every z on the unit circle it holds that lim Y Yn(1) (z)c1 − Yn(2) (z)c2  = 0

n→∞

Proof. By definition,

Yn(2) (z) =

n−1  ℓ=0

(1) Yn (z)

(1 − |ρℓ |2 )


n z = 0


1 0

0 z −n



c2 = V (z)c1 .

⇐⇒

c1 . On the other hand,


n 0 2 −1 z H (z ) n z2 0

0 z −n



H0 (z 2 )




 1 0 c . 0 z −2 2

The result follows since limn→∞ Hn (z 2 )−1 = I2 for z on the unit circle.




Analogs of Canonical Systems with Pseudo-exponential Potential

41

The other spectral functions of the systems (3.2) and (3.1) are also related by the transformation z → z 2 . The definitions and results are identical to the one-sided case. Theorem 4.4. Let ρn , n = 0, 1, . . . be a strictly pseudo-exponential sequence of the form (3.3). The reflection coefficient function of the associated discrete system (3.1) is given by the formula:  −1 R(z) = c (I − ∆a∗ Ωa) − z 2 (I − ∆Ω)a b. (4.2) The scattering function is defined as follows. We look for the C2 -valued solution of the system (3.2), with the boundary conditions  1 −1 Y0 (z) = 0,  0 1 Yn (z) = z −n + o(n).

Then the limit

 lim 1

n→∞

0 Yn (z)z −n

exists and is called the scattering function of the system (3.1). It is related to the scattering function of the system (3.2) by the map z → z 2 . We also mention that J-inner polynomials are now replaced by J-unitary functions with possibly poles at the origin and at infinity, but with constant determinant. 4.2. An illustrative example As a simple example we take a = α ∈ (0, 1), b = 1 and c = c∗ . Then ∆= and

1 , 1 − α2

ρn = −αn

Ω=

c2 , 1 − α2

c 1−

c2 α2n+2 (1−α2 )2

.

(4.3)

The numbers c and α need to satisfy (3.6), that is (1 − α2 )2 > c2 . Note that this condition implies that c c  < |ρ0 | = < 1, c2 1 − α2 2 1 − α (1−α2 )2 and more generally,

|ρn | =

1−

c2 α2n+2 (1−α2 )2 n

α c 1 − α2n+2 αn (1 − α2 ) c αn = < < 1, 2n+2 2 2 1−α 1−α 1 + α + · · · + α2n ≤

as it should be.

αn c

42

D. Alpay and I. Gohberg Continuous case iJf ′ − V f = zf

The system Special solutions Potential

Solution asymptotic to the solution with k ≡ 0

Entire J-inner functions
 0 k(x) v(x) = k(x)∗ 0   −1 ita −2ixa∗ k(x) = −2ce Ip + Ω Y − e Y e2ixa Theorem 2.1

−k is also a potential

Theorem 2.26

Asymptotic property

Formula (2.4)

Reflection coefficient

Formulas (2.11) and (2.10)

Weyl function

Formula (2.14)

Weyl function for −k(x)

Theorem 2.26

Factorization of the asymptotic equivalence matrix

Theorem 2.6

Asymptotic behavior of the orthogonal polynomial

Equation (2.21) Table 1

The reflection coefficient is equal to: R(z) =

1−

α2 c2 (1−α2 )2

c − zα(1 −

c2 (1−α2 )2 )

.

We check directly that it is indeed a Schur function as follows: we have for |z| ≤ 1 c . |R(z)| ≤ α2 c2 c2 1 − (1−α2 )2 − α(1 − (1−α 2 )2 ) We thus need to check that c≤1− that is, with T =

c (1−α2 ) ,

α2 c2 c2 − α(1 − ), (1 − α2 )2 (1 − α2 )2

c ≤ 1 − α2 T 2 − α(1 − T 2 ) = (1 − α)(1 + T 2 α),

Analogs of Canonical Systems with Pseudo-exponential Potential

43

Discrete case (one-sided case)
 z −ρn Yn+1 (z) = Yn (z) −zρ∗n 1

The system Special solutions

J-inner polynomials

Potential: the Schur coefficients ρn

ρn = −can (I − ∆a∗(n+1) Ωan+1 )−1 b

Solution asymptotic to the solution with ρn ≡ 0

Formula (3.14)

−ρn is also pseudo-exponential

Remark 3.1

Asymptotic property

Formula (3.7)

Reflection coefficient

Formulas (3.23) and (3.22)

Weyl function

Formula (3.30)

Weyl function for −ρn

Remark 3.18

Factorization of the asymptotic equivalence matrix

Theorem 3.8

Asymptotic behavior of the orthogonal polynomial

Equation (3.33) Table 2

1 that is, T ≤ 1+α (1 + T 2 α). This last inequality in turn holds since T and α are in (0, 1). Finally, from (3.27) we obtain the expression for the Weyl function:

N (z) = i

1−

1−

α2 c2 (1−α2 )2 α2 c2 (1−α2 )2

− zα(1 −

− zα(1 −

c2 (1−α2 )2 ) c2 (1−α2 )2 )

− zc + zc

.

We summarize the parallels between the continuous case and the one-sided discrete case in Tables 1 and 2.

