Operator Theory: Advances and Applications Vol. 161 Editor: I. Gohberg
Editorial Office: School of Mathematical Scienc...

Author:
Daniel Alpay; Israel Gohberg

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

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)

,

(R+ ) is called the potential. Such systems were introduced by and where k ∈ Ln×n 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 . (1.4) J= and κ(x) = 0 −1 2Z(x) We distinguish two cases: (a) The case where Z(x) > 0 and is continuously diﬀerentiable on R+ . Taking the (inverse) Fourier transform f → f(λ) = R eiλt f (t)dt on both sides we get 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 ∞ eiλt u(t)dt (1.5) f (λ) = D + −∞

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 (resp. W− ) consists of the functions of the dence on R. The sub-algebra W+ 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 f (z) = Z

for which def.

f W =

|ff | < ∞.

Z

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 p and ∪m ∩m =0 ker ca = {0} =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 ia 0 0 f1∗ −c 0 , T = , G = F =i , (2.5) c∗ 0 0 f1 0 −ia∗ 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 In 0 e (2.6) lim U (x, λ) = I2n , λ ∈ R. 0 eixλ In x→∞ 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)

= In + icΩ(λIIp − a∗ )−1 c∗ ,

U21 (0, λ)

= (−b∗ + icΩ)(λIIp − a∗ )−1 c∗ ,

U12 (0, λ)

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

U22 (0, λ)

= 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 0 Ω 0 Ip H= . −i(IIp + Y Ω)Ω−1 Ip −i(IIp + Y Ω)Ω−1 Ip 0 −Ω−1 − Y 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 0 A22 − A21 A−1 I 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 ,

λ ∈ R,

is called the reﬂection coeﬃcient function. 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 B(IIn − D∗ D)−1 B ∗ A + BD∗ (IIn − DD∗ )−1 C 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=

icΩ − b∗ ). −Ω −iIIp

iIIp −Y (IIp + ΩY )−1

.

Then it holds that 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 −Ω 0 Ip 0 0 X= 0 (Ω + ΩY Ω)−1 iΩ−1 Ip 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 V22 (λ) − V12 (λ))−1 W (λ) = (V V22 (λ) − V12 (λ))−∗ (V 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 Θ(x, λ) Θ(x, λ) dx 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)−1 S− (λ)−∗ = S+ (λ)S+ (λ)∗ , = Im N (λ),

S(λ)

= S− (λ)S+ (λ),

R(λ)

= (iN (λ)∗ − In )(iN (λ)∗ + In )−1 ,

N (λ)

= 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

V (λ) 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)

14

D. Alpay and I. Gohberg

Finally I2n − Θ× Θ×

∗

∗

I2n − Θ× Θ×

=

(P − ΘQ)−1 (J − ΘJΘ∗ ) (P − ΘQ)−∗

(2.17)

=

(P + QΘ)−∗ (J − Θ∗ JΘ) (P + QΘ)−1 .

(2.18)

where A ∈ Cn×n . We have: In 0 In P + QΘ = , P − ΘQ = C D 0

Proof. We set Θ =

A C

B C

−B −D

.

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 (2.20) W (λ) = In − eitλ ω(t)dt, λ ∈ R, 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 Γ2t (u, 0)e−iλu du . P (t, λ) = eitλ In + 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 ).

16

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 × PB (2.22) k(x) = 2C P e−2ixA |Im P 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

→ −N −1 (λ),

λ λ

→ −N −1 (−λ∗ )∗ , → N (−λ∗ )∗ ,

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

= =

Im − N (λ)−1 , Im − N (−λ∗ )−∗ ,

W2 (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 c∗

B− = B =

,

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=

a − bc 2 0

0 ∗ (a − bc 2)

a−

bc 2 c c 2 ∗

(a

bb∗ 2 ∗ − bc 2)

,

Z=

0

∗

cc 2

we have A× = D − Z

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 , it holds that P A× |Im P = P (A− − B− C− ) |Im P , i.e., that I 0 I 0 I 0 I 0 = (D − Z) (D + Z) 0 0 0 0 0 0 0 0 for all positive integers . Let = ±1. The expression (D + Z) consists of a sum of terms of the form Dα1 ( Z)β1 Dα2 ( Z)β2 · · · , where the αi and the βi are equal to 1 or 0 and i (αi + βi ) = . Each factor diagonal. We consider two cases, namely Dαi Z βi for which βi = 0 is anti block 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 and have the same value at = 1 and at = −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

−1 P1

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

P1 B1 .

−1 P1

P1∗ C1∗ .

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

= 2CP = c

0 ,

P1 C1∗

1 = −P B = − 2

b 0

,

19

× A∗× 1 = −A ,

which allows to conclude.

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 ∂ ∂ ) 0 ( ∂x + ∂t W (x, t) = 0. ∂ ∂ 0 ( ∂x − ∂t ) 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).

Hence one can write

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.

Indeed, the space C0 is A invariant and H-negative. Furthermore, ∗ Ip −Ω 0 Ip 0 0 −Ω −IIp , = −IIp −a∆a∗ 0 Ω−1 − a∆a∗ Ω−1 Ip Ω−1 Ip p

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 − |ρ | ) H0 (z ∗ )∗ 0 =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

=0

√1

1−|ρ |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 H0 (z) (recall on the unit circle. From the J-unitarity on the unit circle of √ 1 det H0 (z)

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

|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}

(I − ∆Ω)(zI − a)

× (zI − a)−1 (I − ∆Ω)−1 b, 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 (1/z) = can (I − ∆a∗(n+1) Ωan+1 ) − z(I − ∆a∗n Ωan )a b Rn (z) = 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 − |ρ |2 ) Hn (z)−1 0 z 0 =0

the form 1 0 H0 (z) 0 1

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) 0 z 0 1 Yn (z) = (1 − |ρ |2 ) 0 z Hn (z)−1 H0 (z) x(z) . 0 1 z =0

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

n→∞

=0

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

30

D. Alpay and I. Gohberg

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

n→∞

1 0 1 0 x(z) 0 Yn (z)z −n = 1 0 H0 (z) 0 z −1 x(z) 0 z ∞ α0 (z) + β0 (z) 2 z (1 − |ρ | ) 1 0 x(z) = γ0 (z) + δ0z(z) =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

=n−1 1 1 −ρ∗ −1 =0

−ρ 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 22 . 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 = =0 (1 − |ρ |2 ), we have that: cn−1 1 1 iN (z ∗ )∗ 1 0 Un (z) = Hn (z)−1 1 0 z 1 −1 2 n z 0 1 0 1 + iN (z ∗ )∗ × H0 (z) 0 z −1 0 1 −1 + iN (z ∗ )∗ n cn−1 1 1 0 1 0 z = Hn (z)−1 0 1 0 z 1 −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 22 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) =

zβ0 (1/z) − α0 (1/z) zγ0 (z ∗ )∗ − δ0 (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

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

b,

and hence the result.

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 − |ρ |2 ) =0

c = ∞

|z| = 1.

(3.31)

will play an important role in the sequel. 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 1 c , z ∈ T, c = ∞ W (z) = , |S− (1/z)|2 (1 − |ρ |2 ) =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 w eit such that Z |w | < ∞ (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 ⎧ ⎨ CA−1 (I − P )B if = 1, 2, . . . w = D − CP B if = 0 ⎩ −CA−1 P B if = −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 ∞ z (A(I − P )) )B = D + zC( =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

× = w−j + K0j

(m)

= −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 (m)

(m)

(m)

z m pm (1/z) =

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

m

⎞ z j Ω×j ⎠ P × B ⎞

z j Ω×−j ⎠ (I − P × )B

j=0

from which the claim follows. 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)∗ Mn (w)∗ ∗ J − Mn (z)JM =M M , ∗ 1 − zw 1 − zw∗ 2 1 − zw∗ R(z)R(w)∗ 1 N (z) − N (w)∗ = , ∗ ∗ ∗ 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 , k < n % & 1 1 = 2δ,k . Θ , Θk 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 < k. The function k z 0 −1 (Θ Θk )(z) = U U C(ρi ) 0 1 i=+1

Analogs of Canonical Systems with Pseudo-exponential Potential belongs to H2,J and is such that (Θ−1 Θk )(0) Thus,

% Θ

& 1 1 , Θk 1 1 H(Θ

= n)

39

1 0 . = 1 0

% & 1 1 , Θ−1 Θ k 1 1 H(Θ

=0 n)

The proof that the inner product is equal to 2 when = 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 − |ρ | ) Hn (z ) H0 (z ) Y0 (z). Yn (z) = 0 z2 0 z12 0 z −n =0

The solution such that

−n z lim 0 n→∞

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

0 Yn (z) = I2 zn 1 0 2 −1 1 H0 (z ) 0 0 z2

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 = H (z ) . Hn+1 (z 2 ) 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

1 − |ρ |

2

=0

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

1 0

0 2 −1 1 (z ) H 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 Yn(1) (z)c1 − Yn(2) (z)c2 = 0 lim Y

n→∞

Proof. By deﬁnition,

Yn(2) (z) =

n−1

(1) Yn (z)

(1 − |ρ |2 )

=0

n z = 0

1 0

0

z −n

c2 = V (z)c1 .

⇐⇒

c1 . On the other hand,

n 0 2 −1 z (z ) H 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 b. (4.2) R(z) = c (I − ∆a∗ Ωa) − z 2 (I − ∆Ω)a 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 ∆=

1 , 1 − α2

Ω=

and ρn = −αn

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 < < 1, |ρ0 | = c2 1 − α2 2 1 − α (1−α2 )2 and more generally, |ρn | =

αn c 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.

42

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

The system Special solutions

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

Potential

Solution asymptotic to the solution with k ≡ 0

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 =

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

c (1−α2 ) ,

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.

Analogs of Canonical Systems with Pseudo-exponential Potential

45

[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 of Operator Theory: Advances and Applications. Birkhauser ¨ Verlag, Basel, 1996. [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.

46

D. Alpay and I. Gohberg

[29] I. Gohberg, P. Lancaster, and L. Rodman. Matrices and indeﬁnite scalar products, volume 8 of Operator Theory: Advances and Applications. Birkhauser ¨ Verlag, Basel, 1983. [30] I. Gohberg, P. Lancaster, and L. Rodman. Invariant subspaces of matrices with applications. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons Inc., New York, 1986. A Wiley-Interscience Publication. [31] I. Gohberg and Ju. Leiterer. General theorems on the factorization of operatorvalued functions with respect to a contour. I. Holomorphic functions. Acta Sci. Math. (Szeged), 34:103–120, 1973. [32] I. Gohberg and Ju. Leiterer. General theorems on the factorization of operator-valued functions with respect to a contour. II. Generalizations. Acta Sci. Math. (Szeged), 35:39–59, 1973. [33] I. Gohberg and S. Rubinstein. Proper contractions and their unitary minimal completions. In I. Gohberg, editor, Topics in interpolation theory of rational matrix-valued functions, volume 33 of Operator Theory: Advances and Applications, pages 223–247. Birkhauser ¨ Verlag, Basel, 1988. [34] I.C. Gohberg and I.A. Fel dman. Convolution equations and projection methods for their solution. American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by F.M. Goldware, Translations of Mathematical Monographs, Vol. 41. [35] G.J. Groenewald. Toeplitz operators with rational symbols and realizations: an alternative version. Technical Report WS:–362, Vrije Universiteit Amsterdam, 1990. [36] A. Iacob. On the spectral theory of a class of canonical systems of diﬀerential equations. PhD thesis, The Weizmann Institute of Sciences, 1986. [37] M.G. Kre˘n. ˘ Continuous analogues of propositions for polynomials orthogonal on the unit circle. Dokl. Akad. Nauk. SSSR, 105:637–640, 1955. [38] M.G. Kre˘n. Topics in diﬀerential and integral equations and operator theory, volume 7 of Operator theory: Advances and Applications. Birkhauser ¨ Verlag, Basel, 1983. Edited by I. Gohberg, Translated from the Russian by A. Iacob. ¨ [39] M.G. Kre˘n ˘ and H. Langer. Uber die verallgemeinerten Resolventen und die charakteristische Funktion eines isometrischen Operators im Raume Πk . In Hilbert space operators and operator algebras (Proc. Int. Conf. Tihany, 1970), pages 353–399. North-Holland, Amsterdam, 1972. Colloquia Math. Soc. J´ a ´nos Bolyai. [40] L.Golinskii and P. Nevai. Szeg˝ ˝ o diﬀerence equations, transfer matrices and orthogonal polynomials on the unit circle. Comm. Math. Phys., 223(2):223–259, 2001. [41] F.E. Melik-Adamyan. On a class of canonical diﬀerential operators. Izvestya Akademii Nauk. Armyanskoi SSR Matematica, 24:570–592, 1989. English translation in: Soviet Journal of Contemporary Mathematics, vol. 24, pages 48–69 (1989). [42] L. Sakhnovich. Dual discrete canonical systems and dual orthogonal polynomials. In D. Alpay, I. Gohberg, and V. Vinnikov, editors, Interpolation theory, systems theory and related topics (Tel Aviv/Rehovot, 1999), volume 134 of Oper. Theory Adv. Appl., pages 385–401. Birkh¨ a ¨user, Basel, 2002. ¨ [43] I. Schur. Uber die Potenzreihen, die im Innern des Einheitkreises beschr¨ ¨ ankten sind, I. Journal f¨ fur die Reine und Angewandte Mathematik, 147:205–232, 1917. English

Analogs of Canonical Systems with Pseudo-exponential Potential

47

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: dany@math.bgu.ac.il 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: gohberg@post.tau.ac.il

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

52

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

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]), F (z) = D + C(IIγ − zA)−1 zB, where

A C

B D

K(F ) K(F ) : → Cq Cq

is deﬁned by 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.

Matrix-J-unitary Rational Formal Power Series

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

54

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

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 is the Cp×q denotes the set of p × q matrices with complex entries, and (Cr×s ) space of p × q block matrices with block entries in Cr×s . The tensor product A ⊗ B p×q with (i, j)th of matrices A ∈ Cr×s and B ∈ Cp×q is the element of (Cr×s ) 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 ⊗ Eτσ ,

Matrix-J-unitary Rational Formal Power Series

55

where the matrices Eµν ∈ Cr×s and Eτσ ∈ 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, ) = (τ, σ) , τ, k = 1, . . . , p and σ, = 1, . . . q. (Eτ σ )k = 0 if (k, ) = (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 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) T

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

56

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

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, 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 p×q

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 = fwwT ,

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 2 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

of a linear which can be interpreted as a non-commutative transfer function 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 '

Xj , U, Y),

j=1

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 , . . .)

58

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

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 , (CA)gν w = Cν Aνj1 Aj1 j2 · · · Ajk−1 jk , (AB)wgν = Aj1 j2 · · · Ajk−1 jk Ajk ν Bν , (CAB)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 (CA)gν = Cν , (AB)gν = Bν , (CAB)gν = Cν Bν , (CAB)gµ gν = Cµ Aµν Bν , and hence, with the lexicographic order of words in FN , wgk Ok = colw∈F FN (CA)

T

gk w and Ck = roww∈F , FN (AB)

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

(T Tαnc )w = (CAB)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 wT

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 = (CAB)wgk w

T

T

= (CA)wgk (AB)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 2 , 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) =

fw ⊗ Z w .

w∈F FN

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|<pr (CA)wgk , O

*k = row|w| 0 is arbitrary in the case A = 0), and Γn (ε) is deﬁned by (2.9). This GR-node is minimal if both of conditions (3.3) and (3.4) are fulﬁlled. rk

Proof. First, let us remark that for all k = 1, . . . , N the functions ϕk and ψk are well deﬁned in Γn (ε), and holomorphic as functions of matrix entries (Z Zj )µν , j = 1, . . . , N, µ, ν = 1, . . . , n. Second, Theorem 3.1 implies that in Theorem 2.1 applied to ϕk one can choose m = pr−1, and then from (2.10) obtain that observability for a GR-node α is equivalent to condition (3.3). Since α is controllable if and only if α∗ is observable, controllability for α is equivalent to condition (3.4). Since minimality for a GR-node α is equivalent to controllability and observability together, it is in turn equivalent to conditions (3.3) and (3.4) together.

64

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

(N Let α = (N ; A , B , C , D ; Cr = j=1 Crj , Cs , Cp ) and α = (N ; A , B , (N rj q s C , D ; Cr = j=1 C , C , C ) be GR-nodes. For k, j = 1, . . . , N set rj = rj + rj , and Akj Bk Cj Bk D rk ×rj Akj = , Bk = ∈C ∈ Crk ×q , 0 A B (3.7) kj k p×rj p×q C D C Cj = j , D=DD ∈C . j ∈C (N r rj q p Then α = (N ; A, B, C, D; C = j=1 C , C , C ) will be called the product of GR-nodes α and α and denoted by α = α α . A straightforward calculation shows that Tαnc = Tαnc Tαnc . Consider a GR-node N N ' ' Crj , Cq ) := (N ; A, B, C, D; Cr = Cr j , Cq , Cq ) α = (N ; A, B, C, D; Cr = j=1

j=1

(3.8) with invertible operator D. Then α× = (N ; A× , B × , C × , D× ; Cr =

N '

Crj , Cq ),

j=1

with A× = A − BD−1 C,

B × = BD−1 ,

C × = −D−1 C,

D× = D−1 ,

(3.9)

×

will be called the associated GR-node, and A the associated main operator, of α. It is easy to see that, as well as in the one-variable case, (T Tαnc )−1 = Tαnc× . Moreover, × × (α× ) = α (in particular, (A× ) = A), and (α α )× = α× α× up to the natural rj rj identiﬁcation of C ⊕ C with Crj ⊕ Crj , j = 1, . . . , N , which is a similarity transform. Theorem 3.8. A GR-node (3.8) with invertible operator D is minimal if and only if its associated GR-node α× is minimal. Proof. Let a GR-node α of the form (3.8) with invertible operator D be minimal, and x ∈ ker Ok× for some k ∈ {1, . . . , N }, where Ok× is the kth observability matrix × . Then x ∈ ker(C × A× )wgk for every w ∈ FN . Let us show for the GR-node α/ wgk that x ∈ ker Ok = w∈F , i.e, x = 0. FN ker(CA) × For w = ∅, Ck x = 0 means −D−1 Ck x = 0 (see (3.9)), which is equivalent to Ck x = 0. For |w| > 0, w = gi1 · · · gi|w| , (CA)wgk

=

Ci1 Ai1 i2 · · · Ai|w| k

=

−1 −1 −DC Ci×1 (A× Ci2 ) · · · (A× Ck ) i1 i2 + Bi1 D i|w| k + Bi|w| D

=

L0 Ck× +

|w| j=1

× Lj Ci×j A× ij ij+1 · · · Ai|w| k ,

Matrix-J-unitary Rational Formal Power Series

65

with some matrices Lj ∈ Cq×q , j = 0, 1, . . . , |w|. Thus, x ∈ ker(CA)wgk for every w ∈ FN , i.e., x = 0, which means that α× is observable. Since α is controllable if and only if α∗ is observable (see Section 2), and ∗ D is invertible whenever D is invertible, the same is true for α× and (α× )∗ = (α∗ )× . Thus, the controllability of α× follows from the controllability of α. Finally, the minimality of α× follows from the minimality of α. Since (α× )× = α, the minimality of α follows from the minimality of α× . Suppose that for a GR-node (3.8), projections Πk on Crk are deﬁned such that Akj ker Πj ⊂ ker Πk ,

(A× )kj ran Πj ⊂ ran Πk ,

k, j = 1, . . . , N.

We do not assume that Πk are orthogonal. We shall call Πk a kth supporting projection for α. Clearly, the map Π = diag(Π1 , . . . , ΠN ) : Cr → Cr satisﬁes A ker Π ⊂ ker Π,

A× ran Π ⊂ ran Π,

i.e., it is a supporting projection for the one-variable node (1; A, B, C, D; Cr , Cq ) in the sense of [15]. If Π is a supporting projection for α, then Ir − Π is a supporting projection for α× . The following theorem and corollary are analogous to, and are proved in the same way as Theorem 1.1 and its corollary in [15, pp. 7–9] (see also [43, Theorem 2.1]). Theorem 3.9. Let (3.8) be a GR-node with invertible operator D. Let Πk be a projection on Crk , and let (11) (12) (1) Akjj Akjj Bj A= Bj = Ck = Ck(1) Ck(2) (21) (22) , (2) , Akj Akj Bj be the block matrix representations of the operators Akj , Bj and Ck with respect ˙ to the decompositions Crk = ker Πk +ran Πk , for k, j ∈ {1, . . . , N }. Assume that D = D D , where D and D are invertible operators on Cq , and set α = (N ; A(11) , B (1) (D )−1 , C (1) , D ; ker Π =

N '

ker Πk , Cq ),

k=1

α = (N ; A(22) , B (2) , (D )−1 C (2) , D ; ran Π =

N '

ran Πk , Cq ).

k=1

Then α = α α

(up to a similarity which maps C

rk

˙ = ker Πk +ran Πk onto ·

Cdim(ker Πk ) ⊕ Cdim(ranΠk ) (k = 1, . . . , N ) such that ker Πk + {0} is mapped onto ·

Cdim(ker Πk ) ⊕ {0} and {0} + ranΠk is mapped onto {0} ⊕ Cdim(ranΠk ) ) if and only if Π is a supporting projection for α. Corollary 3.10. In the assumptions of Theorem 3.9, Tαnc = F F ,

66

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

where F (z) = D + C(IIr − ∆(z)A)−1 (IIr − Π)∆(z)B(D )−1 , F (z) = D + (D )−1 CΠ(IIr − ∆(z)A)−1 ∆(z)B. We assume now that the external operator of the GR-node (3.8) is equal to D = Iq and that we also take D = D = Iq . Then, the GR-nodes α and α of Theorem 3.9 are called projections of α with respect to the supporting projections Ir − Π and Π, respectively, and we use the notations N ' ker Πk , Cq , α = prIr −Π (α) = N ; A(11) , B (1) , C (1) , D ; ker Π = k=1

α = prΠ (α) =

(22)

N; A

,B

(2)

,C

(2)

, D ; ran Π =

N '

ran Πk , C

q

.

k=1

Let F , F and F be rational FPSs with coeﬃcients in Cq×q such that F = F F .