44

D. Alpay and I. Gohberg

References [1] V.M. Adamyan and S.E. Nechayev. Nuclear Hankel matrices and orthogonal trigonometric polynomials. Contemporary Mathematics, 189:1–15, 1995. [2] D. Alpay, T. Azizov, A. Dijksma, and H. Langer. The Schur algorithm for generalized Schur functions. III. J-unitary matrix polynomials on the circle. Linear Algebra Appl., 369:113–144, 2003. [3] D. Alpay and H. Dym. Hilbert spaces of analytic functions, inverse scattering and operator models, I. Integral Equation and Operator Theory, 7:589–641, 1984. [4] D. Alpay and H. Dym. On applications of reproducing kernel spaces to the Schur algorithm and rational J-unitary factorization. In I. Gohberg, editor, I. Schur methods in operator theory and signal processing, volume 18 of Operator Theory: Advances and Applications, pages 89–159. Birkh¨ auser Verlag, Basel, 1986. [5] D. Alpay and I. Gohberg. Unitary rational matrix functions. In I. Gohberg, editor, Topics in interpolation theory of rational matrix-valued functions, volume 33 of Operator Theory: Advances and Applications, pages 175–222. Birkh¨ a ¨user Verlag, Basel, 1988. [6] D. Alpay and I. Gohberg. Inverse spectral problems for difference operators with rational scattering matrix function. Integral Equations Operator Theory, 20(2):125– 170, 1994. [7] D. Alpay and I. Gohberg. Inverse spectral problem for differential operators with rational scattering matrix functions. Journal of differential equations, 118:1–19, 1995. [8] D. Alpay and I. Gohberg. Inverse scattering problem for differential operators with rational scattering matrix functions. In I. B¨ ¨ ottcher and I. Gohberg, editors, Singular integral operators and related topics (Tel Aviv, 1995), volume 90 of Operator Theory: Advances and Applications, pages 1–18. Birkh¨ auser Verlag, Basel, 1996. ¨ [9] D. Alpay and I. Gohberg. Connections between the Carath´ ´eodory-Toeplitz and the Nehari extension problems: the discrete scalar case. Integral Equations Operator Theory, 37(2):125–142, 2000. [10] D. Alpay and I. Gohberg. Inverse problems associated to a canonical differential system. In L. Kerchy, ´ C. Foias, I. Gohberg, and H. Langer, editors, Recent advances in operator theory and related topics (Szeged, 1999), Operator theory: Advances and Applications, pages 1–27. Birkh¨ auser, Basel, 2001. [11] D. Alpay and I. Gohberg. A trace formula for canonical differential expressions. J. Funct. Anal., 197(2):489–525, 2003. [12] D. Alpay, I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich. Direct and inverse scattering problem for canonical systems with a strictly pseudo-exponential potential. Math. Nachr., 215:5–31, 2000. [13] D. Alpay, I. Gohberg, and L. Sakhnovich. Inverse scattering for continuous transmission lines with rational reflection coefficient function. In I. Gohberg, P. Lancaster, and P.N. Shivakumar, editors, Proceedings of the International Conference on Applications of Operator Theory held in Winnipeg, Manitoba, October 2–6, 1994, volume 87 of Operator theory: Advances and Applications, pages 1–16. Birkh¨ auser Verlag, Basel, 1996.

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[14] H. Bart, I. Gohberg, and M.A. Kaashoek. Minimal factorization of matrix and operator functions, volume 1 of Operator Theory: Advances and Applications. Birkh¨ auser Verlag, Basel, 1979. [15] H. Bart, I. Gohberg, and M.A. Kaashoek. Convolution equations and linear systems. Integral Equations Operator Theory, 5:283–340, 1982. [16] A.M. Bruckstein and T. Kailath. Inverse scattering for discrete transmission-line models. SIAM Rev., 29(3):359–389, 1987. [17] K. Clancey and I. Gohberg. Factorization of matrix functions and singular integral operators, volume 3 of Operator Theory: Advances and Applications. Birkh¨ auser Verlag, Basel, 1981. [18] D de Cogan. Transmission line matrix (LTM) techniques for diffusion applications. Gordon and Breach Science Publishers, 1998. [19] T. Constantinescu. Schur parameters, factorization and dilation problems, volume 82 ¨ Verlag, Basel, 1996. of Operator Theory: Advances and Applications. Birkhauser [20] H. Dym. J-contractive matrix functions, reproducing kernel Hilbert spaces and interpolation. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1989. [21] H. Dym and A. Iacob. Applications of factorization and Toeplitz operators to inverse problems. In I. Gohberg, editor, Toeplitz centennial (Tel Aviv, 1981), volume 4 of Operator Theory: Adv. Appl., pages 233–260. Birkh¨ a ¨user, Basel, 1982. [22] H. Dym and A. Iacob. Positive definite extensions, canonical equations and inverse problems. In H. Dym and I. Gohberg, editors, Proceedings of the workshop on applications of linear operator theory to systems and networks held at Rehovot, June 13–16, 1983, volume 12 of Operator Theory: Advances and Applications, pages 141– 240. Birkhauser ¨ Verlag, Basel, 1984. [23] B. Fritzsche and B. Kirstein, editors. Ausgew¨ ¨ ahlte Arbeiten zu den Urspr¨ ungen ¨ der Schur-Analysis, volume 16 of Teubner-Archiv zur Mathematik. B.G. Teubner Verlagsgesellschaft, Stuttgart–Leipzig, 1991. [24] I. Gohberg, S. Goldberg, and M.A. Kaashoek. Classes of linear operators. Vol. II, I volume 63 of Operator Theory: Advances and Applications. Birkhauser ¨ Verlag, Basel, 1993. [25] I. Gohberg and M.A. Kaashoek. Block Toeplitz operators with rational symbols. In I. Gohberg, J.W. Helton, and L. Rodman, editors, Contributions to operator theory and its applications (Mesa, AZ, 1987), volume 35 of Oper. Theory Adv. Appl., pages 385–440. Birkhauser, ¨ Basel, 1988. [26] I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich. Canonical systems with rational spectral densities: explicit formulas and applications. Math. Nachr., 194:93–125, 1998. [27] I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich. Pseudo-canonical systems with rational Weyl functions: explicit formulas and applications. Journal of differential equations, 146:375–398, 1998. [28] I. Gohberg, M.A. Kaashoek, and F. van Schagen. Szeg¨ ¨ o–Kac–Achiezer formulas in terms of realizations of the symbol. J. Funct. Anal., 74:24–51, 1987.