(3.10)

The factorization (3.10) will be said to be minimal if whenever α and α are minimal GR-realizations of F and F , respectively, α α is a minimal GR-realization of F . In the sequel, we will use the notation N ' γ γk ×γk q α = N ; A, B, C, D; C = C ,C (3.11) k=1

for a minimal GR-realization (i.e., rk = γk for k = 1, . . . , N ) of a rational FPS F in the case when p = q. The following theorem is the multivariable non-commutative version of [15, Theorem 4.8]. It gives a complete description of all minimal factorizations in terms of supporting projections. Theorem 3.11. Let F be a rational FPS with a minimal GR-realization (3.11). Then the following statements hold: (i) if Π = diag(Π1 , . . . , ΠN ) is a supporting projection for α, then F is the transfer function of prIγ −Π (α), F is the transfer function of prΠ (α), and F = F F is a minimal factorization of F ; (ii) if F = F F is a minimal factorization of F , then there exists a uniquely deﬁned supporting projection Π = diag(Π1 , . . . , ΠN ) for the GR-node α such that F and F are the transfer functions of prIγ −Π (α) and prΠ (α), respectively. Proof. (i). Let Π be a supporting projection for α. Then, by Theorem 3.9, α = prIγ −Π (α)prΠ (α).

Matrix-J-unitary Rational Formal Power Series

67

By the assumption, α is minimal. We now show that the GR-nodes α = prIγ −Π (α) and α = prΠ (α) are also minimal. To this end, let x ∈ ran Πk . Then wgk wg wg C (2) A(22) x = (CA) k Πk x = (CA) k x. Thus, if Ok denotes the kth observability matrix of α , then x ∈ ker Ok implies x ∈ ker Ok , and the observability of α implies that α is also observable. Since gk wT g wT A(22) B (2) = Πk (AB) k , one has Ck = Πk Ck , where Ck is the kth controllability matrix of α . Thus, the controllability of α implies the controllability of α . Hence, we have proved the minimality of α . Note that we have used that ker Π = ran (IIγ − Π) is A-invariant. Since ran Π = ker(IIγ − Π) is A× -invariant, by Theorem 3.8 α× is minimal. Using α× = (α α )× = (α )× (α )× , we prove the minimality of (α )× in the same way as that of α . Applying once again Theorem 3.8, we obtain the minimality of α . The dimensions of the state spaces of the minimal GR-nodes α , α and α are related by γk = γk + γk ,

k = 1, . . . , N.

Therefore, given any minimal GR-realizations β and β of F and F , respectively, the same equalities hold for the state space dimensions of β , β and β. Thus, β β is a minimal GR-node, and the factorization F = F F is minimal. (ii). Assume that the factorization F = F F is minimal. Let β and β be minimal GR-realizations of F and F with k-th state space dimensions equal to γk and γk , respectively (k = 1, . . . , N ). Then β β is a minimal GR-realization of F and its kth state space dimension is equal to γk = γk + γk (k = 1, . . . , N ). Hence β β is similar to α. We denote the corresponding GR-node similarity by T = diag(T T1 , . . . , TN ), where

T k : Cγ ⊕ Cγ → Cγ ,

k = 1, . . . N,

is the canonical isomorphism. Let Πk be the projection of Cγk along Tk Cγk onto Tk Cγk , k = 1, . . . , N , and set Π = diag(Π1 , . . . , Πk ). Then Π is a supporting projection for α. Moreover prIγ −Π (α) is similar to β , and prΠ (α) is similar to β . The uniqueness of Π is proved in the same way as in [15, Theorem 4.8]. The uniqueness of the GR-node similarity follows from Theorem 3.5.

4. Matrix-J-unitary formal power series: A multivariable non-commutative analogue of the line case In this section we study a multivariable non-commutative analogue of rational q × q matrix-valued functions which are J-unitary on the imaginary line iR of the complex plane C.

68

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

4.1. Minimal Givone–Roesser realizations and the Lyapunov equation N

Denote by Hn×n the set of Hermitian n × n matrices. Then (iHn×n ) will denote the set of N -tuples of skew-Hermitian matrices. In our paper, the set 0 N iHn×n , JN = n∈N

1 where “ ” stands for a disjoint union, will be a counterpart of the imaginary line iR. Let J ∈ Cq×q be a signature matrix. We will call a rational FPS F ∈ Cq×q z1 , . . . , zN rat matrix-J-unitary on JN if for every n ∈ N, F (Z)(J ⊗ In )F (Z)∗ = J ⊗ In

(4.1)

n×n N

at all points Z ∈ (iH ) where it is deﬁned. For a ﬁxed n ∈ N, F (Z) as a function of matrix entries is rational and holomorphic on some open neighborhood N Γn (ε) of Z = 0, e.g., of the form (2.9), and Γn (ε) ∩ (iHn×n ) is a uniqueness set in n×n N (C ) (see [45] for the uniqueness theorem in several complex variables). Thus, (4.1) implies that (4.2) F (Z)(J ⊗ In )F (−Z ∗ )∗ = J ⊗ In at all points Z ∈ (Cn×n )N where F (Z) is holomorphic and invertible (the set of such points is open and dense, since det F (Z) ≡ 0). The following theorem is a counterpart of Theorem 2.1 in [7]. Theorem 4.1. Let F be a rational FPS with a minimal GR-realization (3.11). Then F is matrix-J-unitary on JN if and only if the following conditions are fulﬁlled: a) D is J-unitary, i.e., DJD∗ = J; b) there exists an invertible Hermitian solution H = diag(H1 , . . . , HN ), with Hk ∈ Cγk ×γk , k = 1, . . . , N , of the Lyapunov equation A∗ H + HA = −C ∗ JC,

(4.3)

B = −H −1 C ∗ JD.

(4.4)

and The property b) is equivalent to b ) there exists an invertible Hermitian matrix H = diag(H1 , . . . , HN ), with Hk ∈ Cγk ×γk , k = 1, . . . , N , such that H −1 A∗ + AH −1 = −BJB ∗ ,

(4.5)

C = −DJB ∗ H.

(4.6)

and Proof. Let F be matrix-J-unitary. Then F is holomorphic at the point Z = 0 in CN , hence D = F (0) is J-unitary (in particular, invertible). Equality (4.2) may be rewritten as (4.7) F (Z)−1 = (J ⊗ In )F (−Z ∗ )∗ (J ⊗ In ).

Matrix-J-unitary Rational Formal Power Series

69

Since (4.7) holds for all n ∈ N, it follows from Corollary 2.2 that the FPSs corresponding to the left and the right sides of equality (4.7) coincide. Due to The(N orem 3.8, α× = (N ; A× , B × , C × , D× ; Cγ = k=1 Cγk , Cq ) with A× , B × , C × , D× given by (3.9) is a minimal GR-realization of F −1 . Due to (4.7), another minimal ˜ B, ˜ C, ˜ D; ˜ Cγ = (N Cγk , Cq ), where ˜ = (N ; A, GR-realization of F −1 is α k=1

∗

A˜ = −A ,

∗

˜ = C J, B

∗

C˜ = −JB ,

˜ = JD∗ J. D

By Theorem 3.5, there exists unique similarity transform T = diag(T T1 , . . . , TN ) which relates α× and α ˜ , where Tk ∈ Cγk ×γk are invertible for k = 1, . . . , N , and T (A − BD−1 C) = −A∗ T,

T BD−1 = C ∗ J,

D−1 C = JB ∗ T.

(4.8)

Note that the relation D−1 = JD∗ J, which means J-unitarity of D, has been already established above. It is easy to check that relations (4.8) are also valid for T ∗ in the place of T . Hence, by the uniqueness of similarity matrix, T = T ∗ . Setting H = −T , we obtain from (4.8) the equalities (4.3) and (4.4), as well as (4.5) and (4.6), by a straightforward calculation. Let us prove now a slightly more general statement than the converse. Let α be a (not necessarily minimal) GR-realization of F of the form (3.8), where D is J-unitary, and let H = diag(H1 , . . . , HN ) with Hk ∈ Crk ×rk , k = 1, . . . , N , be a Hermitian invertible matrix satisfying (4.3) and (4.4). Then in the same way as in [7, Theorem 2.1] for the one-variable case, we obtain for Z, Z ∈ Cn×n : −1

F (Z)(J ⊗ In )F (Z )∗ = J ⊗ In − (C ⊗ In ) (IIr ⊗ In − ∆(Z)(A ⊗ In )) ×∆(Z + Z ∗ )(H −1 ⊗ In ) (IIr ⊗ In − (A∗ ⊗ In )∆(Z ∗ ))

−1

(C ∗ ⊗ In )

(4.9)

−1

(note that ∆(Z) commutes with H ⊗ In ). It follows from (4.9) that F (Z) is (J ⊗ In )-unitary on (iHn×n )N at all points Z where it is deﬁned. Since n ∈ N is arbitrary, F is matrix-J-unitary on JN . Clearly, conditions a) and b’) also imply the matrix-J-unitarity of F on JN . Let us make some remarks. First, it follows from the proof of Theorem 4.1 that the structured solution H = diag(H1 , . . . , HN ) of the Lyapunov equation (4.3) is uniquely determined by a given minimal GR-realization of F . The matrix H = diag(H1 , . . . , HN ) is called the associated structured Hermitian matrix (associated with this minimal GR-realization of F ). The matrix Hk will be called the kth component of the associated Hermitian matrix (k = 1, . . . , N ). The explicit formulas for Hk follow from (3.2): wgk 2 wg 3+ Hk = − col|w|≤qr−1 ((JB ∗ )(−A∗ )) k col|w|≤qr−1 (D−1 C)A× 4 T g wT 5† = −row|w|≤qr−1 ((−A∗ )(C ∗ J))gk w row|w|≤qr−1 A× (BD−1 ) k . Second, let α be a (not necessarily minimal) GR-realization of F of the form (3.8), where D is J-unitary, and let H = diag(H1 , . . . , HN ) with Hk ∈ Crk ×rk , k = 1, . . . , N , be an Hermitian, not necessarily invertible, matrix satisfying (4.3) and

70

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

(4.6). Then in the same way as in [7, Theorem 2.1] for the one-variable case, we obtain for Z, Z ∈ Cn×n : F (Z )∗ (J ⊗ In )F (Z) = J ⊗ In − (B ∗ ⊗ In ) (IIr ⊗ In − ∆(Z ∗ )(A∗ ⊗ In )) ×(H ⊗ In )∆(Z ∗ + Z) (IIr ⊗ In − (A ⊗ In )∆(Z))

−1

(B ⊗ In )

−1

(4.10)

(note that ∆(Z) commutes with H ⊗ In ). It follows from (4.10) that F (Z) is (J ⊗ In )-unitary on (iHn×n )N at all points Z where it is deﬁned. Since n ∈ N is arbitrary, F is matrix-J-unitary on JN . Third, if α is a (not necessarily minimal) GR-realization of F of the form (3.8), where D is J-unitary, and equalities (4.5) and (4.6) are valid with H −1 replaced by some, possibly not invertible, Hermitian matrix Y = diag(Y Y1 , . . . , YN ) with Yk ∈ Crk ×rk , k = 1, . . . , N , then F is matrix-J-unitary on JN . This follows from the fact that (4.9) is valid with H −1 replaced by Y . Theorem 4.2. Let (C, A) be an observable pair of matrices C ∈ Cq×r , A ∈ (N rk and Ok has full column rank for each Cr×r in the sense that Cr = k=1 C k ∈ {1, . . . , N }, and let J ∈ Cq×q be a signature matrix. Then there exists a matrix-J-unitary on JN rational FPS F with a minimal GR-realization (N rk q α = (N ; A, B, C, D; Cr = k=1 C , C ) if and only if the Lyapunov equation (4.3) has a structured solution H = diag(H1 , . . . , HN ) which is both Hermitian and invertible. If such a solution H exists, possible choices of D and B are D0 = Iq ,

B0 = −H −1 C ∗ J.

(4.11)

Finally, for a given such H, all other choices of D and B diﬀer from D0 and B0 by a right multiplicative J-unitary constant matrix. Proof. Let H = diag(H1 , . . . , HN ) be a structured solution of the Lyapunov equation (4.3) which is both Hermitian and invertible. We ﬁrst check that the pair (A, −H −1 C ∗ J) is controllable, or equivalently, that the pair (−JCH −1 , A∗ ) is observable. Using the Lyapunov equation (4.3), one can see that for any k ∈ {1, . . . , N } and w = gi1 · · · gi|w| ∈ FN there exist matrices K0 , . . . , K|w|−1 such that (CA)wgk

= (−1)|w|−1 J((−JCH −1 )A∗ )wgk Hk + K0 J(−JC Ci2 Hi−1 (A∗ )i2 i3 · · · (A∗ )i|w| k )Hk + · · · 2 + K|w|−2 J(−JC Ci|w| (A∗ )i|w| k )Hk + K|w|−1 J(−JCk Hk−1 )Hk .

Thus, if x ∈ ker((−JCH −1 )A∗ )wgk for all of w ∈ FN then Hk−1 x ∈ ker Ok , and the observability of the pair (C, A) implies that x = 0. Therefore, the pair (−JCH −1 , A∗ ) is observable, and the pair (A, −H −1 C ∗ J) is controllable. By Theorem 4.1 we obtain that F0 (z) = Iq − C(IIr − ∆(z)A)−1 ∆(z)H −1 C ∗ J

(4.12)

is a matrix-J-unitary on JN rational FPS, which has a minimal GR-realization (N α0 = (N : A, −H −1 C ∗ J, C, Iq ; Cr = k=1 Crk , Cq ) with the associated structured Hermitian matrix H.

Matrix-J-unitary Rational Formal Power Series

71

(N Conversely, let α = (N ; A, B, C, D; Cr = k=1 Crk , Cq ) be a minimal GRnode. Then by Theorem 4.1 there exists an Hermitian and invertible matrix H = diag(H1 , . . . , HN ) which solves (4.3). Given H = diag(H1 , . . . , HN ), let B, D be any solution of the inverse problem, (N rk q i.e., α = (N ; A, B, C, D; Cr = k=1 C , C ) is a minimal GR-node with the associated structured Hermitian matrix H. Then for F0 = Tαnc0 and F = Tαnc we obtain from (4.9) that F (Z)(J ⊗ In )F (Z )∗ = F0 (Z)(J ⊗ In )F F0 (Z )∗ for any n ∈ N, at all points Z, Z ∈ (Cn×n )N where both F and F0 are deﬁned. By the uniqueness theorem in several complex variables (matrix entries for Zk ’s and Z ∗k ’s, k = 1, . . . , N ), we obtain that F (Z) and F0 (Z) diﬀer by a right multiplicative (J ⊗ In )-unitary constant, which clearly has to be D ⊗ In , i.e., F (Z) = F0 (Z)(D ⊗ In ). Since n ∈ N is arbitrary, by Corollary 2.2 we obtain F (z) = F0 (z)D. Equating the coeﬃcients of these two FPSs, we easily deduce using the observability of the pair (C, A) that B = −H −1 C ∗ JD. The following dual theorem is proved analogously. Theorem 4.3. Let (A, B) be a controllable pair of matrices A ∈ Cr×r , B ∈ Cr×q in (N the sense that Cr = k=1 Crk and Ck has full row rank for each k ∈ {1, . . . , N }, q×q and let J ∈ C be a signature matrix. Then there exists a matrix-J-unitary on JN rational FPS F with a minimal GR-realization α = (N ; A, B, C, D; Cr = (N rk q k=1 C , C ) if and only if the Lyapunov equation GA∗ + AG = −BJB ∗ has a structured solution G = diag(G1 , . . . , GN ) which is both Hermitian and invertible. If such a solution G exists, possible choices of D and C are D0 = Iq ,

C0 = −JB ∗ G−1 .

(4.13)

Finally, for a given such G, all other choices of D and C diﬀer from D0 and C0 by a left multiplicative J-unitary constant matrix. Theorem 4.4. Let F be a matrix-J-unitary on JN rational FPS, and α be its GRrealization. Let H = diag(H1 , . . . , HN ) with Hk ∈ Crk ×rk , k = 1, . . . , N , be an Hermitian invertible matrix satisfying (4.3) and (4.4), or equivalently, (4.5) and (4.6). Then α is observable if and only if α is controllable. Proof. Suppose that α is observable. Since by Theorem 4.1 D = F∅ is J-unitary, by Theorem 4.2 α is a minimal GR-node. In particular, α is controllable. Suppose that α is controllable. Then by Theorem 4.3 α is minimal, and in particular, observable.

72

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

4.2. The associated structured Hermitian matrix Lemma 4.5. Let F be a matrix-J-unitary on JN rational FPS, and let α(i) = (N (N ; A(i) , B (i) , C (i) , D; Cγ = k=1 Cγk , Cq ) be minimal GR-realizations of F , with (i) (i) the associated structured Hermitian matrices H (i) = diag(H1 , . . . , HN ), i = 1, 2. (1) (2) Then α and α are similar, i.e., (2.8) holds with a uniquely deﬁned invertible matrix T = diag(T T1 , . . . , TN ), and (1)

Hk

In particular, the matrices

= Tk∗ Hk Tk , (2)

(1) Hk

and

(2) Hk

k = 1, . . . , N.

(4.14)

have the same signature.

The proof is easy and analogous to the proof of Lemma 2.1 in [7]. Remark 4.6. The similarity matrix T = diag(T T1 , . . . , TN ) is a unitary map(N γk ping from Cγ = C endowed with the inner product [ · , · ]H (1) onto k=1 (N γ γk C = k=1 C endowed with the inner product [ · , · ]H (2) , where [x, y]H (i) = H (i) x, yCγ ,

x, y ∈ Cγ , i = 1, 2,

that is, [x, y]H (i) =

N

[xk , yk ]H (i) ,

k=1

i = 1, 2,

k

where xk , yk ∈ Cγk , x = colk=1,...,N (xk ), y = colk=1,...,N (yk ), and (i)

[xk , yk ]H (i) = Hk xk , yk Cγk ,

k = 1, . . . , N, i = 1, 2.

k

Recall the following deﬁnition [37]. Let Kw,w be a Cq×q -valued function deKw,w )∗ = Kw ,w . Then Kw,w is ﬁned for w and w in some set E and such that (K called a kernel with κ negative squares if for any m ∈ N, any points w1 , . . . , wm in E, and any vectors c1 , . . . , cm in Cq the matrix (c∗j Kwj ,wi ci )i,j=1,...,m ∈ Hm×m has at most κ negative eigenvalues, and has exactly κ negative eigenvalues for some choice of m, w1 , . . . , wm , c1 , . . . , cm . We will use this deﬁnition to give a characterization of the number of negative eigenvalues of the kth component Hk , k = 1, . . . , N , of the associated structured Hermitian matrix H. Theorem 4.7. Let F be a matrix-J-unitary on JN rational FPS, and let α be its minimal GR-realization of the form (3.11), with the associated structured Hermitian matrix H = diag(H1 , . . . , HN ). Then for k = 1, . . . , N the number of negative eigenvalues of the matrix Hk is equal to the number of negative squares of each of the kernels F,k Kw,w ∗

F ,k Kw,w

T

= (CA)wgk Hk−1 (A∗ C ∗ )gk w , T

= (B ∗ A∗ )wgk Hk (AB)gk w ,

w, w ∈ FN , w, w ∈ FN ,

∗

(4.15) (4.16)

For k = 1, . . . , N , denote by Kk (F ) (resp., Kk (F )) the linear span of the functions F,k F ∗ ,k q w → Kw,w c (resp., w → Kw,w c) where w ∈ FN and c ∈ C . Then dim Kk (F ) = dim Kk (F ∗ ) = γk .

Matrix-J-unitary Rational Formal Power Series

73

Proof. Let m ∈ N, w1 , . . . , wm ∈ FN , and c1 , . . . , cm ∈ Cq . Then the matrix equality F,k c) = X ∗ Hk−1 X, (c∗j Kw j ,wi i i,j=1,...,m

with

T X = row1≤i≤m (A∗ C ∗ )gk wi ci ,

F,k implies that the kernel Kw,w has at most κk negative squares, where κk denotes the number of negative eigenvalues of Hk . The pair (C, A) is observable, hence we T can choose a basis of Cq of the form xi = (A∗ C ∗ )gk wi ci , i = 1, . . . , q. Since the matrix X = rowi=1,...,q (xi ) is non-degenerate, and therefore the matrix X ∗ Hk−1 X F,k has exactly κk negative eigenvalues, the kernel Kw,w has κk negative squares. Analogously, from the controllability of the pair (A, B) one can obtain that the kernel Kk (F ∗ ) has κk negative squares. Since Kk (F ) is the span of functions (of variable w ∈ FN ) of the form (CA)wgk y, y ∈ Cγk , it follows that dim Kk (F ) ≤ γk . From the observability of the pair (C, A) we obtain that (CA)wgk y ≡ 0 implies y = 0, thus dim Kk (F ) = γk . In the same way we obtain that the controllability of the pair (A, B) implies that dim Kk (F ∗ ) = γk .