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translation in: I. Schur methods in operator theory and signal processing. (Operator theory: Advances and Applications OT 18 (1986), Birkh¨ ¨ auser Verlag), Basel. [44] B. Simon. Analogs of the m-function in the theory of orthogonal polynomials on the unit circle. J. Comput. Appl. Math., 171(1-2):411–424, 2004. [45] F. Wenger, T. Gustafsson, and L. Svensson. Perturbation theory for inhomogeneous transmission lines. IEEE Trans. Circuits Systems I Fund. Theory Appl., 49(3):289– 297, 2002. [46] A. Yagle and B. Levy. The Schur algorithm and its applications. Acta Applicandae Mathematicae, 3:255–284, 1985. Daniel Alpay Department of Mathematics Ben–Gurion University of the Negev Beer-Sheva 84105 Israel e-mail: [email protected] Israel Gohberg School of Mathematical Sciences The Raymond and Beverly Sackler Faculty of Exact Sciences Tel–Aviv University Tel–Aviv, Ramat–Aviv 69989 Israel e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 161, 49–113 c 2005 Birkhauser
¨ Verlag Basel/Switzerland

Matrix-J-unitary Non-commutative Rational Formal Power Series D. Alpay and D.S. Kalyuzhny˘ı-Verbovetzki˘ Abstract. Formal power series in N non-commuting indeterminates can be considered as a counterpart of functions of one variable holomorphic at 0, and some of their properties are described in terms of coefficients. However, really fruitful analysis begins when one considers for them evaluations on N -tuples of n × n matrices (with n = 1, 2, . . .) or operators on an infinite-dimensional separable Hilbert space. Moreover, such evaluations appear in control, optimization and stabilization problems of modern system engineering. In this paper, a theory of realization and minimal factorization of rational matrix-valued functions which are J-unitary on the imaginary line or on the unit circle is extended to the setting of non-commutative rational formal power series. The property of J-unitarity holds on N -tuples of n × n skew-Hermitian versus unitary matrices (n = 1, 2, . . .), and a rational formal power series is called matrix-J-unitary in this case. The close relationship between minimal realizations and structured Hermitian solutions H of the Lyapunov or Stein equations is established. The results are specialized for the case of matrix-J-inner rational formal power series. In this case H > 0, however the proof of that is more elaborated than in the one-variable case and involves a new technique. For the rational matrix-inner case, i.e., when J = I, the theorem of Ball, Groenewald and Malakorn on unitary realization of a formal power series from the non-commutative Schur–Agler class admits an improvement: the existence of a minimal (thus, finite-dimensional) such unitary realization and its uniqueness up to a unitary similarity is proved. A version of the theory for matrix-selfadjoint rational formal power series is also presented. The concept of non-commutative formal reproducing kernel Pontryagin spaces is introduced, and in this framework the backward shift realization of a matrix-J-unitary rational formal power series in a finite-dimensional non-commutative de Branges–Rovnyak space is described. Mathematics Subject Classification (2000). Primary 47A48; Secondary 13F25, 46C20, 46E22, 93B20, 93D05.

The second author was supported by the Center for Advanced Studies in Mathematics, BenGurion University of the Negev.

50 Keywords. J-unitary matrix functions, non-commutative, rational, formal power series, minimal realizations, Lyapunov equation, Stein equation, minimal factorizations, Schur–Agler class, reproducing kernel Pontryagin spaces, backward shift, de Branges–Rovnyak space.

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3 More on observability, controllability, and minimality in the non-commutative setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 Matrix-J-unitary formal power series: A multivariable non-commutative analogue of the line case . . . . . . . . . . . . . 67 4.1 Minimal Givone–Roesser realizations and the Lyapunov equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 The associated structured Hermitian matrix . . . . . . . . . . . . . . . . . . . . . . 72 4.3 Minimal matrix-J-unitary factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.4 Matrix-unitary rational formal power series . . . . . . . . . . . . . . . . . . . . . . . 75 5 Matrix-J-unitary formal power series: A multivariable non-commutative analogue of the circle case . . . . . . . . . . . 77 5.1 Minimal Givone–Roesser realizations and the Stein equation . . . . . . 77 5.2 The associated structured Hermitian matrix . . . . . . . . . . . . . . . . . . . . . . 83 5.3 Minimal matrix-J-unitary factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.4 Matrix-unitary rational formal power series . . . . . . . . . . . . . . . . . . . . . . . 85 6 Matrix-J-inner rational formal power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.1 A multivariable non-commutative analogue of the half-plane case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.2 A multivariable non-commutative analogue of the disk case . . . . . . . 91 7 Matrix-selfadjoint rational formal power series . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.1 A multivariable non-commutative analogue of the line case . . . . . . . . 96 7.2 A multivariable non-commutative analogue of the circle case . . . . . 100 8 Finite-dimensional de Branges–Rovnyak spaces and backward shift realizations: The multivariable non-commutative setting . . . . . . . . . 102 8.1 Non-commutative formal reproducing kernel Pontryagin spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.2 Minimal realizations in non-commutative de Branges–Rovnyak spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Matrix-J-unitary Rational Formal Power Series