We will denote by νk (F ) the number of negative squares of either the kernel F ∗ ,k or the kernel Kw,w deﬁned by (4.15) and (4.16), respectively.

F,k Kw,w

Theorem 4.8. Let F (i) be matrix-J-unitary on JN rational FPSs, with minimal (N (i) (i) γk GR-realizations α(i) = (N ; A(i) , B (i) , C (i) , D(i) ; Cγ = , Cq ) and the k=1 C (i) (i) associated structured Hermitian matrices H (i) = diag(H1 , . . . , HN ), respectively, (1) (2) i = 1, 2. Suppose that the product α = α α is a minimal GR-node. Then the matrix H = diag(H1 , . . . , HN ), with (1) (1) (2) (1) (2) Hk 0 Hk = (4.17) ∈ C(γk +γk )×(γk +γk ) , k = 1, . . . , N, (2) 0 Hk is the associated structured Hermitian matrix for α = α(1) α(2) . Proof. It suﬃces to check that (4.3) and (4.4) hold for the matrices A, B, C, D deﬁned as in (3.7), and H = diag(H1 , . . . , HN ) where Hk , k = 1, . . . , N , are deﬁned in (4.17). This is an easy computation which is omitted. Corollary 4.9. Let F1 and F2 be matrix-J-unitary on JN rational FPSs, and suppose that the factorization F = F1 F2 is minimal. Then νk (F F1 F2 ) = νk (F F1 ) + νk (F F2 ),

k = 1, . . . , N.

74

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

4.3. Minimal matrix-J-unitary factorizations In this subsection we consider minimal factorizations of rational formal power series which are matrix-J-unitary on JN into factors both of which are also matrix-Junitary on JN . Such factorizations will be called minimal matrix-J-unitary factorizations. Let H ∈ Cr×r be an invertible Hermitian matrix. We denote by [ · , · ]H the Hermitian sesquilinear form [x, y]H = Hx, y where · , · denotes the standard inner product of Cr . Two vectors x and y in Cr are called H-orthogonal if [x, y]H = 0. For any subspace M ⊂ Cr denote M [⊥] = {y ∈ Cr : y, mH = 0 ∀m ∈ M } . The subspace M ⊂ Cr is called non-degenerate if M ∩ M [⊥] = {0}. In this case, ·

M [+]M [⊥] = Cr ·

where [+] denotes the H-orthogonal direct sum. In the case when H = diag(H1 , . . . , HN ) is the structured Hermitian matrix associated with a given minimal GR-realization of a matrix-J-unitary on JN rational FPS F , we will call [ · , · ]H the associated inner product (associated with the given minimal GR-realization of F ). In more details, [x, y]H =

N

[xk , yk ]Hk ,

k=1

where xk , yk ∈ Cγk and x = colk=1,...,N (xk ), y = colk=1,...,N (yk ), and [xk , yk ]Hk = Hk xk , yk Cγk ,

k = 1, . . . , N.

The following theorem (as well as its proof) is analogous to its one-variable counterpart, Theorem 2.6 from [7] (see also [43, Chapter II]). Theorem 4.10. Let F be a matrix-J-unitary on JN rational FPS, and let α be its minimal GR-realization of the form (3.11), with the associated structured Her(N mitian matrix H = diag(H1 , . . . , HN ). Let M = k=1 Mk be an A-invariant subspace such that Mk ⊂ Cγk , k = 1, . . . , N , and M is non-degenerate in the associated inner product [ · , · ]H . Let Π = diag(Π1 , . . . , ΠN ) be the projection deﬁned by ker Π = M,

ran Π = M[⊥] ,

or in more details, ker Πk = Mk ,

[⊥]

ran Πk = Mk ,

k = 1, . . . , N.

Matrix-J-unitary Rational Formal Power Series

75

Let D = D1 D2 be a factorization of D into two J-unitary factors. Then the factorization F = F1 F2 where F1 (z) =

D1 + C(IIγ − ∆(z)A)−1 ∆(z)(IIγ − Π)BD2−1 ,

F2 (z) =

D2 + D1−1 CΠ(IIγ − ∆(z)A)−1 ∆(z)B,

is a minimal matrix-J-unitary factorization of F . Conversely, any minimal matrix-J-unitary factorization of F can be obtained in such a way. For a ﬁxed J-unitary decomposition D = D1 D2 , the correspondence between minimal matrix-J-unitary factorizations of F and non(N degenerate A-invariant subspaces of the form M = k=1 Mk , where Mk ⊂ Cγk for k = 1, . . . , N , is one-to-one. Remark 4.11. We omit here the proof, which can be easily restored, with making use of Theorem 3.9 and Corollary 3.10. Remark 4.12. Minimal matrix-J-unitary factorizations do not always exist, even for N = 1. Examples of J-unitary on iR rational functions which have non-trivial minimal factorizations but lack minimal J-unitary factorizations can be found in [4] and [7]. 4.4. Matrix-unitary rational formal power series In this subsection we specialize some of the preceding results to the case J = Iq . We call the corresponding rational formal power series matrix-unitary on JN . Theorem 4.13. Let F be a rational FPS and α be its minimal GR-realization of the form (3.11). Then F is matrix-unitary on JN if and only if the following conditions are fulﬁlled: a) D is a unitary matrix, i.e., DD∗ = Iq ; b) there exists an Hermitian solution H = diag(H1 , . . . , HN ), with Hk ∈ Cγk ×γk , k = 1, . . . , N , of the Lyapunov equation A∗ H + HA = −C ∗ C, and

(4.18)

C = −D−1 B ∗ H.

(4.19) The property b) is equivalent to b ) there exists an Hermitian solution G = diag(G1 , . . . , GN ), with Gk ∈ Cγk ×γk , k = 1, . . . , N , of the Lyapunov equation and

GA∗ + AG = −BB ∗ ,

(4.20)

B = −GC ∗ D−1 .

(4.21)

Proof. To obtain Theorem 4.13 from Theorem 4.1 it suﬃces to show that any structured Hermitian solution to the Lyapunov equation (4.18) (resp., (4.20)) is invertible. Let H = diag(H1 , . . . , HN ) be a structured Hermitian solution to (4.18), and x ∈ ker H, i.e., x = col1≤k≤N (xk ) and xk ∈ ker Hk , k = 1, . . . , N . Then

76

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

HAx, x = Ax, Hx = 0, and equation (4.18) implies Cx = 0. In particular, ˜ = col(0, . . . , 0, xk , 0, . . . , 0) where xk ∈ for every k ∈ {1, . . . , N } one can deﬁne x ker Hk is on the kth block entry of x ˜, and from C x ˜ = 0 get Ck xk = 0. Thus, ker Hk ⊂ ker Ck , k = 1, . . . , N . Consider the following block representations with respect to the decompositions Cγk = ker Hk ⊕ ran Hk : (11) (12) 0 0 Aijj Aijj (2) , H Aij = , C = = 0 C k k (22) , (21) (22) k 0 Hk Aij Aij where i, j, k = 1, . . . , N . Then (4.18) implies (A∗ H + HA)ij

(12)

and

(21) Aji

(21) ∗

= (A∗ji Hj + Hi Aij )(12) = (Aji ) Hj

(22)

= 0,

= 0, i, j = 1, . . . , N . Therefore, for any w ∈ FN we have (CA)wgk = 0 (C (2) A(22) )wgk , k = 1, . . . , N, (2)

(22)

where C (2) = row1≤k≤N (Ck ), A(22) = (Aij )i,j=1,...,N . If there exists k ∈ {1, . . . , N } such that ker Hk = {0}, then the pair (C, A) is not observable, which contradicts to the assumption on α. Thus, H is invertible. In a similar way one can show that any structured Hermitian solution G = diag(G1 , . . . , GN ) of the Lyapunov equation (4.20) is invertible. A counterpart of Theorem 4.2 in the present case is the following theorem. Theorem 4.14. Let (C, A) be an observable pair of matrices C ∈ Cq×r , A ∈ Cr×r (N rk in the sense that Cr = and Ok has full column rank for each k ∈ k=1 C {1, . . . , N }. Then there exists a matrix-unitary on JN rational FPS F with a mini(N mal GR-realization α = (N ; A, B, C, D; Cr = k=1 Crk , Cq ) if and only if the Lyapunov equation (4.18) has a structured Hermitian solution H = diag(H1 , . . . , HN ). If such a solution H exists, it is invertible, and possible choices of D and B are D0 = Iq ,

B0 = −H −1 C ∗ .

(4.22)

Finally, for a given such H, all other choices of D and B diﬀer from D0 and B0 by a right multiplicative unitary constant matrix. The proof of Theorem 4.14 is a direct application of Theorem 4.2 and Theorem 4.13. One can prove analogously the following theorem which is a counterpart of Theorem 4.3. Theorem 4.15. Let (A, B) be a controllable pair of matrices A ∈ Cr×r , B ∈ Cr×q (N in the sense that Cr = k=1 Crk and Ck has full row rank for each k ∈ {1, . . . , N }. Then there exists a matrix-unitary on JN rational FPS F with a minimal GR(N rk q realization α = (N ; A, B, C, D; Cr = k=1 C , C ) if and only if the Lyapunov equation (4.20) has a structured Hermitian solution G = diag(G1 , . . . , GN ). If such a solution G exists, it is invertible, and possible choices of D and C are D0 = Iq ,

C0 = −B ∗ G−1 .

(4.23)

Matrix-J-unitary Rational Formal Power Series

77

Finally, for a given such G, all other choices of D and C diﬀer from D0 and C0 by a left multiplicative unitary constant matrix. Let A = (A1 , . . . , AN ) be an N -tuple of r × r matrices. A non-zero vector x ∈ Cr is called a common eigenvector for A if there exists λ = (λ1 , . . . , λN ) ∈ CN (which is called a common eigenvalue for A) such that Ak x = λk x,

k = 1, . . . , N.

The following theorem, which is a multivariable non-commutative counterpart of statements a) and b) of Theorem 2.10 in [7], gives a necessary condition on a minimal GR-realization of a matrix-unitary on JN rational FPS. Theorem 4.16. Let F be a matrix-unitary on JN rational FPS and α be its minimal GR-realization, with the associated structured Hermitian matrix H = diag(H1 , . . . , HN ) and the associated inner products [ · , · ]Hk , k = 1, . . . , N . Let Pk denote the orthogonal projection in Cγ onto the subspace {0} ⊕ · · · ⊕ {0} ⊕ Cγk ⊕ {0} ⊕ · · · ⊕ {0}, and Ak = AP Pk , k = 1, . . . , N . If x ∈ Cγ is a common eigenvector for A corresponding to a common eigenvalue λ ∈ CN then there exists Pj x, Pj x]Hj = 0. In particular, A has no j ∈ {1, . . . , N } such that Re λj = 0 and [P common eigenvalues on (iR)N . Proof. By (4.18), we have for every k ∈ {1, . . . , N }, Pk x, Pk x]Hk = − CP Pk x, CP Pk x . (λk + λk )[P Suppose that for all k ∈ {1, . . . , N } the left-hand side of this equality is zero, then CP Pk x = 0. Since for ∅ = w = gi1 · · · gi|w| ∈ FN , Pi1 Ai2 · · · Ai|w| · Ak x = λi2 · · · λi|w| λk CP Pi1 x = 0, (CA)wgk Pk x = CP the observability of the pair (C, A) implies Pk x = 0, k = 1, . . . , N , i.e., x = 0 which contradicts to the assumption that x is a common eigenvector for A. Thus, there exists j ∈ {1, . . . , N } such that (λj + λj )[P Pj x, Pj x]Hj = 0, as desired.

5. Matrix-J-unitary formal power series: A multivariable non-commutative analogue of the circle case In this section we study a multivariable non-commutative analogue of rational Cq×q -valued functions which are J-unitary on the unit circle T. 5.1. Minimal Givone–Roesser realizations and the Stein equation Let n ∈ N. We denote by Tn×n the matrix unit circle Tn×n = W ∈ Cn×n : W W ∗ = In , i.e., the family of unitary n × n complex matrices. We will call the set (Tn×n ) the matrix unit torus. The set 0 N TN = Tn×n n∈N

N

78

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

serves as a multivariable non-commutative counterpart of the unit circle. Let J = J −1 = J ∗ ∈ Cq×q . We will say that a rational FPS f is matrix-J-unitary on TN if for every n ∈ N, f (W )(J ⊗ In )f (W )∗ = J ⊗ In N

at all points W = (W W1 , . . . , WN ) ∈ (Tn×n ) where it is deﬁned. In the following theorem we establish the relationship between matrix-J-unitary rational FPSs on JN and on TN , their minimal GR-realizations, and the structured Hermitian solutions of the corresponding Lyapunov and Stein equations. Theorem 5.1. Let f be a matrix-J-unitary on TN rational FPS, with a minimal GR-realization α of the form (3.11), and let a ∈ T be such that −¯ a ∈ σ(A). Then F (z) = f (a(z1 − 1)(z1 + 1)−1 , . . . , a(zN − 1)(zN + 1)−1 )

(5.1)

is well deﬁned as a rational FPS which is matrix-J-unitary on JN , and F = Tβnc, (N where β = (N ; Aa , Ba , Ca , Da ; Cγ = k=1 Cγk , Cq ), with √ −1 Aa = (aA , Ba = 2(aA + Iγ )−1 aB, γ) √− Iγ )(aA + I−1 (5.2) Ca = 2C(aA + Iγ ) , Da = D − C(aA + Iγ )−1 aB. A GR-node β is minimal, and its associated structured Hermitian matrix H = diag(H1 , . . . , HN ) is the unique invertible structured Hermitian solution of ∗ A B H 0 A B H 0 = . (5.3) C D 0 J C D 0 J Proof. For any a ∈ T and n ∈ N the Cayley transform Z0 −→ W0 = a(Z0 − In )(Z0 + In )−1 maps iHn×n onto Tn×n , thus its simultaneous application to each matrix variable maps (iHn×n )N onto (Tn×n )N . Since the simultaneous application of the Cayley transform to each formal variable in a rational FPS gives a rational FPS, (5.1) deﬁnes a rational FPS F. Since f is matrix-J-unitary on TN , F is matrix-J-unitary on JN . Moreover, −1 F (z) = D + C Iγ − a(∆(z) − Iγ )(∆(z) + Iγ )−1 A ×a(∆(z) − Iγ )(∆(z) + Iγ )−1 B = D + C (∆(z) + Iγ − a(∆(z) − Iγ )A)−1 a(∆(z) − Iγ )B = D + C (aA + Iγ − ∆(z)(aA − Iγ ))−1 a(∆(z) − Iγ )B −1 ∆(z)aB = D + C(aA + Iγ )−1 Iγ − ∆(z)(aA − Iγ )(aA + Iγ )−1 −1 −C(aA + Iγ )−1 Iγ − ∆(z)(aA − Iγ )(aA + Iγ )−1 aB = D − C(aA + Iγ )−1 aB + C(aA + Iγ )−1 −1 × Iγ − ∆(z)(aA − Iγ )(aA + Iγ )−1 ×∆(z) Iγ − (aA − Iγ )(aA + Iγ )−1 aB = Da + Ca (IIγ − ∆(z)Aa )−1 ∆(z)Ba .

Matrix-J-unitary Rational Formal Power Series

79

Thus, F = Tβnc. Let us remark that the FPS

ϕak (z) = Ca (IIγ − ∆(z)Aa )−1 -Cγk

(c.f. (3.5)) has the coeﬃcients Ca Aa )wgk , (ϕak )w = (C

w ∈ FN .

Remark also that ϕ˜k (z) : = ϕk a(z1 − 1)(z1 + 1)−1 , . . . , a(zN − 1)(zN + 1)−1 −1 - γ = C Iγ − a(∆(z) − Iγ )(∆(z) + Iγ )−1 A C k −1 = C ((∆(z) + Iγ ) − a(∆(z) − Iγ )A) (∆(z) + Iγ )-Cγk −1 = C ((aA + Iγ ) − ∆(z)(aA − Iγ )) (∆(z) + Iγ )-Cγk −1 = C(aA + Iγ )−1 Iγ − ∆(z)(aA − Iγ )(aA + Iγ )−1 (∆(z) + Iγ )-Cγk - 1 = √ Ca (IIγ − ∆(z)Aa )−1 -Cγk (zk + 1) 2 1 = √ (ϕak (z) · zk + ϕak (z)) . 2 qγ−1

Let k ∈ {1, . . . , N } be ﬁxed. Suppose that/n ∈ N, n ≥ (qγ − 1) (for qγ − 1 = 0 choose arbitrary n ∈ N), and x ∈ Z∈Γn (ε) ker ϕak (Z), where Γn (ε) is a neighborhood of the origin of Cn×n where ϕak (Z) is well deﬁned, e.g., of the form (2.9) with ε = Aa −1 . Then, by Theorem 3.1 and Theorem 2.1, one has ⎞ ⎛ ) ) ker ϕak (Z) = ⎝ ker (ϕak )w ⎠ ⊗ Cn ⎛ =⎝

w∈F FN : |w|≤qγ−1

Z∈Γn (ε)

)

⎞

˜k (β) ⊗ Cn . ker (C Ca Aa )wgk ⎠ ⊗ Cn = ker O

w∈F FN : |w|≤qγ−1

˜k (β), {y (µ) }lµ=1 ⊂ Cn such that Thus, there exist l ∈ N, {u(µ) }lµ=1 ⊂ ker O x=

l

u(µ) ⊗ y (µ) .

(5.4)

µ=1

Since (ϕak (z) · zk )wgk = (C Ca Aa )wgk for w ∈ FN , and (ϕak (z) · zk )w = 0 for w = wgk with any w ∈ FN , (5.4) implies that ϕak (Z)(IIγk ⊗ Zk )x ≡ 0. Thus, 1 ϕ˜k (Z)x = √ (ϕak (Z)(IIγk ⊗ Zk ) + ϕak (Z)) x ≡ 0. 2 Since the Cayley transform a(∆(z)−IIγ )(∆(z)+IIγ )−1 maps an open and dense subset of the set of matrices of the form ∆(Z) = diag (Z1 , . . . , ZN ), Zj ∈ Cγj ×γj , j = 1, . . . , N , onto an open and dense subset of the same set, ϕk (Z)x = (C ⊗ In )(IIγ − ∆(Z)(A ⊗ In ))−1 x ≡ 0.

80

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

Since the GR-node α is observable, by Theorem 3.7 we get x = 0. Therefore, ) ker ϕak (Z) = 0, k = 1, . . . , N. Z∈Γn (ε)

Applying Theorem 3.7 once again, we obtain the observability of the GR-node β. In the same way one can prove the controllability of β. Thus, β is minimal. Note that ∗ H 0 A B H 0 A B − = 0 J C D 0 J C D ∗ A∗ HB + C ∗ JD A HA + C ∗ JC − H . (5.5) = B ∗ HA + D∗ JC B ∗ HB + D∗ JD − J Since −a ¯∈ / σ(A), the matrix (aA + Iγ )−1 is well deﬁned, as well as Aa = (aA − Iγ )(aA + Iγ )−1 , and Iγ − Aa = 2(aA + Iγ )−1 is invertible. Having this in mind, one can deduce from (5.2) the following relations: A∗ HA + C ∗ JC − H = 2(IIγ − A∗a )−1 (A∗a H + HAa + Ca∗ JC Ca )(IIγ − Aa )−1 B ∗ HA + D∗ JC

√ 2(Ba∗ H + Da∗ JC Ca )(IIγ − Aa )−1 √ ∗ + 2Ba (IIγ − A∗a )−1 (A∗a H + HAa + Ca∗ JC Ca )(IIγ − Aa )−1 =

B ∗ HB + D∗ JD − J =

Ba∗ (IIγ − A∗a )−1 (A∗a H + HAa + Ca∗ JC Ca )(IIγ − Aa )−1 Ba

+

(Ba∗ H + Da∗ JC Ca )(IIγ − Aa )−1 Ba + Ba∗ (IIγ − A∗a )−1 (C Ca∗ JDa + HBa ).

Thus, A, B, C, D, H satisfy (5.3) if and only if Aa , Ba , Ca , Da , H satisfy (4.3) and (4.4) (in the place of A, B, C, D, H therein), which completes the proof. We will call the invertible Hermitian solution H = diag(H1 , . . . , HN ) of (5.3), which is determined uniquely by a minimal GR-realization α of a matrix-J-unitary on TN rational FPS f , the associated structured Hermitian matrix (associated with a minimal GR-realization α of f ). Let us note also that since for the GR-node β from Theorem 5.1 a pair of the equalities (4.3) and (4.4) is equivalent to a pair of the equalities (4.5) and (4.6), the equality (5.3) is equivalent to ∗ −1 −1 H H 0 A B 0 A B = . (5.6) C D 0 J C D 0 J Remark 5.2. Equality (5.3) can be replaced by the following three equalities: H − A∗ HA D∗ JC J − D∗ JD

= C ∗ JC, = −B ∗ HA,

(5.7) (5.8)

= B ∗ HB,

(5.9)

Matrix-J-unitary Rational Formal Power Series

81

and equality (5.6) can be replaced by H −1 − AH −1 A∗ DJB

∗

J − DJD

∗

= = =

BJB ∗ , −CH CH

−1

−1

(5.10) ∗

A ,

(5.11)

∗

(5.12)

C .

Theorem 5.1 allows to obtain a counterpart of the results from Section 4 in the setting of rational FPSs which are matrix-J-unitary on TN . We will skip the proofs when it is clear how to get them. Theorem 5.3. Let f be a rational FPS and α be its minimal GR-realization of the form (3.11). Then f is matrix-J-unitary on TN if and only if there exists an invertible Hermitian matrix H = diag(H1 , . . . , HN ), with Hk ∈ Cγk ×γk , k = 1, . . . , N , which satisﬁes (5.3), or equivalently, (5.6). Remark 5.4. In the same way as in [7, Theorem 3.1] one can show that if a rational FPS f has a (not necessarily minimal) GR-realization (3.8) which satisﬁes (5.3) (resp., (5.6)), with an Hermitian invertible matrix H = diag(H1 , . . . , HN ), then for any n ∈ N, f (Z )∗ (J ⊗ In )f (Z) =

−1

J ⊗ In − (B ∗ ⊗ In ) (IIγ ⊗ In − ∆(Z ∗ )(A∗ ⊗ In ))

×

(H ⊗ In )(IIγ ⊗ In − ∆(Z )∗ ∆(Z))

×

(IIγ ⊗ In − (A ⊗ In )∆(Z))

−1

(B ⊗ In )

(5.13)

and respectively, f (Z)(J ⊗ In )f (Z )∗

= J ⊗ In − (C ⊗ In ) (IIγ ⊗ In − ∆(Z)(A ⊗ In ))−1 × (IIγ ⊗ In − ∆(Z)∆(Z )∗ )(H −1 ⊗ In ) −1

× (IIγ ⊗ In − (A∗ ⊗ In )∆(Z )∗ )

(C ∗ ⊗ In ),

(5.14)

N

at all the points Z, Z ∈ (Cn×n ) where it is deﬁned, which implies that f is matrix-J-unitary on TN . Moreover, the same statement holds true if H = diag(H1 , . . . , HN ) in (5.3) and (5.13) is not supposed to be invertible, and if −1 ) in (5.6) and (5.14) is replaced by any Hermitian, H −1 = diag(H1−1 , . . . , HN Y1 , . . . , YN ). not necessarily invertible matrix Y = diag(Y Theorem 5.5. Let f be a matrix-J-unitary on TN rational FPS, and α be its GRrealization. Let H = diag(H1 , . . . , HN ) with Hk ∈ Crk ×rk , k = 1, . . . , N , be an Hermitian invertible matrix satisfying (5.3) or, equivalently, (5.6). Then α is observable if and only if α is controllable. Proof. Let a ∈ T, −a ¯ ∈ / σ(A). Then F deﬁned by (5.1) is a matrix-J-unitary on JN rational FPS, and (5.2) is its GR-realization. As shown in the proof of Theorem 5.1, α is observable (resp., controllable) if and only if so is β. Since by Theorem 5.1 the GR-node β satisﬁes (4.3) and (4.4) (equivalently, (4.5) and (4.6)), Theorem 4.4 implies the statement of the present theorem.

82

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

Theorem 5.6. Let f be a matrix-J-unitary on TN rational FPS and α be its minimal GR-realization of the form (3.11), with the associated structured Hermitian matrix H. If D = f∅ is invertible then so is A, and A−1 = H −1 (A× )∗ H.

(5.15)

Proof. It follows from (5.8) that C = −JD−∗ B ∗ HA. Then (5.7) turns into H − A∗ HA = C ∗ J(−JD−∗ B ∗ HA) = −C ∗ D−∗ B ∗ HA, which implies that H = (A× )∗ HA, and (5.15) follows.

The following two lemmas, which are used in the sequel, can be found in [7]. Lemma 5.7. Let A ∈ Cr×r , C ∈ Cq×r , where A is invertible. Let H be an invertible Hermitian matrix and J be a signature matrix such that H − A∗ HA = C ∗ JC. Let a ∈ T, a ∈ / σ(A). Deﬁne Da

=

Ba Then

A C

=

Ba Da

Iq − CH −1 (IIr − aA∗ )−1 C ∗ J, −H

∗ H 0

−1

−∗

A

0 J

(5.16)

∗

C JDa .

A C

Ba Da

(5.17) H = 0

0 . J

Lemma 5.8. Let A ∈ Cr×r , B ∈ Cr×q , where A is invertible. Let H be an invertible Hermitian matrix and J be a signature matrix such that H −1 − AH −1 A∗ = BJB ∗ . Let a ∈ T, a ∈ / σ(A). Deﬁne Da Ca Then

A Ca

B Da

=

Iq − JB ∗ (IIr − aA∗ )−1 HB,

(5.18)

=

−Da JB ∗ A−∗ H.

(5.19)

−1 H 0

0 J

A Ca

B Da

∗

−1 H = 0

0 . J

Theorem 5.9. Let (C, A) be an observable pair of matrices C ∈ Cq×r , A ∈ Cr×r (N rk in the sense that Cr = and Ok has full column rank for each k ∈ k=1 C {1, . . . , N }. Let A be invertible and J ∈ Cq×q be a signature matrix. Then there exists a matrix-J-unitary on TN rational FPS f with a minimal GR-realization (N rk q α = (N ; A, B, C, D; Cr = k=1 C , C ) if and only if the Stein equation (5.7) has a structured solution H = diag(H1 , . . . , HN ) which is both Hermitian and invertible. If such a solution H exists, possible choices of D and B are Da and Ba deﬁned in (5.16) and (5.17), respectively. For a given such H, all other choices of D and B diﬀer from Da and Ba by a right multiplicative J-unitary constant matrix.

Matrix-J-unitary Rational Formal Power Series

83

Proof. Let H = diag(H1 , . . . , HN ) be a structured solution of the Stein equation (5.7) which is both Hermitian and invertible, Da and Ba are deﬁned as in (5.16) and (5.17), respectively, where a ∈ T, a ∈ / σ(A). Set αa = (N ; A, Ba , C, Da ; Cr = (N rk q nc k=1 C , C ). By Lemma 5.7 and due to Remark 5.4, the transfer function Tα of αa is a matrix-J-unitary on TN rational FPS. Since αa is observable, by Theorem 5.5 αa is controllable, and thus, minimal. (N rk q Conversely, if α = (N ; A, B, C, D; Cr = k=1 C , C ) is a minimal GRnode whose transfer function is matrix-J-unitary on TN then by Theorem 5.3 there exists a solution H = diag(H1 , . . . , HN ) of the Stein equation (5.7) which is both Hermitian and invertible. The rest of the proof is analogous to the one of Theorem 4.2. Analogously, one can obtain the following. Theorem 5.10. Let (A, B) be a controllable pair of matrices A ∈ Cr×r , B ∈ Cr×q in (N the sense that Cr = k=1 Crk and Ck has full row rank for each k ∈ {1, . . . , N }. Let A be invertible and J ∈ Cq×q be a signature matrix. Then there exists a matrix-J-unitary on TN rational FPS f with a minimal GR-realization α = (N (N ; A, B, C, D; Cr = k=1 Crk , Cq ) if and only if the Stein equation G − AGA∗ = BJB ∗

(5.20)

has a structured solution G = diag(G1 , . . . , GN ) which is both Hermitian and invertible. If such a solution G exists, possible choices of D and C are Da and Ca deﬁned in (5.16) and (5.17), respectively, where H = G−1 . For a given such G, all other choices of D and C diﬀer from Da and Ca by a left multiplicative J-unitary constant matrix. 5.2. The associated structured Hermitian matrix In this subsection we give the analogue of the results of Section 4.2. The proofs are similar and will be omitted. Lemma 5.11. Let f be a matrix-J-unitary on TN rational FPS and α(i) = (N γk q (N ; A(i) , B (i) , C (i) , D; Cγ = k=1 C , C ) be its minimal GR-realizations, with (i) (i) the associated structured Hermitian matrices H (i) = diag(H1 , . . . , HN ), i = 1, 2. (1) (2) Then α and α are similar, that is C (1) = C (2) T,

T A(1) = A(2) T,

and

T B (1) = B (2) ,

for a uniquely deﬁned invertible matrix T = diag (T T1 , . . . , TN ) ∈ Cγ×γ and (1)

Hk

In particular, the matrices

= Tk∗ Hk Tk , (2)

(1) Hk

and

(2) Hk

k = 1, . . . , N. have the same signature.

Theorem 5.12. Let f be a matrix-J-unitary on TN rational FPS, and let α be its minimal GR-realization of the form (3.11), with the associated structured Hermitian matrix H = diag(H1 , . . . , HN ). Then for each k ∈ {1, . . . , N } the number of

84

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

negative eigenvalues of the matrix Hk is equal to the number of negative squares of each of the kernels (on FN ): T

f,k wgk −1 Kw,w Hk (A∗ C ∗ )gk w , = (CA) ∗

(5.21)

T

f ,k ∗ ∗ wgk Kw,w Hk (AB)gk w . = (B A )

Finally, for k ∈ {1, . . . , N } let Kk (f ) (resp., Kk (f ∗ )) be the span of the functions f,k f ∗ ,k q w → Kw,w c (resp., w → Kw,w c) where w ∈ FN and c ∈ C . Then dim Kk (f ) = dim Kk (f ∗ ) = γk . We will denote by νk (f ) the number of negative squares of either of the functions deﬁned in (5.21). Theorem 5.13. Let fi , i = 1, 2, be two matrix-J-unitary on TN rational FPSs, with minimal GR-realizations N ' (i) (i) (i) (i) (i) γ (i) γk q = C ,C α = N ; A , B , C , D; C k=1 (i)

(i)

and the associated structured Hermitian matrices H (i) = diag(H1 , . . . , HN ). Assume that the product α = α(1) α(2) is a minimal GR-node. Then, for each k ∈ {1, . . . , N } the matrix (1) (1) (2) (1) (2) Hk 0 Hk = ∈ C(γk +γk )×(γk +γk ) (5.22) (2) 0 Hk is the associated kth Hermitian matrix for α = α(1) α(2) . Corollary 5.14. Let f1 and f2 be two matrix-J-unitary on TN rational FPSs, and assume that the factorization f = f1 f2 is minimal. Then, ν(f1 f2 ) = ν(f1 ) + ν(ff2 ). 5.3. Minimal matrix-J-unitary factorizations In this subsection we consider minimal factorizations of matrix-J-unitary on TN rational FPSs into two factors, both of which are also matrix-J-unitary on TN rational FPSs. Such factorizations will be called minimal matrix-J-unitary factorizations. The following theorem is analogous to its one-variable counterpart [7, Theorem 3.7] and proved in the same way. Theorem 5.15. Let f be a matrix-J-unitary on TN rational FPS and α be its minimal GR-realization of the form (3.11), with the associated structured Hermitian matrix H = diag(H1 , . . . , HN ), and assume that D is invertible. Let (N γ M = k=1 Mk be an A-invariant subspace of C , which is non-degenerate in the associated inner product [ · , · ]H and such that Mk ⊂ Cγk , k = 1, . . . , N . Let Π = diag(Π1 , . . . , ΠN ) be a projection deﬁned by ker Π = M,

and

ran Π = M [⊥] ,

Matrix-J-unitary Rational Formal Power Series

85

that is [⊥]

f or k = 1, . . . , N. ker Πk = Mk , and ran Πk = Mk Then f (z) = f1 (z)ff2 (z), where 2 3 f1 (z) = Iq + C(IIγ − ∆(z)A)−1 ∆(z)(IIγ − Π)BD−1 D1 , 3 2 f2 (z) = D2 Iq + D−1 CΠ(IIγ − ∆(z)A)−1 ∆(z)B ,

(5.23) (5.24)

with D1 = Iq − CH −1 (IIγ − aA∗ )−1 C ∗ J, D = D1 D2 , where a ∈ T belongs to the resolvent set of A1 , and where C1 = C - , A1 = A- , H1 = PM H M

M

M

(with PM being the orthogonal projection onto M in the standard metric of Cγ ), is a minimal matrix-J-unitary factorization of f . Conversely, any minimal matrix-J-unitary factorization of f can be obtained in such a way, and the correspondence between minimal matrix-J-unitary factorizations of f with f1 (a, . . . , a) = Iq and non-degenerate subspaces of A of the form (N M = k=1 Mk , with Mk ⊂ Cγk , k = 1, . . . , N , is one-to-one. Remark 5.16. In the proof of Theorem 5.15, as well as of Theorem 4.10, we make use of Theorem 3.9 and Corollary 3.10. Remark 5.17. Minimal matrix-J-unitary factorizations do not always exist, even in the case N = 1. See [7] for examples in that case. 5.4. Matrix-unitary rational formal power series In this subsection we specialize some of the results in the present section to the case J = Iq . We shall call corresponding rational FPSs matrix-unitary on TN . Theorem 5.18. Let f be a rational FPS and α be its minimal GR-realization of the form (3.11). Then f is matrix-unitary on TN if and only if: (a) There exists an Hermitian matrix H = diag(H1 , . . . , HN ) (with Hk ∈ Cγk ×γk , k = 1, . . . , N ) such that ∗ A B H 0 H 0 A B = . (5.25) C D 0 Iq C D 0 Iq Condition (a) is equivalent to: (a ) There exists an Hermitian matrix G = diag (G1 , . . . , GN ) (with Gk ∈ γk ×γk C , k = 1, . . . , N ) such that ∗ G 0 A B G 0 A B = . (5.26) 0 Iq C D 0 Iq C D Proof. The necessity follows from Theorem 5.1. To prove the suﬃciency, suppose that the Hermitian matrix H = diag(H1 , . . . , HN ) satisﬁes (5.25) and let a ∈ T be such that −a ∈ σ(A). Then, H satisﬁes conditions (4.18) and (4.19) for the GR(N node β = (N ; Aa , Ba , Ca , Da ; Cγ = k=1 Cγk , Cq ) deﬁned by (5.2) (this follows

86

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

from the proof of Theorem 5.1). Thus, from Theorem 4.13 and Theorem 5.1 we obtain that f is matrix-unitary on TN . Analogously, condition (a ) implies that the FPS f is matrix-unitary on TN . A counterpart of Theorem 4.14 in the present case is the following theorem: Theorem 5.19. Let (C, A) be an observable pair of matrices in the sense that Ok has full column rank for each k = 1, . . . , N . Assume that A ∈ Cr×r is invertible. Then there exists a matrix-unitary on TN rational FPS f with a minimal GR-realization (N α = (N ; A, B, C, D; Cr = k=1 Crk , Cq ) if and only if the Stein equation H − A∗ HA = C ∗ C

(5.27) rk ×rk

, k = has an Hermitian solution H = diag(H1 , . . . , HN ), with Hk ∈ C 1, . . . , N . If such a matrix H exists, it is invertible, and possible choices of D and B are Da and Ba given by (5.16) and (5.17) with J = Iq . Finally, for a given H = diag(H1 , . . . , HN ), all other choices of D and B diﬀer from Da and Ba by a right multiplicative unitary constant. A counterpart of Theorem 4.15 is the following theorem: Theorem 5.20. Let (A, B) be a controllable pair of matrices, in the sense that Ck has full row rank for each k = 1, . . . , N . Assume that A ∈ Cr×r is invertible. Then there exists a matrix-unitary on TN rational FPS f with a minimal GR-realization (N α = (N ; A, B, C, D; Cr = k=1 Crk , Cq ) if and only if the Stein equation G − AGA∗ = BB ∗

(5.28) rk ×rk

has an Hermitian solution G = diag(G1 , . . . , GN ) with Gk ∈ G , k = 1, . . . , N . If such a matrix G exists, it is invertible, and possible choices of D and C are Da and Ca given by (5.18) and (5.19) with H = G−1 and J = Iq . Finally, for a given G = diag(G1 , . . . , GN ), all other choices of D and C diﬀer from Da and Ca by a left multiplicative unitary constant. A counterpart of Theorem 4.16 in the present case is the following: Theorem 5.21. Let f be a matrix-unitary on TN rational FPS and α be its minimal GR-realization of the form (3.11), with the associated structured Hermitian matrix H = diag(H1 , . . . , HN ) and the associated kth inner products [·, ·]Hk , k = 1, . . . , N . Let Pk denote the orthogonal projection in Cγ onto the subspace {0} ⊕ · · · ⊕ {0} ⊕ γ Cγk ⊕ {0} ⊕ · · · ⊕ {0}, and set Ak = AP Pk for k = 1, . . . , N . If x ∈ C is a common eigenvector for A = A1 , . . . , AN corresponding to a common eigenvalue λ = (λ1 , . . . , λN ) ∈ CN , then there exists j ∈ {1, . . . , N } such that |λj | = 1 and [P Pj x, Pj x]Hj = 0. In particular A has no common eigenvalues on TN . The proof of this theorem relies on the equality (1 − |λk |2 )[P Pk x, Pk x]Hk = CP Pk x, CP Pk x,

k = 1, . . . , N,

and follows the same argument as the proof of Theorem 4.16.

Matrix-J-unitary Rational Formal Power Series

87

6. Matrix-J-inner rational formal power series 6.1. A multivariable non-commutative analogue of the half-plane case Let n ∈ N. We deﬁne the matrix open right poly-half-plane as the set 7 N n×n N 6 = Z = (Z1 , . . . , ZN ) ∈ Cn×n : Zk + Zk∗ > 0, k = 1, . . . , N , Π and the matrix closed right poly-half-plane as the set N N = clos Πn×n clos Πn×n 6 7 N = Z = (Z1 , . . . , ZN ) ∈ Cn×n : Zk + Zk∗ ≥ 0, k = 1, . . . , N . We also introduce PN =

0 N Πn×n

and clos PN =

n∈N

It is clear that

0

N clos Πn×n .

n∈N

n×n N N iH ⊂ clos Πn×n N

is the essential (or Shilov ) boundary of the matrix poly-half-plane (Πn×n ) (see 1 N [45]) and that JN ⊂ clos PN (recall that JN = n∈N (iHn×n ) ). Let J = J −1 = J ∗ ∈ Cq×q . A matrix-J-unitary on JN rational FPS F is called matrix-J-inner (in PN ) if for each n ∈ N: F (Z)(J ⊗ In )F (Z)∗ ≤ J ⊗ In

(6.1)

N

at those points Z ∈ clos (Πn×n ) where it is deﬁned (the set of such points is N open and dense, in the relative topology, in clos (Πn×n ) since F (Z) is a rational matrix-valued function of the complex variables (Zk )ij , k = 1, . . . , N, i, j = 1, . . . , n). The following theorem is a counterpart of part a) of Theorem 2.16 of [7]. Theorem 6.1. Let F be a matrix-J-unitary on JN rational FPS and α be its minimal GR-realization of the form (3.11). Then F is matrix-J-inner in PN if and only if the associated structured Hermitian matrix H = diag(H1 , . . . , HN ) is strictly positive. Proof. Let n ∈ N. Equality (4.9) can be rewritten as ∗

∗

J ⊗ In − F (Z)(J ⊗ In )F (Z ) = ϕ(Z)∆(Z + Z )(H −1 ⊗ In )ϕ(Z ) where ϕ is a FPS deﬁned by ϕ(z) := C(IIγ − ∆(z)A)−1 ∈ Cq×γ z1 , . . . , zN rat , and (6.2) is well deﬁned at all points Z, Z ∈ (Cn×n )N for which 1 ∈ σ (∆(Z)(A ⊗ In )) ,

1 ∈ σ (∆(Z )(A ⊗ In )) .

∗

(6.2)

88

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

Set ϕk (z) := C(IIγ − ∆(z)A)−1 -Cγk ∈ Cq×γk z1 , . . . , zN rat , k = 1, . . . , N. Then (6.2) becomes: ∗

J ⊗ In − F (Z)(J ⊗ In )F (Z ) =

N

∗

∗

ϕk (Z)(Hk−1 ⊗ (Zk + Zk ))ϕk (Z ) .

(6.3)

k=1

Let X ∈ Hn×n be some positive semideﬁnite matrix, let Y ∈ (Hn×n )N be such that 1 ∈ σ(∆(iY )(A ⊗ In )), and set for k = 1, . . . , N : ek := (0, 0, . . . , 0, 1, 0, . . . , 0) ∈ CN with 1 at the kth place. Then for λ ∈ C set (k)

Y1 , . . . , iY Yk−1 , λX + iY Yk , iY Yk+1 , . . . , iY YN ). ZX,Y (λ) := λX ⊗ ek + iY = (iY Now, (6.3) implies that ∗

ZX,Y (λ))(J ⊗ In )F (Z ZX,Y (λ )) J ⊗ In − F (Z (k)

(k)

∗

= (λ + λ )ϕk (Z ZX,Y (λ))(Hk−1 ⊗ X)ϕk (Z ZX,Y (λ )) . (k)

(k)

(6.4)

(k)

The function h(λ) = F (Z ZX,Y (λ)) is a rational function of λ ∈ C. It is easily seen from (6.4) that h is (J ⊗ In )-inner in the open right half-plane. In particular, it is (J ⊗ In )-contractive in the closed right half-plane (this also follows directly from (6.1)). Therefore (see, e.g., [22]) the function Ψ(λ, λ ) =

∗

ZX,Y (λ))(J ⊗ In )F (Z ZX,Y )(λ ) J ⊗ In − F (Z (k)

(k)

(6.5) λ + λ is a positive semideﬁnite kernel on C: for every choice of r ∈ N, of points λ1 , . . . , λr ∈ C for which the matrices Ψ(λj , λi ) are well deﬁned, and vectors c1 , . . . , cr ∈ Cq ⊗ Cn one has r

c∗j Ψ(λj , λi )ci ≥ 0,

i,j=1 (k)

ZX,Y (0)) = i.e., the matrix (Ψ(λj , λi ))i,j=1,...,r is positive semideﬁnite. Since ϕk (Z ϕk (iY ) is well deﬁned, we obtain from (6.4) that Ψ(0, 0) is also well deﬁned and Ψ(0, 0) = ϕk (iY )(Hk−1 ⊗ X)ϕk (iY )∗ ≥ 0. This inequality holds for every n ∈ N, every positive semideﬁnite X ∈ Hn×n and every Y ∈ (Hn×n )N . Thus, for an arbitrary r ∈ N we can deﬁne n

= nr, Y = (1) (r) (j) n

× n N (Y 1 , . . . , Y N ) ∈ (H ) , where Y k = diag(Y Yk , . . . , Yk ) and Yk ∈ Hn×n , k = 1, . . . , N, j = 1, . . . , r, such that ϕk (iY ) is well deﬁned, ⎛ ⎞ In · · · In . .. ⎟ ∈ Cn×n ⊗ Cr×r ∼ Cn × n

=⎜ X = ⎝ .. .⎠ In

···

In

Matrix-J-unitary Rational Formal Power Series

89

and get

k (iY )∗ 0 ≤ ϕk (iY )(Hk−1 ⊗ X)ϕ = diag(ϕk (iY (1) ), . . . , ϕk (iY (r) ))× ⎛ ⎞ ⎛ ⎞ In ⎟ ⎜ ⎜ ⎟ × ⎝Hk−1 ⊗ ⎝ ... ⎠ In · · · In ⎠ diag(ϕk (iY (1) )∗ , . . . , ϕk (iY (r) )∗ ) In ⎛ ⎞ ϕk (iY (1) ) ⎜ ⎟ −1 .. =⎝ ⎠ (Hk ⊗ In ) ϕk (iY (1) )∗ . ϕk (iY (r) ) = ϕk (iY (µ) )(Hk−1 ⊗ In )ϕk (iY (ν) )∗

ϕk (iY (r) )∗

···

.