51

1. Introduction In the present paper we study a non-commutative analogue of rational matrixvalued functions which are J-unitary on the imaginary line or on the unit circle and, as a special case, J-inner ones. Let J ∈ Cq×q be a signature matrix, i.e., a matrix which is both self-adjoint and unitary. A Cq×q -valued rational function F is J-unitary on the imaginary line if F (z)JF (z)∗ = J

(1.1)

at every point of holomorphy of F on the imaginary line. It is called J-inner if moreover F (z)JF (z)∗ ≤ J (1.2)

at every point of holomorphy of F in the open right half-plane Π. Replacing the imaginary line by the unit circle T in (1.1) and the open right half-plane Π by the open unit disk D in (1.2), one defines J-unitary functions on the unit circle (resp., J-inner functions in the open unit disk). These classes of rational functions were studied in [7] and [6] using the theory of realizations of rational matrix-valued functions, and in [4] using the theory of reproducing kernel Pontryagin spaces. The circle and line cases were studied in a unified way in [5]. We mention also the earlier papers [36, 23] that inspired much of investigation of these and other classes of rational matrix-valued functions with symmetries. We now recall some of the arguments in [7], then explain the difficulties appearing in the several complex variables setting, and why the arguments of [7] extend to the non-commutative framework. So let F be a rational function which is J-unitary on the imaginary line, and assume that F is holomorphic in a neighborhood of the origin. It then admits a minimal realization F (z) = D + C(IIγ − zA)−1 zB where D = F (0), and A, B, C are matrices of appropriate sizes (the size γ × γ of the square matrix A is minimal possible for such a realization). Rewrite (1.1) as F (z) = JF (−z)−∗ J,

(1.3)

where z is in the domain of holomorphy of both F (z) and F (−z)−∗ . We can rewrite (1.3) as  D + C(IIγ − zA)−1 zB = J D−∗ + D−∗ B ∗ (IIγ + z(A − BD−1 C)∗ )−1 zC ∗ D−∗ J.

The above equality gives two minimal realizations of a given rational matrix-valued function. These realizations are therefore similar, and there is a uniquely defined matrix (which, for convenience, we denote by −H) such that 



A B −(A∗ − C ∗ D−∗ B ∗ ) C ∗ D−∗ J −H 0 −H 0 = . (1.4) C D JD−∗ B ∗ JD−∗ J 0 Iq 0 Iq

The matrix −H ∗ in the place of −H also satisfies (1.4), and by uniqueness of the similarity matrix we have H = H ∗ , which leads to the following theorem.

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Theorem 1.1. Let F be a rational matrix-valued function holomorphic in a neighborhood of the origin and let F (z) = D + C(IIγ − zA)−1 zB be a minimal realization of F . Then F is J-unitary on the imaginary line if and only if the following conditions hold: (1) D is J-unitary, that is, DJD∗ = J; (2) there exists an Hermitian invertible matrix H such that A∗ H + HA = B

=

−C ∗ JC, −H

−1

∗

C JD.

(1.5) (1.6)

The matrix H is uniquely determined by a given minimal realization (it is called the associated Hermitian matrix to this realization). It holds that J − F (z)JF (z ′ )∗ = C(IIγ − zA)−1 H −1 (IIγ − z ′ A)−∗ C ∗ . z + z′ In particular, F is J-inner if and only if H > 0.

(1.7)

The finite-dimensional reproducing kernel Pontryagin space K(F ) with reproducing kernel J − F (z)JF (z ′ )∗ K F (z, z ′ ) = (z + z ′ ) provides a minimal state space realization for F : more precisely (see [4]), where

is defined by

F (z) = D + C(IIγ − zA)−1 zB,


A C

B D





 K(F ) K(F ) : → Cq Cq

f (z) − f (0) F (z) − F (0) u, Cf = f (0), Dx = F (0)x. , Bu = z z Another topic considered in [7] and [4] is J-unitary factorization. Given a matrix-valued function F which is J-unitary on the imaginary line one looks for all minimal factorizations of F (see [15]) into factors which are themselves Junitary on the imaginary line. There are two equivalent characterizations of these factorizations: the first one uses the theory of realization and the second one uses the theory of reproducing kernel Pontryagin spaces. (Af )(z) = (R0 f )(z) :=

Theorem 1.2. Let F be a rational matrix-valued function which is J-unitary on the imaginary line and holomorphic in a neighborhood of the origin, and let F (z) = D + C(IIγ − zA)−1 zB be a minimal realization of F , with the associated Hermitian matrix H. There is a one-to-one correspondence between minimal J-unitary factorizations of F (up to a multiplicative J-unitary constant) and Ainvariant subspaces which are non-degenerate in the (possibly, indefinite) metric induced by H. In general, F may fail to have non-trivial J-unitary factorizations.