µ,ν=1,...,r

Therefore, the function Kk (iY, iY ) = ϕk (iY )(Hk−1 ⊗ In )ϕk (iY )∗ is a positive semideﬁnite kernel on any subset of (iHn×n )N where it is deﬁned, and in particular in some neighborhood of the origin. One can extend this function to Kk (Z, Z ) = ϕk (Z)(Hk−1 ⊗ In )ϕk (Z )∗

(6.6)

at those points Z, Z ∈ (C ) × (C ) where ϕk is deﬁned. Thus, on some neighborhood Γ of the origin in (Cn×n )N × (Cn×n )N , the function Kk (Z, Z ) is holomorphic in Z and anti-holomorphic in Z . On the other hand, it is well known (see, e.g., [9]) that one can construct a reproducing kernel Hilbert space (which we will denote by H(Kk )) with reproducing kernel Kk (iY, iY ), which is obtained as the completion of H0 = span Kk (·, iY )x ; iY ∈ (iHn×n )N ∩ Γ, x ∈ Cq ⊗ Cn n×n N

n×n N

with respect to the inner product 8 r 9 (µ) (ν) Kk (·, iY )xµ , Kk (·, iY )xν µ=1

=

ν=0 r :

Kk (iY (ν) , iY (µ) )xµ , xν

µ=1 ν=1

H0

; Cq ⊗Cn

.

The reproducing kernel property reads: f (·), Kk (·, iY )xH(Kk ) = f (iY ), xCq ⊗Cn , ∗

and thus Kk (iY, iY ) = Φ(iY )Φ(iY ) where Φ(iY ) : f (·) → f (iY ) is the evaluation map. In view of (6.6), the kernel Kk (·, ·) is extendable on Γ × Γ to the function K(Z, Z ) which is holomorphic in Z and antiholomorphic in Z ,

90

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

all the elements of H(Kk ) have holomorphic continuations to Γ, and so has the function Φ(·). Thus, Kk (Z, Z ) = Φ(Z)Φ(Z )∗ and so Kk (Z, Z ) is a positive semideﬁnite kernel on Γ. (We could also use [3, Theorem 1.1.4, p.10] to obtain this conclusion.) Therefore, for any choice of ∈ N and Z (1) , . . . , Z () ∈ Γ the matrix ϕk (Z (µ) )(Hk−1 ⊗ In )ϕk (Z (ν) )∗ µ,ν=1,..., ⎞ ⎛ ϕk (Z (1) ) (6.7) ⎟ ⎜ .. −1 (1) ∗ () ∗ =⎝ ⊗ I ) · · (H ϕ (Z ) · · · ϕ (Z ) ⎠ n k k k . ϕk (Z () ) is positive semideﬁnite. Since the coeﬃcients of the FPS ϕk are (ϕk )w = (CA)wgk , w ∈ FN , and since α is an observable GR-node, we have ) ker(CA)wgk = {0} . w∈F FN

Hence, by Theorem 2.1 we can chose n, ∈ N and Z (1) , . . . , Z () ∈ Γ such that )

ker ϕk (Z (j) ) = {0} .

j=1

Thus the matrix colj=1,..., ϕk (Z (j) ) has full column rank. (We could also use Theorem 3.7.) From (6.7) it then follows that Hk−1 > 0. Since this holds for all k ∈ {1, . . . , N }, we get H > 0. Conversely, if H > 0 then it follows from (6.2) that for every n ∈ N and N Z ∈ (Πn×n ) for which 1 ∈ σ(∆(Z)(A ⊗ In )), one has J ⊗ In − F (Z)(J ⊗ In )F (Z)∗ ≥ 0. Therefore F is matrix-J-inner in PN , and the proof is complete.

Theorem 6.2. Let F ∈ C z1 , . . . , zN rat be matrix-J-inner in PN . Then F has a minimal GR-realization of the form (3.11) with the associated structured Hermitian matrix H = Iγ . This realization is unique up to a unitary similarity. q×q

Proof. Let α◦ = (N ; A◦ , B ◦ , C ◦ , D; Cγ =

N '

Cγ k , Cq )

k=1

be a minimal GR-realization of F , with the associated structured Hermitian ma◦ trix H ◦ = diag(H1◦ , . . . , HN ). By Theorem 6.1 the matrix H ◦ is strictly positive. ◦ 1/2 ◦ 1/2 Therefore, (H ) = diag((H1◦ )1/2 , . . . , (H HN ) ) is well deﬁned and strictly positive, and N ' Cγk , Cq ), α = (N ; A, B, C, D; Cγ = k=1

Matrix-J-unitary Rational Formal Power Series

91

where A = (H ◦ )1/2 A◦ (H ◦ )−1/2 ,

B = (H ◦ )1/2 B ◦ ,

C = C ◦ (H ◦ )−1/2 ,

(6.8)

is a minimal GR-realization of F satisfying A∗ + A =

−C ∗ JC,

(6.9)

=

∗

−C JD,

(6.10)

A∗ + A = C =

−BJB ∗ , −DJB ∗ ,

(6.11) (6.12)

B or equivalently,

and thus having the associated structured Hermitian matrix H = Iγ . Since in this case the inner product [ · , · ]H coincides with the standard inner product · , · of Cγ , by Remark 4.6 this minimal GR-realization with the property H = Iγ is unique up to unitary similarity. We remark that a one-variable counterpart of the latter result is essentially contained in [20], [38] (see also [10, Section 4.2]). 6.2. A multivariable non-commutative analogue of the disk case Let n ∈ N. We deﬁne the matrix open unit polydisk as 7 N n×n N 6 D = W = (W W1 , . . . , WN ) ∈ Cn×n : Wk Wk∗ < In , k = 1, . . . , N , and the matrix closed unit polydisk as N N clos Dn×n = clos Dn×n 6 7 N = W = (W W1 , . . . , WN ) ∈ Cn×n : Wk Wk∗ ≤ In , k = 1, . . . , N . N

The matrix unit torus (Tn×n ) is the essential (or Shilov) boundary of (Dn×n ) (see [45]). In our setting, the set 0 0 n×n N n×n N DN = D resp., clos DN = clos D n∈N

N

n∈N

is a multivariable non-commutative counterpart of the open (resp., closed) unit disk. Let J = J −1 = J ∗ ∈ Cq×q . A rational FPS f which is matrix-J-unitary on TN is called matrix-J-inner in DN if for every n ∈ N: f (W )(J ⊗ In )f (W )∗ ≤ J ⊗ In N

(6.13)

at those points W ∈ clos (Dn×n ) where it is deﬁned. We note that the set of N such points is open and dense (in the relative topology) in clos (Dn×n ) since f (W ) is a rational matrix-valued function of the complex variables (W Wk )ij , k = 1, . . . , N, i, j = 1, . . . , n.

92

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

Theorem 6.3. Let f be a rational FPS which is matrix-J-unitary on TN , and let α be its minimal GR-realization of the form (3.11). Then f is matrix-J-inner in DN if and only if the associated structured Hermitian matrix H = diag(H1 , . . . , HN ) is strictly positive. Proof. The statement of this theorem follows from Theorem 6.1 and Theorem 5.1, since the Cayley transform deﬁned in Theorem 5.1 maps each open matrix unit polydisk (Dn×n )N onto the open right matrix poly-half-plane (Πn×n )N , and the inequality (6.13) turns into (6.1) for the function F deﬁned in (5.1). The following theorem is an analogue of Theorem 6.2. Theorem 6.4. Let f be a rational FPS which is matrix-J-inner in DN . Then there exists its minimal GR-realization α of the form (3.11), with the associated structured Hermitian matrix H = Iγ . Such a realization is unique up to a unitary similarity. In the special case of Theorem 6.4 where J = Iq the FPS f is called matrixinner, and the GR-node α satisﬁes ∗ A B A B = Iγ +q , C D C D i.e., α is a unitary GR-node, which has been considered ﬁrst by J. Agler in [1]. In what follows we will show that Theorem 6.4 for J = Iq is a special case of the theorem of J. A. Ball, G. Groenewald and T. Malakorn on unitary GR-realizations of FPSs from the non-commutative Schur–Agler class [12], which becomes in several aspects stronger in this special case. Let U and Y be Hilbert spaces. Denote by L(U, Y) the Banach space of bounded linear operators from U into Y. A GR-node in the general setting of Hilbert spaces is α = (N ; A, B, C, D; X =

N '

Xk , U, Y),

k=1

i.e., a collection of Hilbert spaces X , X1 , . . . , XN , U, Y and operators A ∈ L(X ) = L(X , X ), B ∈ L(U, X ), C ∈ L(X , Y), and D ∈ L(U, Y). Such a GR-node α is called unitary if ∗ ∗ A B A B A B A B = IX ⊕Y , = IX ⊕U , C D C D C D C D

A B i.e., C is a unitary operator from X ⊕ U onto X ⊕ Y. The non-commutative D transfer function of α is

Tαnc (z) = D + C(I − ∆(z)A)−1 ∆(z)B,

(6.14)

Matrix-J-unitary Rational Formal Power Series

93

where the expression (6.14) is understood as a FPS from L(U, Y) z1 , . . . , zN given by ∞ w k (CAB) z w = D + C (∆(z)A) ∆(z)B. (6.15) Tαnc (z) = D + w∈F FN \{∅}

k=0

The non-commutative Schur–Agler class SAnc N (U, Y) consists of all FPSs f ∈ L(U, Y) z1 , . . . , zN such that for any separable Hilbert space K and any N tuple δ = (δ1 , . . . , δN ) of strict contractions in K the limit in the operator norm topology fw ⊗ δ w f (δ) = lim m→∞

w∈F FN : |w|≤m

exists and deﬁnes a contractive operator f (δ) ∈ L(U ⊗ K, Y ⊗ K). We note that the non-commutative Schur–Agler class was deﬁned in [12] also for a more general class of operator N -tuples δ. ). Consider another set of non-commuting indeterminates z = (z1 , . . . , zN For f (z) ∈ L(V, Y) z1 , . . . , zN and f (z ) ∈ L(V, U) z1 , . . . , zN we deﬁne a FPS ∗ f (z)f (z ) ∈ L(U, Y) z1 , . . . , zN , z1 , . . . , zN by w T ∗ ∗ f (z)f (z ) = fw (ffw ) z w z . (6.16) w,w ∈F FN

In [12] the class

SAnc N (U, Y)

was characterized as follows:

Theorem 6.5. Let f ∈ L(U, Y) z1 , . . . , zN . The following statements are equivalent: (1) f ∈ SAnc N (U, Y); (2) there exist auxiliary Hilbert spaces H, H1 , . . . , HN which are related by H = (N k=1 Hk , and a FPS ϕ ∈ L(H, Y) z1 , . . . , zN such that ∗

∗

IY − f (z)f (z ) = ϕ(z)(IIH − ∆(z)∆(z )∗ )ϕ(z ) ; (6.17) (N (3) there exists a unitary GR-node α = (N ; A, B, C, D; X = k=1 Xk , U, Y) such that f = Tαnc. We now give another characterization of the Schur–Agler class SAnc N (U, Y). Theorem 6.6. A FPS f belongs to SAnc N (U, Y) if and only if for every n ∈ N and W ∈ (Dn×n )N the limit in the operator norm topology f (W ) = lim fw ⊗ W w (6.18) m→∞

w∈F FN : |w|≤m

exists and f (W ) ≤ 1. Proof. The necessity is clear. We prove the suﬃciency. We set fk (z) = fw z w , k = 0, 1, . . . . w∈F FN : |w|=k

94

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

Then for every n ∈ N and W ∈ (Dn×n )N , (6.18) becomes f (W ) = lim

m→∞

m

fk (W ),

(6.19)

k=0

where the limit is taken in the operator norm topology. Let r ∈ (0, 1) and choose τ > 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 W x λW x = λ fk 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

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)

,

(R+ ) is called the potential. Such systems were introduced by and where k ∈ Ln×n 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 . (1.4) J= and κ(x) = 0 −1 2Z(x) We distinguish two cases: (a) The case where Z(x) > 0 and is continuously diﬀerentiable on R+ . Taking the (inverse) Fourier transform f → f(λ) = R eiλt f (t)dt on both sides we get 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 ∞ eiλt u(t)dt (1.5) f (λ) = D + −∞

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 (resp. W− ) consists of the functions of the dence on R. The sub-algebra W+ 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 f (z) = Z

for which def.

f W =

|ff | < ∞.

Z

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 p and ∪m ∩m =0 ker ca = {0} =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 ia 0 0 f1∗ −c 0 , T = , G = F =i , (2.5) c∗ 0 0 f1 0 −ia∗ 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 In 0 e (2.6) lim U (x, λ) = I2n , λ ∈ R. 0 eixλ In x→∞ 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)

= In + icΩ(λIIp − a∗ )−1 c∗ ,

U21 (0, λ)

= (−b∗ + icΩ)(λIIp − a∗ )−1 c∗ ,

U12 (0, λ)

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

U22 (0, λ)

= 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 0 Ω 0 Ip H= . −i(IIp + Y Ω)Ω−1 Ip −i(IIp + Y Ω)Ω−1 Ip 0 −Ω−1 − Y 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 0 A22 − A21 A−1 I 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 ,

λ ∈ R,

is called the reﬂection coeﬃcient function. 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 B(IIn − D∗ D)−1 B ∗ A + BD∗ (IIn − DD∗ )−1 C 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=

icΩ − b∗ ). −Ω −iIIp

iIIp −Y (IIp + ΩY )−1

.

Then it holds that 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 −Ω 0 Ip 0 0 X= 0 (Ω + ΩY Ω)−1 iΩ−1 Ip 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 V22 (λ) − V12 (λ))−1 W (λ) = (V V22 (λ) − V12 (λ))−∗ (V 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 Θ(x, λ) Θ(x, λ) dx 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)−1 S− (λ)−∗ = S+ (λ)S+ (λ)∗ , = Im N (λ),

S(λ)

= S− (λ)S+ (λ),

R(λ)

= (iN (λ)∗ − In )(iN (λ)∗ + In )−1 ,

N (λ)

= 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

V (λ) 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)

14

D. Alpay and I. Gohberg

Finally I2n − Θ× Θ×

∗

∗

I2n − Θ× Θ×

=

(P − ΘQ)−1 (J − ΘJΘ∗ ) (P − ΘQ)−∗

(2.17)

=

(P + QΘ)−∗ (J − Θ∗ JΘ) (P + QΘ)−1 .

(2.18)

where A ∈ Cn×n . We have: In 0 In P + QΘ = , P − ΘQ = C D 0

Proof. We set Θ =

A C

B C

−B −D

.

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 (2.20) W (λ) = In − eitλ ω(t)dt, λ ∈ R, 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 Γ2t (u, 0)e−iλu du . P (t, λ) = eitλ In + 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 ).

16

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 × PB (2.22) k(x) = 2C P e−2ixA |Im P 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

→ −N −1 (λ),

λ λ

→ −N −1 (−λ∗ )∗ , → N (−λ∗ )∗ ,

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

= =

Im − N (λ)−1 , Im − N (−λ∗ )−∗ ,

W2 (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 c∗

B− = B =

,

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=

a − bc 2 0

0 ∗ (a − bc 2)

a−

bc 2 c c 2 ∗

(a

bb∗ 2 ∗ − bc 2)

,

Z=

0

∗

cc 2

we have A× = D − Z

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 , it holds that P A× |Im P = P (A− − B− C− ) |Im P , i.e., that I 0 I 0 I 0 I 0 = (D − Z) (D + Z) 0 0 0 0 0 0 0 0 for all positive integers . Let = ±1. The expression (D + Z) consists of a sum of terms of the form Dα1 ( Z)β1 Dα2 ( Z)β2 · · · , where the αi and the βi are equal to 1 or 0 and i (αi + βi ) = . Each factor diagonal. We consider two cases, namely Dαi Z βi for which βi = 0 is anti block 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 and have the same value at = 1 and at = −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

−1 P1

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

P1 B1 .

−1 P1

P1∗ C1∗ .

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

= 2CP = c

0 ,

P1 C1∗

1 = −P B = − 2

b 0

,

19

× A∗× 1 = −A ,

which allows to conclude.

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 ∂ ∂ ) 0 ( ∂x + ∂t W (x, t) = 0. ∂ ∂ 0 ( ∂x − ∂t ) 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).

Hence one can write

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.

Indeed, the space C0 is A invariant and H-negative. Furthermore, ∗ Ip −Ω 0 Ip 0 0 −Ω −IIp , = −IIp −a∆a∗ 0 Ω−1 − a∆a∗ Ω−1 Ip Ω−1 Ip p

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 − |ρ | ) H0 (z ∗ )∗ 0 =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

=0

√1

1−|ρ |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 H0 (z) (recall on the unit circle. From the J-unitarity on the unit circle of √ 1 det H0 (z)

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

|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}

(I − ∆Ω)(zI − a)

× (zI − a)−1 (I − ∆Ω)−1 b, 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 (1/z) = can (I − ∆a∗(n+1) Ωan+1 ) − z(I − ∆a∗n Ωan )a b Rn (z) = 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 − |ρ |2 ) Hn (z)−1 0 z 0 =0

the form 1 0 H0 (z) 0 1

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) 0 z 0 1 Yn (z) = (1 − |ρ |2 ) 0 z Hn (z)−1 H0 (z) x(z) . 0 1 z =0

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

n→∞

=0

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

30

D. Alpay and I. Gohberg

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

n→∞

1 0 1 0 x(z) 0 Yn (z)z −n = 1 0 H0 (z) 0 z −1 x(z) 0 z ∞ α0 (z) + β0 (z) 2 z (1 − |ρ | ) 1 0 x(z) = γ0 (z) + δ0z(z) =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

=n−1 1 1 −ρ∗ −1 =0

−ρ 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 22 . 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 = =0 (1 − |ρ |2 ), we have that: cn−1 1 1 iN (z ∗ )∗ 1 0 Un (z) = Hn (z)−1 1 0 z 1 −1 2 n z 0 1 0 1 + iN (z ∗ )∗ × H0 (z) 0 z −1 0 1 −1 + iN (z ∗ )∗ n cn−1 1 1 0 1 0 z = Hn (z)−1 0 1 0 z 1 −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 22 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) =

zβ0 (1/z) − α0 (1/z) zγ0 (z ∗ )∗ − δ0 (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

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

b,

and hence the result.

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 − |ρ |2 ) =0

c = ∞

|z| = 1.

(3.31)

will play an important role in the sequel. 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 1 c , z ∈ T, c = ∞ W (z) = , |S− (1/z)|2 (1 − |ρ |2 ) =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 w eit such that Z |w | < ∞ (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 ⎧ ⎨ CA−1 (I − P )B if = 1, 2, . . . w = D − CP B if = 0 ⎩ −CA−1 P B if = −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 ∞ z (A(I − P )) )B = D + zC( =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

× = w−j + K0j

(m)

= −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 (m)

(m)

(m)

z m pm (1/z) =

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

m

⎞ z j Ω×j ⎠ P × B ⎞

z j Ω×−j ⎠ (I − P × )B

j=0

from which the claim follows. 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)∗ Mn (w)∗ ∗ J − Mn (z)JM =M M , ∗ 1 − zw 1 − zw∗ 2 1 − zw∗ R(z)R(w)∗ 1 N (z) − N (w)∗ = , ∗ ∗ ∗ 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 , k < n % & 1 1 = 2δ,k . Θ , Θk 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 < k. The function k z 0 −1 (Θ Θk )(z) = U U C(ρi ) 0 1 i=+1

Analogs of Canonical Systems with Pseudo-exponential Potential belongs to H2,J and is such that (Θ−1 Θk )(0) Thus,

% Θ

& 1 1 , Θk 1 1 H(Θ

= n)

39

1 0 . = 1 0

% & 1 1 , Θ−1 Θ k 1 1 H(Θ

=0 n)

The proof that the inner product is equal to 2 when = 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 − |ρ | ) Hn (z ) H0 (z ) Y0 (z). Yn (z) = 0 z2 0 z12 0 z −n =0

The solution such that

−n z lim 0 n→∞

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

0 Yn (z) = I2 zn 1 0 2 −1 1 H0 (z ) 0 0 z2

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 = H (z ) . Hn+1 (z 2 ) 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

1 − |ρ |

2

=0

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

1 0

0 2 −1 1 (z ) H 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 Yn(1) (z)c1 − Yn(2) (z)c2 = 0 lim Y

n→∞

Proof. By deﬁnition,

Yn(2) (z) =

n−1

(1) Yn (z)

(1 − |ρ |2 )

=0

n z = 0

1 0

0

z −n

c2 = V (z)c1 .