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53

Theorem 1.3. Let F be a rational matrix-valued function which is J-unitary on the imaginary line and holomorphic in a neighborhood of the origin. There is a one-to-one correspondence between minimal J-unitary factorizations of F (up to a multiplicative J-unitary constant) and R0 -invariant non-degenerate subspaces of K(F ). The arguments in the proof of Theorem 1.1 do not go through in the several complex variables context. Indeed, uniqueness, up to a similarity, of minimal realizations doesn’t hold anymore (see, e.g., [27, 25, 33]). On the other hand, the notion of realization still makes sense in the non-commutative setting, namely for non-commutative rational formal power series (FPSs in short), and there is a uniqueness result for minimal realizations in this case (see [16, 39, 11]). The latter allows us to extend the notion and study of J-unitary matrix-valued functions to the non-commutative case. We introduce the notion of a matrix-J-unitary rational FPS as a formal power series in N non-commuting indeterminates which is J ⊗ In -unitary on N -tuples of n × n skew-Hermitian versus unitary matrices for n = 1, 2, . . .. We extend to this case the theory of minimal realizations, minimal J-unitary factorizations, and backward shift models in finite-dimensional de Branges–Rovnyak spaces. We also introduce, in a similar way, the notion of matrixselfadjoint rational formal power series, and show how to deduce the related theory for them from the theory of matrix-J-unitary ones. We now turn to the outline of this paper. It consists of eight sections. Section 1 is this introduction. In Section 2 we review various results in the theory of FPSs. Let us note that the theorem on null spaces for matrix substitutions and its corollary, from our paper [8], which are recollected in the end of Section 2, become an important tool in our present work on FPSs. In Section 3 we study the properties of observability, controllability and minimality of Givone-Roesser nodes in the non-commutative setting and give the corresponding criteria in terms of matrix evaluations for their “formal transfer functions”. We also formulate a theorem on minimal factorizations of a rational FPS. In Section 4 we define the non-commutative analogue of the imaginary line and study matrix-J-unitary FPSs for this case. We in particular obtain a non-commutative version of Theorem 1.1. We obtain a counterpart of the Lyapunov equation (1.5) and of Theorem 1.2 on minimal J-unitary factorizations. The unique solution of the Lyapunov equation has in this case a block diagonal structure: H = diag(H1 , . . . , HN ), and is said to be the associated structured Hermitian matrix (associated with a given minimal realization of a matrix-J-unitary FPS). Section 5 contains the analogue of the previous section for the case of a non-commutative counterpart of the unit circle. These two sections do not take into account a counterpart of condition (1.2), which is considered in Section 6 where we study matrix-J-inner rational FPSs. In particular, we show that the associated structured Hermitian matrix H = diag(H1 , . . . , HN ) is strictly positive in this case, which generalizes the statement in Theorem 1.1 on J-inner functions. We define non-commutative counterparts of the right half-plane and the unit disk, and formulate our results for both of these domains. The second

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one is the disjoint union of the products of N copies of n × n matrix unit disks, n = 1, 2, . . ., and plays a role of a “non-commutative polydisk”. In Theorem 6.6 we show that any (not necessarily rational) FPS with operator coefficients, which takes contractive values in this domain, belongs to the non-commutative Schur– Agler class, defined by J.A. Ball, G. Groenewald and T. Malakorn in [12]. (The opposite is trivial: any function from this class has the above-mentioned property.) In other words, the contractivity of values of a FPS on N -tuples of strictly contractive n × n matrices, n = 1, 2, . . ., is sufficient for the contractivity of its values on N -tuples of strictly contractive operators in an infinite-dimensional separable Hilbert space. Thus, matrix-inner rational FPSs (i.e., matrix-J-inner ones for the case J = Iq ) belong to the non-commutative Schur–Agler class. For this case, we recover the theorem on unitary realizations for FPSs from the latter class which was obtain in [12]. Moreover, our Theorem 6.4 establishes the existence of a minimal, thus finite-dimensional, unitary Givone–Roesser realization of a rational matrix-inner FPS and the uniqueness of such a realization up to a unitary similarity. This implies, in particular, non-commutative Lossless Bounded Real Lemma (see [41, 7] for its one-variable counterpart). A non-commutative version of standard Bounded Real Lemma (see [47]) has been presented recently in [13]. In Section 7 we study matrix-selfadjoint rational FPSs. In Section 8 we introduce non-commutative formal reproducing kernel Pontryagin spaces in a way which extends one that J.A. Ball and V. Vinnikov have introduced in [14] non-commutative formal reproducing kernel Hilbert spaces. We describe minimal backward shift realizations in non-commutative formal reproducing kernel Pontryagin spaces which serve as a counterpart of finite-dimensional de Branges–Rovnyak spaces. Let us note that we derive an explicit formula (8.12) for the corresponding reproducing kernels. In the last subsection of Section 8 we present examples of matrix-inner rational FPSs with scalar coefficients, in two non-commuting indeterminates, and the corresponding reproducing kernels computed by formula (8.12).

2. Preliminaries In this section we introduce the notations which will be used throughout this paper and review some definitions from the theory of formal power series. The symbol p×q Cp×q denotes the set of p × q matrices with complex entries, and (Cr×s ) is the space of p × q block matrices with block entries in Cr×s . The tensor product A ⊗ B p×q of matrices A ∈ Cr×s and B ∈ Cp×q is the element of (Cr×s ) with (i, j)th r×s p×q block entry equal to Abij . The tensor product C ⊗ C is the linear span of n finite sums of the form C = k=1 Ak ⊗ Bk where Ak ∈ Cr×s and Bk ∈ Cp×q . One p×q identifies Cr×s ⊗ Cp×q with (Cr×s ) . Different representations for an element C ∈ Cr×s ⊗ Cp×q can be reduced to a unique one: C=

p  q r  s  

µ=1 ν=1 τ =1 σ=1

′ cµντ σ Eµν ⊗ Eτ′′σ ,

Matrix-J-unitary Rational Formal Power Series

55

′ where the matrices Eµν ∈ Cr×s and Eτ′′σ ∈ Cp×q are given by

 ′ 1 if (i, j) = (µ, ν) Eµν ij = , µ, i = 1, . . . , r and ν, j = 1, . . . s, 0 if (i, j) = (µ, ν)

1 if (k, ℓ) = (τ, σ) ′′ (Eτ σ )kℓ = , τ, k = 1, . . . , p and σ, ℓ = 1, . . . q. 0 if (k, ℓ) = (τ, σ)

We denote by FN the free semigroup with N generators g1 , . . . , gN and the identity element ∅ with respect to the concatenation product. This means that the generic element of FN is a word w = gi1 · · · gin , where iν ∈ {1, . . . , N } for ν = 1, . . . , n, the identity element ∅ corresponds to the empty word, and for another word w′ = gj1 · · · gjm , one defines the product as T

ww′ = gi1 · · · gin gj1 · · · gjm ,

w∅ = ∅w = w.

We denote by w = gin · · · gi1 ∈ FN the transpose of w = gi1 · · · gin ∈ FN and by |w| = n the length of the word w. Correspondingly, ∅T = ∅, and |∅| = 0. A formal power series (FPS in short) in non-commuting indeterminates z1 , . . . , zN with coefficients in a linear space E is given by  f (z) = fw z w , fw ∈ E, (2.1) w∈F FN

where for w = gi1 · · · gin and z = (z1 , . . . , zN ) we set z w = zi1 · · · zin , and z ∅ = 1. We denote by E z1 , . . . , zN  the linear space of FPSs in non-commuting indeterminates z1 , . . . , zN with coefficients in E. A series f ∈ Cp×q z1 , . . . , zN  of the form (2.1) can also be viewed as a p × q matrix whose entries are formal power series with coefficients in C, i.e., belong to the space C z1 , . . . , zN , which has an additional structure of non-commutative ring (we assume that the indeterminates zj formally commute with the coefficients fw ). The support of a FPS f given by (2.1) is the set supp f = {w ∈ FN : fw = 0} . Non-commutative polynomials are formal power series with finite support. We denote by E z1 , . . . , zN  the subspace in the space E z1 , . . . , zN  consisting of non-commutative polynomials. Clearly, a FPS is determined by its coefficients fw . Sums and products of two FPSs f and g with matrix coefficients of compatible sizes (or with operator coefficients) are given by  (f + g)w = fw + gw , (f g)w = fw′ gw′′ . (2.2) w ′ w ′′ =w

A FPS f with coefficients in C is invertible if and only if f∅ = 0. Indeed, assume that f is invertible. From the definition of the product of two FPSs in (2.2) we get f∅ (f −1 )∅ = 1, and hence f∅ = 0. On the other hand, if f∅ = 0 then f −1 is given by ∞  k  f −1 (z) = 1 − f∅−1 f (z) f∅−1 . k=0

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The formal power series in the right-hand side is well defined since the expansion  k of 1 − f∅−1 f contains words of length at least k, and thus the coefficients (f −1 )w are finite sums. A FPS with coefficients in C is called rational if it can be expressed as a finite number of sums, products and inversions of non-commutative polynomials. A formal power series with coefficients in Cp×q is called rational if it is a p × q matrix whose all entries are rational FPSs with coefficients in C. We will denote by Cp×q z1 , . . . , zN rat the linear space of rational FPSs with coefficients in Cp×q . Define the product of f ∈ Cp×q z1 , . . . , zN rat and p ∈ C z1 , . . . , zN  as follows: 1. f · 1 = f for every f ∈ Cp×q z1 , . . . , zN rat ; 2. For every word w′ ∈ FN and every f ∈ Cp×q z1 , . . . , zN rat ,   ′ ′ f · zw = fw z ww = fv z w w∈F FN

w

where the last sum is taken over all w which can be written as w = vw′ for some v ∈ FN ; 3. For every f ∈ Cp×q z1 , . . . , zN rat , p1 , p2 ∈ C z1 , . . . , zN  and α1 , α2 ∈ C, p×q

f · (α1 p1 + α2 p2 ) = α1 (f · p1 ) + α2 (f · p2 ).

The space C z1 , . . . , zN rat is a right module over the ring C z1 , . . . , zN  with respect to this product. A structure of left C z1 , . . . , zN -module can be defined in a similar way since the indeterminates commute with coefficients. Formal power series are used in various branches of mathematics, e.g., in abstract algebra, enumeration problems and combinatorics; rational formal power series have been extensively used in theoretical computer science, mostly in automata u ¨ tzenberger theorem [35, 44] theory and language theory (see [18]). The Kleene–Sch¨ (see also [24]) says that a FPS f with coefficients in Cp×q is rational if and only if it is recognizable, i.e., there exist r ∈ N and matrices C ∈ Cp×r , A1 , . . . , AN ∈ Cr×r and B ∈ Cr×q such that for every word w = gi1 · · · gin ∈ FN one has fw = CAw B,

where Aw = Ai1 . . . Ain .

(2.3)

Let Hf be the Hankel matrix whose rows and columns are indexed by the words of FN and defined by (Hf )w,w′ = fww′T ,

w, w′ ∈ FN .