⇐⇒

c1 . On the other hand,

n 0 2 −1 z (z ) H 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 b. (4.2) R(z) = c (I − ∆a∗ Ωa) − z 2 (I − ∆Ω)a 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 ∆=

1 , 1 − α2

Ω=

and ρn = −αn

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 < < 1, |ρ0 | = c2 1 − α2 2 1 − α (1−α2 )2 and more generally, |ρn | =

αn c 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.

42

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

The system Special solutions

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

Potential

Solution asymptotic to the solution with k ≡ 0

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 =

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

c (1−α2 ) ,

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.

Analogs of Canonical Systems with Pseudo-exponential Potential

45

[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 of Operator Theory: Advances and Applications. Birkhauser ¨ Verlag, Basel, 1996. [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.

46

D. Alpay and I. Gohberg

[29] I. Gohberg, P. Lancaster, and L. Rodman. Matrices and indeﬁnite scalar products, volume 8 of Operator Theory: Advances and Applications. Birkhauser ¨ Verlag, Basel, 1983. [30] I. Gohberg, P. Lancaster, and L. Rodman. Invariant subspaces of matrices with applications. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons Inc., New York, 1986. A Wiley-Interscience Publication. [31] I. Gohberg and Ju. Leiterer. General theorems on the factorization of operatorvalued functions with respect to a contour. I. Holomorphic functions. Acta Sci. Math. (Szeged), 34:103–120, 1973. [32] I. Gohberg and Ju. Leiterer. General theorems on the factorization of operator-valued functions with respect to a contour. II. Generalizations. Acta Sci. Math. (Szeged), 35:39–59, 1973. [33] I. Gohberg and S. Rubinstein. Proper contractions and their unitary minimal completions. In I. Gohberg, editor, Topics in interpolation theory of rational matrix-valued functions, volume 33 of Operator Theory: Advances and Applications, pages 223–247. Birkhauser ¨ Verlag, Basel, 1988. [34] I.C. Gohberg and I.A. Fel dman. Convolution equations and projection methods for their solution. American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by F.M. Goldware, Translations of Mathematical Monographs, Vol. 41. [35] G.J. Groenewald. Toeplitz operators with rational symbols and realizations: an alternative version. Technical Report WS:–362, Vrije Universiteit Amsterdam, 1990. [36] A. Iacob. On the spectral theory of a class of canonical systems of diﬀerential equations. PhD thesis, The Weizmann Institute of Sciences, 1986. [37] M.G. Kre˘n. ˘ Continuous analogues of propositions for polynomials orthogonal on the unit circle. Dokl. Akad. Nauk. SSSR, 105:637–640, 1955. [38] M.G. Kre˘n. Topics in diﬀerential and integral equations and operator theory, volume 7 of Operator theory: Advances and Applications. Birkhauser ¨ Verlag, Basel, 1983. Edited by I. Gohberg, Translated from the Russian by A. Iacob. ¨ [39] M.G. Kre˘n ˘ and H. Langer. Uber die verallgemeinerten Resolventen und die charakteristische Funktion eines isometrischen Operators im Raume Πk . In Hilbert space operators and operator algebras (Proc. Int. Conf. Tihany, 1970), pages 353–399. North-Holland, Amsterdam, 1972. Colloquia Math. Soc. J´ a ´nos Bolyai. [40] L.Golinskii and P. Nevai. Szeg˝ ˝ o diﬀerence equations, transfer matrices and orthogonal polynomials on the unit circle. Comm. Math. Phys., 223(2):223–259, 2001. [41] F.E. Melik-Adamyan. On a class of canonical diﬀerential operators. Izvestya Akademii Nauk. Armyanskoi SSR Matematica, 24:570–592, 1989. English translation in: Soviet Journal of Contemporary Mathematics, vol. 24, pages 48–69 (1989). [42] L. Sakhnovich. Dual discrete canonical systems and dual orthogonal polynomials. In D. Alpay, I. Gohberg, and V. Vinnikov, editors, Interpolation theory, systems theory and related topics (Tel Aviv/Rehovot, 1999), volume 134 of Oper. Theory Adv. Appl., pages 385–401. Birkh¨ a ¨user, Basel, 2002. ¨ [43] I. Schur. Uber die Potenzreihen, die im Innern des Einheitkreises beschr¨ ¨ ankten sind, I. Journal f¨ fur die Reine und Angewandte Mathematik, 147:205–232, 1917. English

Analogs of Canonical Systems with Pseudo-exponential Potential

47

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: dany@math.bgu.ac.il 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: gohberg@post.tau.ac.il

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

52

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

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]), F (z) = D + C(IIγ − zA)−1 zB, where

A C

B D

K(F ) K(F ) : → Cq Cq

is deﬁned by 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.

Matrix-J-unitary Rational Formal Power Series

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

54

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

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 is the Cp×q denotes the set of p × q matrices with complex entries, and (Cr×s ) space of p × q block matrices with block entries in Cr×s . The tensor product A ⊗ B p×q with (i, j)th of matrices A ∈ Cr×s and B ∈ Cp×q is the element of (Cr×s ) 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 ⊗ Eτσ ,

Matrix-J-unitary Rational Formal Power Series

55

where the matrices Eµν ∈ Cr×s and Eτσ ∈ 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, ) = (τ, σ) , τ, k = 1, . . . , p and σ, = 1, . . . q. (Eτ σ )k = 0 if (k, ) = (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 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) T

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

56

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

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, 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 p×q

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 = fwwT ,

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 2 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

of a linear which can be interpreted as a non-commutative transfer function 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 '

Xj , U, Y),

j=1

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 , . . .)

58

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

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 , (CA)gν w = Cν Aνj1 Aj1 j2 · · · Ajk−1 jk , (AB)wgν = Aj1 j2 · · · Ajk−1 jk Ajk ν Bν , (CAB)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 (CA)gν = Cν , (AB)gν = Bν , (CAB)gν = Cν Bν , (CAB)gµ gν = Cµ Aµν Bν , and hence, with the lexicographic order of words in FN , wgk Ok = colw∈F FN (CA)

T

gk w and Ck = roww∈F , FN (AB)

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

(T Tαnc )w = (CAB)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 wT

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 = (CAB)wgk w

T

T

= (CA)wgk (AB)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 2 , 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) =

fw ⊗ Z w .

w∈F FN

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|<pr (CA)wgk , O

*k = row|w| 0 is arbitrary in the case A = 0), and Γn (ε) is deﬁned by (2.9). This GR-node is minimal if both of conditions (3.3) and (3.4) are fulﬁlled. rk

Proof. First, let us remark that for all k = 1, . . . , N the functions ϕk and ψk are well deﬁned in Γn (ε), and holomorphic as functions of matrix entries (Z Zj )µν , j = 1, . . . , N, µ, ν = 1, . . . , n. Second, Theorem 3.1 implies that in Theorem 2.1 applied to ϕk one can choose m = pr−1, and then from (2.10) obtain that observability for a GR-node α is equivalent to condition (3.3). Since α is controllable if and only if α∗ is observable, controllability for α is equivalent to condition (3.4). Since minimality for a GR-node α is equivalent to controllability and observability together, it is in turn equivalent to conditions (3.3) and (3.4) together.

64

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

(N Let α = (N ; A , B , C , D ; Cr = j=1 Crj , Cs , Cp ) and α = (N ; A , B , (N rj q s C , D ; Cr = j=1 C , C , C ) be GR-nodes. For k, j = 1, . . . , N set rj = rj + rj , and Akj Bk Cj Bk D rk ×rj Akj = , Bk = ∈C ∈ Crk ×q , 0 A B (3.7) kj k p×rj p×q C D C Cj = j , D=DD ∈C . j ∈C (N r rj q p Then α = (N ; A, B, C, D; C = j=1 C , C , C ) will be called the product of GR-nodes α and α and denoted by α = α α . A straightforward calculation shows that Tαnc = Tαnc Tαnc . Consider a GR-node N N ' ' Crj , Cq ) := (N ; A, B, C, D; Cr = Cr j , Cq , Cq ) α = (N ; A, B, C, D; Cr = j=1

j=1

(3.8) with invertible operator D. Then α× = (N ; A× , B × , C × , D× ; Cr =

N '

Crj , Cq ),

j=1

with A× = A − BD−1 C,

B × = BD−1 ,

C × = −D−1 C,

D× = D−1 ,

(3.9)

×

will be called the associated GR-node, and A the associated main operator, of α. It is easy to see that, as well as in the one-variable case, (T Tαnc )−1 = Tαnc× . Moreover, × × (α× ) = α (in particular, (A× ) = A), and (α α )× = α× α× up to the natural rj rj identiﬁcation of C ⊕ C with Crj ⊕ Crj , j = 1, . . . , N , which is a similarity transform. Theorem 3.8. A GR-node (3.8) with invertible operator D is minimal if and only if its associated GR-node α× is minimal. Proof. Let a GR-node α of the form (3.8) with invertible operator D be minimal, and x ∈ ker Ok× for some k ∈ {1, . . . , N }, where Ok× is the kth observability matrix × . Then x ∈ ker(C × A× )wgk for every w ∈ FN . Let us show for the GR-node α/ wgk that x ∈ ker Ok = w∈F , i.e, x = 0. FN ker(CA) × For w = ∅, Ck x = 0 means −D−1 Ck x = 0 (see (3.9)), which is equivalent to Ck x = 0. For |w| > 0, w = gi1 · · · gi|w| , (CA)wgk

=

Ci1 Ai1 i2 · · · Ai|w| k

=

−1 −1 −DC Ci×1 (A× Ci2 ) · · · (A× Ck ) i1 i2 + Bi1 D i|w| k + Bi|w| D

=

L0 Ck× +

|w| j=1

× Lj Ci×j A× ij ij+1 · · · Ai|w| k ,

Matrix-J-unitary Rational Formal Power Series

65

with some matrices Lj ∈ Cq×q , j = 0, 1, . . . , |w|. Thus, x ∈ ker(CA)wgk for every w ∈ FN , i.e., x = 0, which means that α× is observable. Since α is controllable if and only if α∗ is observable (see Section 2), and ∗ D is invertible whenever D is invertible, the same is true for α× and (α× )∗ = (α∗ )× . Thus, the controllability of α× follows from the controllability of α. Finally, the minimality of α× follows from the minimality of α. Since (α× )× = α, the minimality of α follows from the minimality of α× . Suppose that for a GR-node (3.8), projections Πk on Crk are deﬁned such that Akj ker Πj ⊂ ker Πk ,

(A× )kj ran Πj ⊂ ran Πk ,

k, j = 1, . . . , N.

We do not assume that Πk are orthogonal. We shall call Πk a kth supporting projection for α. Clearly, the map Π = diag(Π1 , . . . , ΠN ) : Cr → Cr satisﬁes A ker Π ⊂ ker Π,

A× ran Π ⊂ ran Π,

i.e., it is a supporting projection for the one-variable node (1; A, B, C, D; Cr , Cq ) in the sense of [15]. If Π is a supporting projection for α, then Ir − Π is a supporting projection for α× . The following theorem and corollary are analogous to, and are proved in the same way as Theorem 1.1 and its corollary in [15, pp. 7–9] (see also [43, Theorem 2.1]). Theorem 3.9. Let (3.8) be a GR-node with invertible operator D. Let Πk be a projection on Crk , and let (11) (12) (1) Akjj Akjj Bj A= Bj = Ck = Ck(1) Ck(2) (21) (22) , (2) , Akj Akj Bj be the block matrix representations of the operators Akj , Bj and Ck with respect ˙ to the decompositions Crk = ker Πk +ran Πk , for k, j ∈ {1, . . . , N }. Assume that D = D D , where D and D are invertible operators on Cq , and set α = (N ; A(11) , B (1) (D )−1 , C (1) , D ; ker Π =

N '

ker Πk , Cq ),

k=1

α = (N ; A(22) , B (2) , (D )−1 C (2) , D ; ran Π =

N '

ran Πk , Cq ).

k=1

Then α = α α

(up to a similarity which maps C

rk

˙ = ker Πk +ran Πk onto ·

Cdim(ker Πk ) ⊕ Cdim(ranΠk ) (k = 1, . . . , N ) such that ker Πk + {0} is mapped onto ·

Cdim(ker Πk ) ⊕ {0} and {0} + ranΠk is mapped onto {0} ⊕ Cdim(ranΠk ) ) if and only if Π is a supporting projection for α. Corollary 3.10. In the assumptions of Theorem 3.9, Tαnc = F F ,

66

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

where F (z) = D + C(IIr − ∆(z)A)−1 (IIr − Π)∆(z)B(D )−1 , F (z) = D + (D )−1 CΠ(IIr − ∆(z)A)−1 ∆(z)B. We assume now that the external operator of the GR-node (3.8) is equal to D = Iq and that we also take D = D = Iq . Then, the GR-nodes α and α of Theorem 3.9 are called projections of α with respect to the supporting projections Ir − Π and Π, respectively, and we use the notations N ' ker Πk , Cq , α = prIr −Π (α) = N ; A(11) , B (1) , C (1) , D ; ker Π = k=1

α = prΠ (α) =

(22)

N; A

,B

(2)

,C

(2)

, D ; ran Π =

N '

ran Πk , C

q

.

k=1

Let F , F and F be rational FPSs with coeﬃcients in Cq×q such that F = F F .

(3.10)

The factorization (3.10) will be said to be minimal if whenever α and α are minimal GR-realizations of F and F , respectively, α α is a minimal GR-realization of F . In the sequel, we will use the notation N ' γ γk ×γk q α = N ; A, B, C, D; C = C ,C (3.11) k=1

for a minimal GR-realization (i.e., rk = γk for k = 1, . . . , N ) of a rational FPS F in the case when p = q. The following theorem is the multivariable non-commutative version of [15, Theorem 4.8]. It gives a complete description of all minimal factorizations in terms of supporting projections. Theorem 3.11. Let F be a rational FPS with a minimal GR-realization (3.11). Then the following statements hold: (i) if Π = diag(Π1 , . . . , ΠN ) is a supporting projection for α, then F is the transfer function of prIγ −Π (α), F is the transfer function of prΠ (α), and F = F F is a minimal factorization of F ; (ii) if F = F F is a minimal factorization of F , then there exists a uniquely deﬁned supporting projection Π = diag(Π1 , . . . , ΠN ) for the GR-node α such that F and F are the transfer functions of prIγ −Π (α) and prΠ (α), respectively. Proof. (i). Let Π be a supporting projection for α. Then, by Theorem 3.9, α = prIγ −Π (α)prΠ (α).

Matrix-J-unitary Rational Formal Power Series

67

By the assumption, α is minimal. We now show that the GR-nodes α = prIγ −Π (α) and α = prΠ (α) are also minimal. To this end, let x ∈ ran Πk . Then wgk wg wg C (2) A(22) x = (CA) k Πk x = (CA) k x. Thus, if Ok denotes the kth observability matrix of α , then x ∈ ker Ok implies x ∈ ker Ok , and the observability of α implies that α is also observable. Since gk wT g wT A(22) B (2) = Πk (AB) k , one has Ck = Πk Ck , where Ck is the kth controllability matrix of α . Thus, the controllability of α implies the controllability of α . Hence, we have proved the minimality of α . Note that we have used that ker Π = ran (IIγ − Π) is A-invariant. Since ran Π = ker(IIγ − Π) is A× -invariant, by Theorem 3.8 α× is minimal. Using α× = (α α )× = (α )× (α )× , we prove the minimality of (α )× in the same way as that of α . Applying once again Theorem 3.8, we obtain the minimality of α . The dimensions of the state spaces of the minimal GR-nodes α , α and α are related by γk = γk + γk ,

k = 1, . . . , N.

Therefore, given any minimal GR-realizations β and β of F and F , respectively, the same equalities hold for the state space dimensions of β , β and β. Thus, β β is a minimal GR-node, and the factorization F = F F is minimal. (ii). Assume that the factorization F = F F is minimal. Let β and β be minimal GR-realizations of F and F with k-th state space dimensions equal to γk and γk , respectively (k = 1, . . . , N ). Then β β is a minimal GR-realization of F and its kth state space dimension is equal to γk = γk + γk (k = 1, . . . , N ). Hence β β is similar to α. We denote the corresponding GR-node similarity by T = diag(T T1 , . . . , TN ), where

T k : Cγ ⊕ Cγ → Cγ ,

k = 1, . . . N,

is the canonical isomorphism. Let Πk be the projection of Cγk along Tk Cγk onto Tk Cγk , k = 1, . . . , N , and set Π = diag(Π1 , . . . , Πk ). Then Π is a supporting projection for α. Moreover prIγ −Π (α) is similar to β , and prΠ (α) is similar to β . The uniqueness of Π is proved in the same way as in [15, Theorem 4.8]. The uniqueness of the GR-node similarity follows from Theorem 3.5.

4. Matrix-J-unitary formal power series: A multivariable non-commutative analogue of the line case In this section we study a multivariable non-commutative analogue of rational q × q matrix-valued functions which are J-unitary on the imaginary line iR of the complex plane C.

68

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

4.1. Minimal Givone–Roesser realizations and the Lyapunov equation N

Denote by Hn×n the set of Hermitian n × n matrices. Then (iHn×n ) will denote the set of N -tuples of skew-Hermitian matrices. In our paper, the set 0 N iHn×n , JN = n∈N

1 where “ ” stands for a disjoint union, will be a counterpart of the imaginary line iR. Let J ∈ Cq×q be a signature matrix. We will call a rational FPS F ∈ Cq×q z1 , . . . , zN rat matrix-J-unitary on JN if for every n ∈ N, F (Z)(J ⊗ In )F (Z)∗ = J ⊗ In

(4.1)

n×n N

at all points Z ∈ (iH ) where it is deﬁned. For a ﬁxed n ∈ N, F (Z) as a function of matrix entries is rational and holomorphic on some open neighborhood N Γn (ε) of Z = 0, e.g., of the form (2.9), and Γn (ε) ∩ (iHn×n ) is a uniqueness set in n×n N (C ) (see [45] for the uniqueness theorem in several complex variables). Thus, (4.1) implies that (4.2) F (Z)(J ⊗ In )F (−Z ∗ )∗ = J ⊗ In at all points Z ∈ (Cn×n )N where F (Z) is holomorphic and invertible (the set of such points is open and dense, since det F (Z) ≡ 0). The following theorem is a counterpart of Theorem 2.1 in [7]. Theorem 4.1. Let F be a rational FPS with a minimal GR-realization (3.11). Then F is matrix-J-unitary on JN if and only if the following conditions are fulﬁlled: a) D is J-unitary, i.e., DJD∗ = J; b) there exists an invertible Hermitian solution H = diag(H1 , . . . , HN ), with Hk ∈ Cγk ×γk , k = 1, . . . , N , of the Lyapunov equation A∗ H + HA = −C ∗ JC,

(4.3)

B = −H −1 C ∗ JD.

(4.4)

and The property b) is equivalent to b ) there exists an invertible Hermitian matrix H = diag(H1 , . . . , HN ), with Hk ∈ Cγk ×γk , k = 1, . . . , N , such that H −1 A∗ + AH −1 = −BJB ∗ ,

(4.5)

C = −DJB ∗ H.

(4.6)

and Proof. Let F be matrix-J-unitary. Then F is holomorphic at the point Z = 0 in CN , hence D = F (0) is J-unitary (in particular, invertible). Equality (4.2) may be rewritten as (4.7) F (Z)−1 = (J ⊗ In )F (−Z ∗ )∗ (J ⊗ In ).

Matrix-J-unitary Rational Formal Power Series

69

Since (4.7) holds for all n ∈ N, it follows from Corollary 2.2 that the FPSs corresponding to the left and the right sides of equality (4.7) coincide. Due to The(N orem 3.8, α× = (N ; A× , B × , C × , D× ; Cγ = k=1 Cγk , Cq ) with A× , B × , C × , D× given by (3.9) is a minimal GR-realization of F −1 . Due to (4.7), another minimal ˜ B, ˜ C, ˜ D; ˜ Cγ = (N Cγk , Cq ), where ˜ = (N ; A, GR-realization of F −1 is α k=1

∗

A˜ = −A ,

∗

˜ = C J, B

∗

C˜ = −JB ,

˜ = JD∗ J. D

By Theorem 3.5, there exists unique similarity transform T = diag(T T1 , . . . , TN ) which relates α× and α ˜ , where Tk ∈ Cγk ×γk are invertible for k = 1, . . . , N , and T (A − BD−1 C) = −A∗ T,

T BD−1 = C ∗ J,

D−1 C = JB ∗ T.

(4.8)

Note that the relation D−1 = JD∗ J, which means J-unitarity of D, has been already established above. It is easy to check that relations (4.8) are also valid for T ∗ in the place of T . Hence, by the uniqueness of similarity matrix, T = T ∗ . Setting H = −T , we obtain from (4.8) the equalities (4.3) and (4.4), as well as (4.5) and (4.6), by a straightforward calculation. Let us prove now a slightly more general statement than the converse. Let α be a (not necessarily minimal) GR-realization of F of the form (3.8), where D is J-unitary, and let H = diag(H1 , . . . , HN ) with Hk ∈ Crk ×rk , k = 1, . . . , N , be a Hermitian invertible matrix satisfying (4.3) and (4.4). Then in the same way as in [7, Theorem 2.1] for the one-variable case, we obtain for Z, Z ∈ Cn×n : −1

F (Z)(J ⊗ In )F (Z )∗ = J ⊗ In − (C ⊗ In ) (IIr ⊗ In − ∆(Z)(A ⊗ In )) ×∆(Z + Z ∗ )(H −1 ⊗ In ) (IIr ⊗ In − (A∗ ⊗ In )∆(Z ∗ ))

−1

(C ∗ ⊗ In )

(4.9)

−1

(note that ∆(Z) commutes with H ⊗ In ). It follows from (4.9) that F (Z) is (J ⊗ In )-unitary on (iHn×n )N at all points Z where it is deﬁned. Since n ∈ N is arbitrary, F is matrix-J-unitary on JN . Clearly, conditions a) and b’) also imply the matrix-J-unitarity of F on JN . Let us make some remarks. First, it follows from the proof of Theorem 4.1 that the structured solution H = diag(H1 , . . . , HN ) of the Lyapunov equation (4.3) is uniquely determined by a given minimal GR-realization of F . The matrix H = diag(H1 , . . . , HN ) is called the associated structured Hermitian matrix (associated with this minimal GR-realization of F ). The matrix Hk will be called the kth component of the associated Hermitian matrix (k = 1, . . . , N ). The explicit formulas for Hk follow from (3.2): wgk 2 wg 3+ Hk = − col|w|≤qr−1 ((JB ∗ )(−A∗ )) k col|w|≤qr−1 (D−1 C)A× 4 T g wT 5† = −row|w|≤qr−1 ((−A∗ )(C ∗ J))gk w row|w|≤qr−1 A× (BD−1 ) k . Second, let α be a (not necessarily minimal) GR-realization of F of the form (3.8), where D is J-unitary, and let H = diag(H1 , . . . , HN ) with Hk ∈ Crk ×rk , k = 1, . . . , N , be an Hermitian, not necessarily invertible, matrix satisfying (4.3) and

70

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

(4.6). Then in the same way as in [7, Theorem 2.1] for the one-variable case, we obtain for Z, Z ∈ Cn×n : F (Z )∗ (J ⊗ In )F (Z) = J ⊗ In − (B ∗ ⊗ In ) (IIr ⊗ In − ∆(Z ∗ )(A∗ ⊗ In )) ×(H ⊗ In )∆(Z ∗ + Z) (IIr ⊗ In − (A ⊗ In )∆(Z))

−1

(B ⊗ In )

−1

(4.10)

(note that ∆(Z) commutes with H ⊗ In ). It follows from (4.10) that F (Z) is (J ⊗ In )-unitary on (iHn×n )N at all points Z where it is deﬁned. Since n ∈ N is arbitrary, F is matrix-J-unitary on JN . Third, if α is a (not necessarily minimal) GR-realization of F of the form (3.8), where D is J-unitary, and equalities (4.5) and (4.6) are valid with H −1 replaced by some, possibly not invertible, Hermitian matrix Y = diag(Y Y1 , . . . , YN ) with Yk ∈ Crk ×rk , k = 1, . . . , N , then F is matrix-J-unitary on JN . This follows from the fact that (4.9) is valid with H −1 replaced by Y . Theorem 4.2. Let (C, A) be an observable pair of matrices C ∈ Cq×r , A ∈ (N rk and Ok has full column rank for each Cr×r in the sense that Cr = k=1 C k ∈ {1, . . . , N }, and let J ∈ Cq×q be a signature matrix. Then there exists a matrix-J-unitary on JN rational FPS F with a minimal GR-realization (N rk q α = (N ; A, B, C, D; Cr = k=1 C , C ) if and only if the Lyapunov equation (4.3) has a structured solution H = diag(H1 , . . . , HN ) which is both Hermitian and invertible. If such a solution H exists, possible choices of D and B are D0 = Iq ,

B0 = −H −1 C ∗ J.

(4.11)

Finally, for a given such H, all other choices of D and B diﬀer from D0 and B0 by a right multiplicative J-unitary constant matrix. Proof. Let H = diag(H1 , . . . , HN ) be a structured solution of the Lyapunov equation (4.3) which is both Hermitian and invertible. We ﬁrst check that the pair (A, −H −1 C ∗ J) is controllable, or equivalently, that the pair (−JCH −1 , A∗ ) is observable. Using the Lyapunov equation (4.3), one can see that for any k ∈ {1, . . . , N } and w = gi1 · · · gi|w| ∈ FN there exist matrices K0 , . . . , K|w|−1 such that (CA)wgk

= (−1)|w|−1 J((−JCH −1 )A∗ )wgk Hk + K0 J(−JC Ci2 Hi−1 (A∗ )i2 i3 · · · (A∗ )i|w| k )Hk + · · · 2 + K|w|−2 J(−JC Ci|w| (A∗ )i|w| k )Hk + K|w|−1 J(−JCk Hk−1 )Hk .

Thus, if x ∈ ker((−JCH −1 )A∗ )wgk for all of w ∈ FN then Hk−1 x ∈ ker Ok , and the observability of the pair (C, A) implies that x = 0. Therefore, the pair (−JCH −1 , A∗ ) is observable, and the pair (A, −H −1 C ∗ J) is controllable. By Theorem 4.1 we obtain that F0 (z) = Iq − C(IIr − ∆(z)A)−1 ∆(z)H −1 C ∗ J

(4.12)

is a matrix-J-unitary on JN rational FPS, which has a minimal GR-realization (N α0 = (N : A, −H −1 C ∗ J, C, Iq ; Cr = k=1 Crk , Cq ) with the associated structured Hermitian matrix H.

Matrix-J-unitary Rational Formal Power Series

71

(N Conversely, let α = (N ; A, B, C, D; Cr = k=1 Crk , Cq ) be a minimal GRnode. Then by Theorem 4.1 there exists an Hermitian and invertible matrix H = diag(H1 , . . . , HN ) which solves (4.3). Given H = diag(H1 , . . . , HN ), let B, D be any solution of the inverse problem, (N rk q i.e., α = (N ; A, B, C, D; Cr = k=1 C , C ) is a minimal GR-node with the associated structured Hermitian matrix H. Then for F0 = Tαnc0 and F = Tαnc we obtain from (4.9) that F (Z)(J ⊗ In )F (Z )∗ = F0 (Z)(J ⊗ In )F F0 (Z )∗ for any n ∈ N, at all points Z, Z ∈ (Cn×n )N where both F and F0 are deﬁned. By the uniqueness theorem in several complex variables (matrix entries for Zk ’s and Z ∗k ’s, k = 1, . . . , N ), we obtain that F (Z) and F0 (Z) diﬀer by a right multiplicative (J ⊗ In )-unitary constant, which clearly has to be D ⊗ In , i.e., F (Z) = F0 (Z)(D ⊗ In ). Since n ∈ N is arbitrary, by Corollary 2.2 we obtain F (z) = F0 (z)D. Equating the coeﬃcients of these two FPSs, we easily deduce using the observability of the pair (C, A) that B = −H −1 C ∗ JD. The following dual theorem is proved analogously. Theorem 4.3. Let (A, B) be a controllable pair of matrices A ∈ Cr×r , B ∈ Cr×q in (N the sense that Cr = k=1 Crk and Ck has full row rank for each k ∈ {1, . . . , N }, q×q and let J ∈ C be a signature matrix. Then there exists a matrix-J-unitary on JN rational FPS F with a minimal GR-realization α = (N ; A, B, C, D; Cr = (N rk q k=1 C , C ) if and only if the Lyapunov equation GA∗ + AG = −BJB ∗ has a structured solution G = diag(G1 , . . . , GN ) which is both Hermitian and invertible. If such a solution G exists, possible choices of D and C are D0 = Iq ,

C0 = −JB ∗ G−1 .

(4.13)

Finally, for a given such G, all other choices of D and C diﬀer from D0 and C0 by a left multiplicative J-unitary constant matrix. Theorem 4.4. Let F be a matrix-J-unitary on JN rational FPS, and α be its GRrealization. Let H = diag(H1 , . . . , HN ) with Hk ∈ Crk ×rk , k = 1, . . . , N , be an Hermitian invertible matrix satisfying (4.3) and (4.4), or equivalently, (4.5) and (4.6). Then α is observable if and only if α is controllable. Proof. Suppose that α is observable. Since by Theorem 4.1 D = F∅ is J-unitary, by Theorem 4.2 α is a minimal GR-node. In particular, α is controllable. Suppose that α is controllable. Then by Theorem 4.3 α is minimal, and in particular, observable.

72

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

4.2. The associated structured Hermitian matrix Lemma 4.5. Let F be a matrix-J-unitary on JN rational FPS, and let α(i) = (N (N ; A(i) , B (i) , C (i) , D; Cγ = k=1 Cγk , Cq ) be minimal GR-realizations of F , with (i) (i) the associated structured Hermitian matrices H (i) = diag(H1 , . . . , HN ), i = 1, 2. (1) (2) Then α and α are similar, i.e., (2.8) holds with a uniquely deﬁned invertible matrix T = diag(T T1 , . . . , TN ), and (1)

Hk

In particular, the matrices

= Tk∗ Hk Tk , (2)

(1) Hk

and

(2) Hk

k = 1, . . . , N.

(4.14)

have the same signature.

The proof is easy and analogous to the proof of Lemma 2.1 in [7]. Remark 4.6. The similarity matrix T = diag(T T1 , . . . , TN ) is a unitary map(N γk ping from Cγ = C endowed with the inner product [ · , · ]H (1) onto k=1 (N γ γk C = k=1 C endowed with the inner product [ · , · ]H (2) , where [x, y]H (i) = H (i) x, yCγ ,

x, y ∈ Cγ , i = 1, 2,

that is, [x, y]H (i) =

N

[xk , yk ]H (i) ,

k=1

i = 1, 2,

k

where xk , yk ∈ Cγk , x = colk=1,...,N (xk ), y = colk=1,...,N (yk ), and (i)

[xk , yk ]H (i) = Hk xk , yk Cγk ,

k = 1, . . . , N, i = 1, 2.

k

Recall the following deﬁnition [37]. Let Kw,w be a Cq×q -valued function deKw,w )∗ = Kw ,w . Then Kw,w is ﬁned for w and w in some set E and such that (K called a kernel with κ negative squares if for any m ∈ N, any points w1 , . . . , wm in E, and any vectors c1 , . . . , cm in Cq the matrix (c∗j Kwj ,wi ci )i,j=1,...,m ∈ Hm×m has at most κ negative eigenvalues, and has exactly κ negative eigenvalues for some choice of m, w1 , . . . , wm , c1 , . . . , cm . We will use this deﬁnition to give a characterization of the number of negative eigenvalues of the kth component Hk , k = 1, . . . , N , of the associated structured Hermitian matrix H. Theorem 4.7. Let F be a matrix-J-unitary on JN rational FPS, and let α be its minimal GR-realization of the form (3.11), with the associated structured Hermitian matrix H = diag(H1 , . . . , HN ). Then for k = 1, . . . , N the number of negative eigenvalues of the matrix Hk is equal to the number of negative squares of each of the kernels F,k Kw,w ∗

F ,k Kw,w

T

= (CA)wgk Hk−1 (A∗ C ∗ )gk w , T

= (B ∗ A∗ )wgk Hk (AB)gk w ,

w, w ∈ FN , w, w ∈ FN ,

∗

(4.15) (4.16)

For k = 1, . . . , N , denote by Kk (F ) (resp., Kk (F )) the linear span of the functions F,k F ∗ ,k q w → Kw,w c (resp., w → Kw,w c) where w ∈ FN and c ∈ C . Then dim Kk (F ) = dim Kk (F ∗ ) = γk .

Matrix-J-unitary Rational Formal Power Series

73

Proof. Let m ∈ N, w1 , . . . , wm ∈ FN , and c1 , . . . , cm ∈ Cq . Then the matrix equality F,k c) = X ∗ Hk−1 X, (c∗j Kw j ,wi i i,j=1,...,m

with

T X = row1≤i≤m (A∗ C ∗ )gk wi ci ,

F,k implies that the kernel Kw,w has at most κk negative squares, where κk denotes the number of negative eigenvalues of Hk . The pair (C, A) is observable, hence we T can choose a basis of Cq of the form xi = (A∗ C ∗ )gk wi ci , i = 1, . . . , q. Since the matrix X = rowi=1,...,q (xi ) is non-degenerate, and therefore the matrix X ∗ Hk−1 X F,k has exactly κk negative eigenvalues, the kernel Kw,w has κk negative squares. Analogously, from the controllability of the pair (A, B) one can obtain that the kernel Kk (F ∗ ) has κk negative squares. Since Kk (F ) is the span of functions (of variable w ∈ FN ) of the form (CA)wgk y, y ∈ Cγk , it follows that dim Kk (F ) ≤ γk . From the observability of the pair (C, A) we obtain that (CA)wgk y ≡ 0 implies y = 0, thus dim Kk (F ) = γk . In the same way we obtain that the controllability of the pair (A, B) implies that dim Kk (F ∗ ) = γk .

We will denote by νk (F ) the number of negative squares of either the kernel F ∗ ,k or the kernel Kw,w deﬁned by (4.15) and (4.16), respectively.

F,k Kw,w

Theorem 4.8. Let F (i) be matrix-J-unitary on JN rational FPSs, with minimal (N (i) (i) γk GR-realizations α(i) = (N ; A(i) , B (i) , C (i) , D(i) ; Cγ = , Cq ) and the k=1 C (i) (i) associated structured Hermitian matrices H (i) = diag(H1 , . . . , HN ), respectively, (1) (2) i = 1, 2. Suppose that the product α = α α is a minimal GR-node. Then the matrix H = diag(H1 , . . . , HN ), with (1) (1) (2) (1) (2) Hk 0 Hk = (4.17) ∈ C(γk +γk )×(γk +γk ) , k = 1, . . . , N, (2) 0 Hk is the associated structured Hermitian matrix for α = α(1) α(2) . Proof. It suﬃces to check that (4.3) and (4.4) hold for the matrices A, B, C, D deﬁned as in (3.7), and H = diag(H1 , . . . , HN ) where Hk , k = 1, . . . , N , are deﬁned in (4.17). This is an easy computation which is omitted. Corollary 4.9. Let F1 and F2 be matrix-J-unitary on JN rational FPSs, and suppose that the factorization F = F1 F2 is minimal. Then νk (F F1 F2 ) = νk (F F1 ) + νk (F F2 ),

k = 1, . . . , N.

74

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

4.3. Minimal matrix-J-unitary factorizations In this subsection we consider minimal factorizations of rational formal power series which are matrix-J-unitary on JN into factors both of which are also matrix-Junitary on JN . Such factorizations will be called minimal matrix-J-unitary factorizations. Let H ∈ Cr×r be an invertible Hermitian matrix. We denote by [ · , · ]H the Hermitian sesquilinear form [x, y]H = Hx, y where · , · denotes the standard inner product of Cr . Two vectors x and y in Cr are called H-orthogonal if [x, y]H = 0. For any subspace M ⊂ Cr denote M [⊥] = {y ∈ Cr : y, mH = 0 ∀m ∈ M } . The subspace M ⊂ Cr is called non-degenerate if M ∩ M [⊥] = {0}. In this case, ·

M [+]M [⊥] = Cr ·

where [+] denotes the H-orthogonal direct sum. In the case when H = diag(H1 , . . . , HN ) is the structured Hermitian matrix associated with a given minimal GR-realization of a matrix-J-unitary on JN rational FPS F , we will call [ · , · ]H the associated inner product (associated with the given minimal GR-realization of F ). In more details, [x, y]H =

N

[xk , yk ]Hk ,

k=1

where xk , yk ∈ Cγk and x = colk=1,...,N (xk ), y = colk=1,...,N (yk ), and [xk , yk ]Hk = Hk xk , yk Cγk ,

k = 1, . . . , N.

The following theorem (as well as its proof) is analogous to its one-variable counterpart, Theorem 2.6 from [7] (see also [43, Chapter II]). Theorem 4.10. Let F be a matrix-J-unitary on JN rational FPS, and let α be its minimal GR-realization of the form (3.11), with the associated structured Her(N mitian matrix H = diag(H1 , . . . , HN ). Let M = k=1 Mk be an A-invariant subspace such that Mk ⊂ Cγk , k = 1, . . . , N , and M is non-degenerate in the associated inner product [ · , · ]H . Let Π = diag(Π1 , . . . , ΠN ) be the projection deﬁned by ker Π = M,

ran Π = M[⊥] ,

or in more details, ker Πk = Mk ,

[⊥]

ran Πk = Mk ,

k = 1, . . . , N.

Matrix-J-unitary Rational Formal Power Series

75

Let D = D1 D2 be a factorization of D into two J-unitary factors. Then the factorization F = F1 F2 where F1 (z) =

D1 + C(IIγ − ∆(z)A)−1 ∆(z)(IIγ − Π)BD2−1 ,

F2 (z) =

D2 + D1−1 CΠ(IIγ − ∆(z)A)−1 ∆(z)B,

is a minimal matrix-J-unitary factorization of F . Conversely, any minimal matrix-J-unitary factorization of F can be obtained in such a way. For a ﬁxed J-unitary decomposition D = D1 D2 , the correspondence between minimal matrix-J-unitary factorizations of F and non(N degenerate A-invariant subspaces of the form M = k=1 Mk , where Mk ⊂ Cγk for k = 1, . . . , N , is one-to-one. Remark 4.11. We omit here the proof, which can be easily restored, with making use of Theorem 3.9 and Corollary 3.10. Remark 4.12. Minimal matrix-J-unitary factorizations do not always exist, even for N = 1. Examples of J-unitary on iR rational functions which have non-trivial minimal factorizations but lack minimal J-unitary factorizations can be found in [4] and [7]. 4.4. Matrix-unitary rational formal power series In this subsection we specialize some of the preceding results to the case J = Iq . We call the corresponding rational formal power series matrix-unitary on JN . Theorem 4.13. Let F be a rational FPS and α be its minimal GR-realization of the form (3.11). Then F is matrix-unitary on JN if and only if the following conditions are fulﬁlled: a) D is a unitary matrix, i.e., DD∗ = Iq ; b) there exists an Hermitian solution H = diag(H1 , . . . , HN ), with Hk ∈ Cγk ×γk , k = 1, . . . , N , of the Lyapunov equation A∗ H + HA = −C ∗ C, and

(4.18)

C = −D−1 B ∗ H.

(4.19) The property b) is equivalent to b ) there exists an Hermitian solution G = diag(G1 , . . . , GN ), with Gk ∈ Cγk ×γk , k = 1, . . . , N , of the Lyapunov equation and

GA∗ + AG = −BB ∗ ,

(4.20)

B = −GC ∗ D−1 .

(4.21)

Proof. To obtain Theorem 4.13 from Theorem 4.1 it suﬃces to show that any structured Hermitian solution to the Lyapunov equation (4.18) (resp., (4.20)) is invertible. Let H = diag(H1 , . . . , HN ) be a structured Hermitian solution to (4.18), and x ∈ ker H, i.e., x = col1≤k≤N (xk ) and xk ∈ ker Hk , k = 1, . . . , N . Then

76

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

HAx, x = Ax, Hx = 0, and equation (4.18) implies Cx = 0. In particular, ˜ = col(0, . . . , 0, xk , 0, . . . , 0) where xk ∈ for every k ∈ {1, . . . , N } one can deﬁne x ker Hk is on the kth block entry of x ˜, and from C x ˜ = 0 get Ck xk = 0. Thus, ker Hk ⊂ ker Ck , k = 1, . . . , N . Consider the following block representations with respect to the decompositions Cγk = ker Hk ⊕ ran Hk : (11) (12) 0 0 Aijj Aijj (2) , H Aij = , C = = 0 C k k (22) , (21) (22) k 0 Hk Aij Aij where i, j, k = 1, . . . , N . Then (4.18) implies (A∗ H + HA)ij

(12)

and

(21) Aji

(21) ∗

= (A∗ji Hj + Hi Aij )(12) = (Aji ) Hj

(22)

= 0,

= 0, i, j = 1, . . . , N . Therefore, for any w ∈ FN we have (CA)wgk = 0 (C (2) A(22) )wgk , k = 1, . . . , N, (2)

(22)

where C (2) = row1≤k≤N (Ck ), A(22) = (Aij )i,j=1,...,N . If there exists k ∈ {1, . . . , N } such that ker Hk = {0}, then the pair (C, A) is not observable, which contradicts to the assumption on α. Thus, H is invertible. In a similar way one can show that any structured Hermitian solution G = diag(G1 , . . . , GN ) of the Lyapunov equation (4.20) is invertible. A counterpart of Theorem 4.2 in the present case is the following theorem. Theorem 4.14. Let (C, A) be an observable pair of matrices C ∈ Cq×r , A ∈ Cr×r (N rk in the sense that Cr = and Ok has full column rank for each k ∈ k=1 C {1, . . . , N }. Then there exists a matrix-unitary on JN rational FPS F with a mini(N mal GR-realization α = (N ; A, B, C, D; Cr = k=1 Crk , Cq ) if and only if the Lyapunov equation (4.18) has a structured Hermitian solution H = diag(H1 , . . . , HN ). If such a solution H exists, it is invertible, and possible choices of D and B are D0 = Iq ,

B0 = −H −1 C ∗ .