It follows from (2.3) that if the FPS f is recognizable then (Hf )w,w′ = ′T

CAww B for all w, w′ ∈ FN . M. Fliess has shown in [24] that a FPS f is rational (that is, recognizable) if and only if γ := rank Hf < ∞. In this case the number γ is the smallest possible r for a representation (2.3). In control theory, rational FPSs appear as the input/output mappings of linear systems with structured uncertainties. For instance, in [17] a system matrix

Matrix-J-unitary Rational Formal Power Series is given by

57

 A B ∈ C(r+p)×(r+q) , C D and the uncertainty operator is given by M=




∆(δ) = diag(δ1 Ir1 , . . . , δN IrN ), where r1 + · · · + rN = r. The uncertainties δk are linear operators on ℓ2 representing disturbances or small perturbation parameters which enter the system at different locations. Mathematically, they can be interpreted as non-commuting indeterminates. The input/output map is a linear fractional transformation LF T (M, ∆(δ)) = D + C(IIr − ∆(δ)A)−1 ∆(δ)B,

(2.4) Tαnc

which can be interpreted as a non-commutative transfer function of a linear system α with evolution on FN :

xj (gj w) = Aj1 x1 (w) + · · · + AjN xN (w) + Bj u(w), j = 1, . . . , N, α: (2.5) y(w) = C1 x1 (w) + · · · + CN xN (w) + Du(w), where xj (w) ∈ Crj (j = 1, . . . , N ), u(w) ∈ Cq , y(w) ∈ Cp , and the matrices Ajk , B and C are of appropriate sizes along the decomposition Cr = Cr1 ⊕ · · · ⊕ CrN . Such a system appears in [39, 11, 12, 13] and is known as the non-commutative Givone–Roesser model of multidimensional linear system; see [26, 27, 42] for its commutative counterpart. In this paper we do not consider system evolutions (i.e., equations (2.5)). We will use the terminology N -dimensional Givone–Roesser operator node (for brevity, GR-node) for the collection of data α = (N ; A, B, C, D; Cr =

N '

Crj , Cq , Cp ).

(2.6)

j=1

Sometimes instead of spaces Cr , Crj (j = 1, . . . , N ), Cq and Cp we shall consider abstract finite-dimensional linear spaces X (the state space), Xj (j = 1, . . . , N ), U (the input space) and Y (the output space), respectively, and a node α = (N ; A, B, C, D; X =

N ' j=1

Xj , U, Y),

where A, B, C, D are linear operators in the corresponding pairs of spaces. The non-commutative transfer function of a GR-node α is a rational FPS Tαnc(z) = D + C(IIr − ∆(z)A)−1 ∆(z)B.

(2.7)

Minimal GR-realizations (2.6) of non-commutative rational FPSs, that is, representations of them in the form (2.7), with minimal possible rk for k = 1, . . . , N were studied in [17, 16, 39, 11]. For k = 1, . . . , N , the kth observability matrix is Ok = col(Ck , C1 A1k , . . . , CN AN k , C1 A11 A1k , . . . C1 A1N AN k , . . .)

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and the kth controllability matrix is Ck = row(Bk , Ak1 B1 , . . . , AkN BN , Ak1 A11 B1 , . . . AkN AN 1 B1 , . . .) (note that these are infinite block matrices). A GR-node α is called observable (resp., controllable) if rank Ok = rk (resp., rank Ck = rk ) for k = 1, . . . , N . A GR( rj q p node α = (N ; A, B, C, D; Cr = N j=1 C , C , C ) is observable if and only if its (N adjoint GR-node α∗ = (N ; A∗ , C ∗ , B ∗ , D∗ ; Cr = j=1 Crj , Cp , Cq ) is controllable. (Clearly, (α∗ )∗ = α.) In view of the sequel, we introduce some notations. We set: Awgν = Aj1 j2 Aj2 j3 · · · Ajk−1 jk Ajk ν ,

(C♭A)gν w = Cν Aνj1 Aj1 j2 · · · Ajk−1 jk ,

(A♯B)wgν = Aj1 j2 · · · Ajk−1 jk Ajk ν Bν ,

(C♭A♯B)gµ wgν = Cµ Aµj1 Aj1 j2 · · · Ajk−1 jk Ajk ν Bν , where w = gj1 · · · gjk ∈ FN and µ, ν ∈ {1, . . . , N }. We also define: Agν = A∅ = Iγ

(C♭A)gν = Cν , (A♯B)gν = Bν , (C♭A♯B)gν = Cν Bν , (C♭A♯B)gµ gν = Cµ Aµν Bν , and hence, with the lexicographic order of words in FN , wgk Ok = colw∈F FN (C♭A)

T

gk w and Ck = roww∈F , FN (A♯B)

and the coefficients of the FPS Tαnc (defined by (2.7)) are given by (T Tαnc )∅ = D,

(T Tαnc )w = (C♭A♯B)w

for

w = gj1 · · · gjn ∈ FN .

The kth Hankel matrix associated with a FPS f is defined in [39] (see also [11]) as (Hf,k )w,w′ gk = fwgk w′T

with

w, w′ ∈ FN ,

that is, the rows of Hf,k are indexed by all the words of FN and the columns of Hf,k are indexed by all the words of FN ending by gk , provided the lexicographic order is used. If a GR-node α defines a realization of f , that is, f = Tαnc, then (Hf,k )w,w′ gk = (C♭A♯B)wgk w

′T

′T

= (C♭A)wgk (A♯B)gk w ,

i.e., Hf,k = Ok Ck . Hence, the node α is minimal if and only if α is both observable and controllable, i.e., γk := rank Hf,k = rk

for all k ∈ {1, . . . , N } .

This last set of conditions is an analogue of the above mentioned result of Fliess on minimal recognizable representations of rational formal power series. Every non-commutative rational FPS has a minimal GR-realization.

Matrix-J-unitary Rational Formal Power Series

59

Finally, we note (see [17, 39]) that two minimal GR-realizations of a given (N rational FPS are similar : if α(i) = (N ; A(i) , B (i) , C (i) , D; Cγ = k=1 Cγk , Cq , Cp ) (i=1,2) are minimal GR-nodes such that Tαnc(1) = Tαnc(2) then there exists a block diagonal invertible matrix T = diag(T T1 , . . . , TN ) (with Tk ∈ Cγk ×γk ) such that A(1) = T −1 A(2) T,

B (1) = T −1 B (2) ,

C (1) = C (2) T.

(2.8)

Of course, the converse is also true, moreover, any two similar (not necessarily minimal) GR-nodes have the same transfer functions. Now we turn to the discussion on substitutions of matrices for indeterminates in formal power series. Many properties of non-commutative FPSs or noncommutative polynomials are described in terms of matrix substitutions, e.g., matrix-positivity of non-commutative polynomials (non-commutative Positivstellensatz) [29, 40, 31, 32], matrix-positivity of FPS kernels [34], matrix-convexity [21, 30]. The non-commutative Schur–Agler class, i.e., the class of FPSs with operator coefficients, which take contractive values on all N -tuples of strictly contractive operators on ℓ2 , was studied in [12] 1 ; we will show in Section 6 that in order that a FPS belongs to this class it suffices to check its contractivity on N -tuples of strictly contractive n × n matrices, for all n ∈ N. The notions of matrix-Junitary (in particular, matrix-J-inner) and matrix-selfadjoint rational FPS, which will be introduced and studied in the present paper, are also defined in terms of substitutions of matrices (of a certain class) for indeterminates.  w ∈ C z1 , . . . , zN . For n ∈ N and an N -tuple of Let p(z) = |w|≤m pw z N

matrices Z = (Z1 , . . . , ZN ) ∈ (Cn×n ) , set  p(Z) = pw Z w , |w|≤m

where Z w = Zi1 · · · Zi|w| for w = gi1 · · · gi|w| ∈ FN , and Z ∅ = In . Then for any N

rational expression for a FPS f ∈ C z1 , . . . , zN rat its value at Z ∈ (Cn×n ) is well defined provided all of the inversions of polynomials p(j) ∈ C z1 , . . . , zN  in this expression are well defined at Z. The latter is the case at least in some (j) neighborhood of Z = 0, since p∅ = 0. N

Now, if f ∈ Cp×q z1 , . . . , zN rat then the value f (Z) at some Z ∈ (Cn×n ) is well defined whenever the values of matrix entries (ffij (Z)) (i = 1, . . . , p; j = 1, . . . , q) are well defined at Z. As a function of matrix entries (Zk )ij (k = 1, . . . , N ; i, j = 1, . . . , n), f (Z) is rational Cp×q ⊗ Cn×n -valued function, which is holomorphic on an open and dense set in Cn×n . The latter set contains some neighborhood  N : Zk  < ε, k = 1, . . . , N } (2.9) Γn (ε) := {Z ∈ Cn×n 1 In fact, a more general class was studied in [12], however for our purposes it is enough to consider here only the case mentioned above.

60

D. Alpay and D.S. Kalyuzhny˘-Verbovetzki˘ ˘

of Z = 0, where f (Z) is given by f (Z) =



w∈F FN

fw ⊗ Z w .

The following results from [8] on matrix substitutions are used in the sequel. Theorem 2.1. Let f ∈ Cp×q z1 , . . . , zN rat , and m ∈ Z+ be such that ) ) ker fw = ker fw . w∈F FN

w∈F FN :|w|≤m

Then there exists ε > 0 such that for every n ∈ N : n ≥ mm (in the case m = 0, for every n ∈ N), ⎛ ⎞ ) ) ker f (Z) = ⎝ ker fw ⎠ ⊗ Cn , (2.10) Z∈Γn (ε)

w∈F FN : |w|≤m

and moreover, there exist l ∈ N : l ≤ qn, and N -tuples of matrices Z (1) , . . . , Z (l) from Γn (ε) such that ⎛ ⎞ l ) ) (j) ker f (Z ) = ⎝ ker fw ⎠ ⊗ Cn . j=1

w∈F FN : |w|≤m

Corollary 2.2. In conditions of Theorem 2.1, if for some n ∈ N : n ≥ mm (in the case m = 0, for some n ∈ N) one has f (Z) = 0, ∀Z ∈ Γn (ε), then f = 0.

3. More on observability, controllability, and minimality in the non-commutative setting In this section we prove a number of results on observable, controllable and minimal GR-nodes in the multivariable non-commutative setting, which generalize some well-known statements for one-variable nodes (see [15]).

k and the kth trunLet us introduce the kth truncated observability matrix O

cated controllability matrix Ck of a GR-node (2.6) by *k = col|w| 0 such that r + τ < 1. Let W ∈ (Dn×n )N be such that W Wj  ≤ r, j = 1, . . . , N . Then, for every x ∈ U ⊗ Cn the series  

∞  r+τ r+τ k λW x = λ fk W x f r r k=0

converges uniformly in λ ∈ clos D to a Y ⊗ Cn -valued function holomorphic on clos D. Furthermore, <
<  < <  
< < < < r+τ −k−1