(4.22)

Finally, for a given such H, all other choices of D and B diﬀer from D0 and B0 by a right multiplicative unitary constant matrix. The proof of Theorem 4.14 is a direct application of Theorem 4.2 and Theorem 4.13. One can prove analogously the following theorem which is a counterpart of Theorem 4.3. Theorem 4.15. Let (A, B) be a controllable pair of matrices A ∈ Cr×r , B ∈ Cr×q (N in the sense that Cr = k=1 Crk and Ck has full row rank for each k ∈ {1, . . . , N }. Then there exists a matrix-unitary on JN rational FPS F with a minimal GR(N rk q realization α = (N ; A, B, C, D; Cr = k=1 C , C ) if and only if the Lyapunov equation (4.20) has a structured Hermitian solution G = diag(G1 , . . . , GN ). If such a solution G exists, it is invertible, and possible choices of D and C are D0 = Iq ,

C0 = −B ∗ G−1 .

(4.23)

Matrix-J-unitary Rational Formal Power Series

77

Finally, for a given such G, all other choices of D and C diﬀer from D0 and C0 by a left multiplicative unitary constant matrix. Let A = (A1 , . . . , AN ) be an N -tuple of r × r matrices. A non-zero vector x ∈ Cr is called a common eigenvector for A if there exists λ = (λ1 , . . . , λN ) ∈ CN (which is called a common eigenvalue for A) such that Ak x = λk x,

k = 1, . . . , N.

The following theorem, which is a multivariable non-commutative counterpart of statements a) and b) of Theorem 2.10 in [7], gives a necessary condition on a minimal GR-realization of a matrix-unitary on JN rational FPS. Theorem 4.16. Let F be a matrix-unitary on JN rational FPS and α be its minimal GR-realization, with the associated structured Hermitian matrix H = diag(H1 , . . . , HN ) and the associated inner products [ · , · ]Hk , k = 1, . . . , N . Let Pk denote the orthogonal projection in Cγ onto the subspace {0} ⊕ · · · ⊕ {0} ⊕ Cγk ⊕ {0} ⊕ · · · ⊕ {0}, and Ak = AP Pk , k = 1, . . . , N . If x ∈ Cγ is a common eigenvector for A corresponding to a common eigenvalue λ ∈ CN then there exists Pj x, Pj x]Hj = 0. In particular, A has no j ∈ {1, . . . , N } such that Re λj = 0 and [P common eigenvalues on (iR)N . Proof. By (4.18), we have for every k ∈ {1, . . . , N }, Pk x, Pk x]Hk = − CP Pk x, CP Pk x . (λk + λk )[P Suppose that for all k ∈ {1, . . . , N } the left-hand side of this equality is zero, then CP Pk x = 0. Since for ∅ = w = gi1 · · · gi|w| ∈ FN , Pi1 Ai2 · · · Ai|w| · Ak x = λi2 · · · λi|w| λk CP Pi1 x = 0, (CA)wgk Pk x = CP the observability of the pair (C, A) implies Pk x = 0, k = 1, . . . , N , i.e., x = 0 which contradicts to the assumption that x is a common eigenvector for A. Thus, there exists j ∈ {1, . . . , N } such that (λj + λj )[P Pj x, Pj x]Hj = 0, as desired.

5. Matrix-J-unitary formal power series: A multivariable non-commutative analogue of the circle case In this section we study a multivariable non-commutative analogue of rational Cq×q -valued functions which are J-unitary on the unit circle T. 5.1. Minimal Givone–Roesser realizations and the Stein equation Let n ∈ N. We denote by Tn×n the matrix unit circle Tn×n = W ∈ Cn×n : W W ∗ = In , i.e., the family of unitary n × n complex matrices. We will call the set (Tn×n ) the matrix unit torus. The set 0 N TN = Tn×n n∈N

N

78

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

serves as a multivariable non-commutative counterpart of the unit circle. Let J = J −1 = J ∗ ∈ Cq×q . We will say that a rational FPS f is matrix-J-unitary on TN if for every n ∈ N, f (W )(J ⊗ In )f (W )∗ = J ⊗ In N

at all points W = (W W1 , . . . , WN ) ∈ (Tn×n ) where it is deﬁned. In the following theorem we establish the relationship between matrix-J-unitary rational FPSs on JN and on TN , their minimal GR-realizations, and the structured Hermitian solutions of the corresponding Lyapunov and Stein equations. Theorem 5.1. Let f be a matrix-J-unitary on TN rational FPS, with a minimal GR-realization α of the form (3.11), and let a ∈ T be such that −¯ a ∈ σ(A). Then F (z) = f (a(z1 − 1)(z1 + 1)−1 , . . . , a(zN − 1)(zN + 1)−1 )

(5.1)

is well deﬁned as a rational FPS which is matrix-J-unitary on JN , and F = Tβnc, (N where β = (N ; Aa , Ba , Ca , Da ; Cγ = k=1 Cγk , Cq ), with √ −1 Aa = (aA , Ba = 2(aA + Iγ )−1 aB, γ) √− Iγ )(aA + I−1 (5.2) Ca = 2C(aA + Iγ ) , Da = D − C(aA + Iγ )−1 aB. A GR-node β is minimal, and its associated structured Hermitian matrix H = diag(H1 , . . . , HN ) is the unique invertible structured Hermitian solution of ∗ A B H 0 A B H 0 = . (5.3) C D 0 J C D 0 J Proof. For any a ∈ T and n ∈ N the Cayley transform Z0 −→ W0 = a(Z0 − In )(Z0 + In )−1 maps iHn×n onto Tn×n , thus its simultaneous application to each matrix variable maps (iHn×n )N onto (Tn×n )N . Since the simultaneous application of the Cayley transform to each formal variable in a rational FPS gives a rational FPS, (5.1) deﬁnes a rational FPS F. Since f is matrix-J-unitary on TN , F is matrix-J-unitary on JN . Moreover, −1 F (z) = D + C Iγ − a(∆(z) − Iγ )(∆(z) + Iγ )−1 A ×a(∆(z) − Iγ )(∆(z) + Iγ )−1 B = D + C (∆(z) + Iγ − a(∆(z) − Iγ )A)−1 a(∆(z) − Iγ )B = D + C (aA + Iγ − ∆(z)(aA − Iγ ))−1 a(∆(z) − Iγ )B −1 ∆(z)aB = D + C(aA + Iγ )−1 Iγ − ∆(z)(aA − Iγ )(aA + Iγ )−1 −1 −C(aA + Iγ )−1 Iγ − ∆(z)(aA − Iγ )(aA + Iγ )−1 aB = D − C(aA + Iγ )−1 aB + C(aA + Iγ )−1 −1 × Iγ − ∆(z)(aA − Iγ )(aA + Iγ )−1 ×∆(z) Iγ − (aA − Iγ )(aA + Iγ )−1 aB = Da + Ca (IIγ − ∆(z)Aa )−1 ∆(z)Ba .

Matrix-J-unitary Rational Formal Power Series

79

Thus, F = Tβnc. Let us remark that the FPS

ϕak (z) = Ca (IIγ − ∆(z)Aa )−1 -Cγk

(c.f. (3.5)) has the coeﬃcients Ca Aa )wgk , (ϕak )w = (C

w ∈ FN .

Remark also that ϕ˜k (z) : = ϕk a(z1 − 1)(z1 + 1)−1 , . . . , a(zN − 1)(zN + 1)−1 −1 - γ = C Iγ − a(∆(z) − Iγ )(∆(z) + Iγ )−1 A C k −1 = C ((∆(z) + Iγ ) − a(∆(z) − Iγ )A) (∆(z) + Iγ )-Cγk −1 = C ((aA + Iγ ) − ∆(z)(aA − Iγ )) (∆(z) + Iγ )-Cγk −1 = C(aA + Iγ )−1 Iγ − ∆(z)(aA − Iγ )(aA + Iγ )−1 (∆(z) + Iγ )-Cγk - 1 = √ Ca (IIγ − ∆(z)Aa )−1 -Cγk (zk + 1) 2 1 = √ (ϕak (z) · zk + ϕak (z)) . 2 qγ−1

Let k ∈ {1, . . . , N } be ﬁxed. Suppose that/n ∈ N, n ≥ (qγ − 1) (for qγ − 1 = 0 choose arbitrary n ∈ N), and x ∈ Z∈Γn (ε) ker ϕak (Z), where Γn (ε) is a neighborhood of the origin of Cn×n where ϕak (Z) is well deﬁned, e.g., of the form (2.9) with ε = Aa −1 . Then, by Theorem 3.1 and Theorem 2.1, one has ⎞ ⎛ ) ) ker ϕak (Z) = ⎝ ker (ϕak )w ⎠ ⊗ Cn ⎛ =⎝

w∈F FN : |w|≤qγ−1

Z∈Γn (ε)

)

⎞

˜k (β) ⊗ Cn . ker (C Ca Aa )wgk ⎠ ⊗ Cn = ker O

w∈F FN : |w|≤qγ−1

˜k (β), {y (µ) }lµ=1 ⊂ Cn such that Thus, there exist l ∈ N, {u(µ) }lµ=1 ⊂ ker O x=

l

u(µ) ⊗ y (µ) .

(5.4)

µ=1

Since (ϕak (z) · zk )wgk = (C Ca Aa )wgk for w ∈ FN , and (ϕak (z) · zk )w = 0 for w = wgk with any w ∈ FN , (5.4) implies that ϕak (Z)(IIγk ⊗ Zk )x ≡ 0. Thus, 1 ϕ˜k (Z)x = √ (ϕak (Z)(IIγk ⊗ Zk ) + ϕak (Z)) x ≡ 0. 2 Since the Cayley transform a(∆(z)−IIγ )(∆(z)+IIγ )−1 maps an open and dense subset of the set of matrices of the form ∆(Z) = diag (Z1 , . . . , ZN ), Zj ∈ Cγj ×γj , j = 1, . . . , N , onto an open and dense subset of the same set, ϕk (Z)x = (C ⊗ In )(IIγ − ∆(Z)(A ⊗ In ))−1 x ≡ 0.

80

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

Since the GR-node α is observable, by Theorem 3.7 we get x = 0. Therefore, ) ker ϕak (Z) = 0, k = 1, . . . , N. Z∈Γn (ε)

Applying Theorem 3.7 once again, we obtain the observability of the GR-node β. In the same way one can prove the controllability of β. Thus, β is minimal. Note that ∗ H 0 A B H 0 A B − = 0 J C D 0 J C D ∗ A∗ HB + C ∗ JD A HA + C ∗ JC − H . (5.5) = B ∗ HA + D∗ JC B ∗ HB + D∗ JD − J Since −a ¯∈ / σ(A), the matrix (aA + Iγ )−1 is well deﬁned, as well as Aa = (aA − Iγ )(aA + Iγ )−1 , and Iγ − Aa = 2(aA + Iγ )−1 is invertible. Having this in mind, one can deduce from (5.2) the following relations: A∗ HA + C ∗ JC − H = 2(IIγ − A∗a )−1 (A∗a H + HAa + Ca∗ JC Ca )(IIγ − Aa )−1 B ∗ HA + D∗ JC

√ 2(Ba∗ H + Da∗ JC Ca )(IIγ − Aa )−1 √ ∗ + 2Ba (IIγ − A∗a )−1 (A∗a H + HAa + Ca∗ JC Ca )(IIγ − Aa )−1 =

B ∗ HB + D∗ JD − J =

Ba∗ (IIγ − A∗a )−1 (A∗a H + HAa + Ca∗ JC Ca )(IIγ − Aa )−1 Ba

+

(Ba∗ H + Da∗ JC Ca )(IIγ − Aa )−1 Ba + Ba∗ (IIγ − A∗a )−1 (C Ca∗ JDa + HBa ).

Thus, A, B, C, D, H satisfy (5.3) if and only if Aa , Ba , Ca , Da , H satisfy (4.3) and (4.4) (in the place of A, B, C, D, H therein), which completes the proof. We will call the invertible Hermitian solution H = diag(H1 , . . . , HN ) of (5.3), which is determined uniquely by a minimal GR-realization α of a matrix-J-unitary on TN rational FPS f , the associated structured Hermitian matrix (associated with a minimal GR-realization α of f ). Let us note also that since for the GR-node β from Theorem 5.1 a pair of the equalities (4.3) and (4.4) is equivalent to a pair of the equalities (4.5) and (4.6), the equality (5.3) is equivalent to ∗ −1 −1 H H 0 A B 0 A B = . (5.6) C D 0 J C D 0 J Remark 5.2. Equality (5.3) can be replaced by the following three equalities: H − A∗ HA D∗ JC J − D∗ JD

= C ∗ JC, = −B ∗ HA,

(5.7) (5.8)

= B ∗ HB,

(5.9)

Matrix-J-unitary Rational Formal Power Series

81

and equality (5.6) can be replaced by H −1 − AH −1 A∗ DJB

∗

J − DJD

∗

= = =

BJB ∗ , −CH CH

−1

−1

(5.10) ∗

A ,

(5.11)

∗

(5.12)

C .

Theorem 5.1 allows to obtain a counterpart of the results from Section 4 in the setting of rational FPSs which are matrix-J-unitary on TN . We will skip the proofs when it is clear how to get them. Theorem 5.3. Let f be a rational FPS and α be its minimal GR-realization of the form (3.11). Then f is matrix-J-unitary on TN if and only if there exists an invertible Hermitian matrix H = diag(H1 , . . . , HN ), with Hk ∈ Cγk ×γk , k = 1, . . . , N , which satisﬁes (5.3), or equivalently, (5.6). Remark 5.4. In the same way as in [7, Theorem 3.1] one can show that if a rational FPS f has a (not necessarily minimal) GR-realization (3.8) which satisﬁes (5.3) (resp., (5.6)), with an Hermitian invertible matrix H = diag(H1 , . . . , HN ), then for any n ∈ N, f (Z )∗ (J ⊗ In )f (Z) =

−1

J ⊗ In − (B ∗ ⊗ In ) (IIγ ⊗ In − ∆(Z ∗ )(A∗ ⊗ In ))

×

(H ⊗ In )(IIγ ⊗ In − ∆(Z )∗ ∆(Z))

×

(IIγ ⊗ In − (A ⊗ In )∆(Z))

−1

(B ⊗ In )

(5.13)

and respectively, f (Z)(J ⊗ In )f (Z )∗

= J ⊗ In − (C ⊗ In ) (IIγ ⊗ In − ∆(Z)(A ⊗ In ))−1 × (IIγ ⊗ In − ∆(Z)∆(Z )∗ )(H −1 ⊗ In ) −1

× (IIγ ⊗ In − (A∗ ⊗ In )∆(Z )∗ )

(C ∗ ⊗ In ),

(5.14)

N

at all the points Z, Z ∈ (Cn×n ) where it is deﬁned, which implies that f is matrix-J-unitary on TN . Moreover, the same statement holds true if H = diag(H1 , . . . , HN ) in (5.3) and (5.13) is not supposed to be invertible, and if −1 ) in (5.6) and (5.14) is replaced by any Hermitian, H −1 = diag(H1−1 , . . . , HN Y1 , . . . , YN ). not necessarily invertible matrix Y = diag(Y Theorem 5.5. Let f be a matrix-J-unitary on TN rational FPS, and α be its GRrealization. Let H = diag(H1 , . . . , HN ) with Hk ∈ Crk ×rk , k = 1, . . . , N , be an Hermitian invertible matrix satisfying (5.3) or, equivalently, (5.6). Then α is observable if and only if α is controllable. Proof. Let a ∈ T, −a ¯ ∈ / σ(A). Then F deﬁned by (5.1) is a matrix-J-unitary on JN rational FPS, and (5.2) is its GR-realization. As shown in the proof of Theorem 5.1, α is observable (resp., controllable) if and only if so is β. Since by Theorem 5.1 the GR-node β satisﬁes (4.3) and (4.4) (equivalently, (4.5) and (4.6)), Theorem 4.4 implies the statement of the present theorem.

82

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

Theorem 5.6. Let f be a matrix-J-unitary on TN rational FPS and α be its minimal GR-realization of the form (3.11), with the associated structured Hermitian matrix H. If D = f∅ is invertible then so is A, and A−1 = H −1 (A× )∗ H.

(5.15)

Proof. It follows from (5.8) that C = −JD−∗ B ∗ HA. Then (5.7) turns into H − A∗ HA = C ∗ J(−JD−∗ B ∗ HA) = −C ∗ D−∗ B ∗ HA, which implies that H = (A× )∗ HA, and (5.15) follows.

The following two lemmas, which are used in the sequel, can be found in [7]. Lemma 5.7. Let A ∈ Cr×r , C ∈ Cq×r , where A is invertible. Let H be an invertible Hermitian matrix and J be a signature matrix such that H − A∗ HA = C ∗ JC. Let a ∈ T, a ∈ / σ(A). Deﬁne Da

=

Ba Then

A C

=

Ba Da

Iq − CH −1 (IIr − aA∗ )−1 C ∗ J, −H

∗ H 0

−1

−∗

A

0 J

(5.16)

∗

C JDa .

A C

Ba Da

(5.17) H = 0

0 . J

Lemma 5.8. Let A ∈ Cr×r , B ∈ Cr×q , where A is invertible. Let H be an invertible Hermitian matrix and J be a signature matrix such that H −1 − AH −1 A∗ = BJB ∗ . Let a ∈ T, a ∈ / σ(A). Deﬁne Da Ca Then

A Ca

B Da

=

Iq − JB ∗ (IIr − aA∗ )−1 HB,

(5.18)

=

−Da JB ∗ A−∗ H.

(5.19)

−1 H 0

0 J

A Ca

B Da

∗

−1 H = 0

0 . J

Theorem 5.9. Let (C, A) be an observable pair of matrices C ∈ Cq×r , A ∈ Cr×r (N rk in the sense that Cr = and Ok has full column rank for each k ∈ k=1 C {1, . . . , N }. Let A be invertible and J ∈ Cq×q be a signature matrix. Then there exists a matrix-J-unitary on TN rational FPS f with a minimal GR-realization (N rk q α = (N ; A, B, C, D; Cr = k=1 C , C ) if and only if the Stein equation (5.7) has a structured solution H = diag(H1 , . . . , HN ) which is both Hermitian and invertible. If such a solution H exists, possible choices of D and B are Da and Ba deﬁned in (5.16) and (5.17), respectively. For a given such H, all other choices of D and B diﬀer from Da and Ba by a right multiplicative J-unitary constant matrix.

Matrix-J-unitary Rational Formal Power Series

83

Proof. Let H = diag(H1 , . . . , HN ) be a structured solution of the Stein equation (5.7) which is both Hermitian and invertible, Da and Ba are deﬁned as in (5.16) and (5.17), respectively, where a ∈ T, a ∈ / σ(A). Set αa = (N ; A, Ba , C, Da ; Cr = (N rk q nc k=1 C , C ). By Lemma 5.7 and due to Remark 5.4, the transfer function Tα of αa is a matrix-J-unitary on TN rational FPS. Since αa is observable, by Theorem 5.5 αa is controllable, and thus, minimal. (N rk q Conversely, if α = (N ; A, B, C, D; Cr = k=1 C , C ) is a minimal GRnode whose transfer function is matrix-J-unitary on TN then by Theorem 5.3 there exists a solution H = diag(H1 , . . . , HN ) of the Stein equation (5.7) which is both Hermitian and invertible. The rest of the proof is analogous to the one of Theorem 4.2. Analogously, one can obtain the following. Theorem 5.10. Let (A, B) be a controllable pair of matrices A ∈ Cr×r , B ∈ Cr×q in (N the sense that Cr = k=1 Crk and Ck has full row rank for each k ∈ {1, . . . , N }. Let A be invertible and J ∈ Cq×q be a signature matrix. Then there exists a matrix-J-unitary on TN rational FPS f with a minimal GR-realization α = (N (N ; A, B, C, D; Cr = k=1 Crk , Cq ) if and only if the Stein equation G − AGA∗ = BJB ∗

(5.20)

has a structured solution G = diag(G1 , . . . , GN ) which is both Hermitian and invertible. If such a solution G exists, possible choices of D and C are Da and Ca deﬁned in (5.16) and (5.17), respectively, where H = G−1 . For a given such G, all other choices of D and C diﬀer from Da and Ca by a left multiplicative J-unitary constant matrix. 5.2. The associated structured Hermitian matrix In this subsection we give the analogue of the results of Section 4.2. The proofs are similar and will be omitted. Lemma 5.11. Let f be a matrix-J-unitary on TN rational FPS and α(i) = (N γk q (N ; A(i) , B (i) , C (i) , D; Cγ = k=1 C , C ) be its minimal GR-realizations, with (i) (i) the associated structured Hermitian matrices H (i) = diag(H1 , . . . , HN ), i = 1, 2. (1) (2) Then α and α are similar, that is C (1) = C (2) T,

T A(1) = A(2) T,

and

T B (1) = B (2) ,

for a uniquely deﬁned invertible matrix T = diag (T T1 , . . . , TN ) ∈ Cγ×γ and (1)

Hk

In particular, the matrices

= Tk∗ Hk Tk , (2)

(1) Hk

and

(2) Hk

k = 1, . . . , N. have the same signature.

Theorem 5.12. Let f be a matrix-J-unitary on TN rational FPS, and let α be its minimal GR-realization of the form (3.11), with the associated structured Hermitian matrix H = diag(H1 , . . . , HN ). Then for each k ∈ {1, . . . , N } the number of

84

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

negative eigenvalues of the matrix Hk is equal to the number of negative squares of each of the kernels (on FN ): T

f,k wgk −1 Kw,w Hk (A∗ C ∗ )gk w , = (CA) ∗

(5.21)

T

f ,k ∗ ∗ wgk Kw,w Hk (AB)gk w . = (B A )

Finally, for k ∈ {1, . . . , N } let Kk (f ) (resp., Kk (f ∗ )) be the span of the functions f,k f ∗ ,k q w → Kw,w c (resp., w → Kw,w c) where w ∈ FN and c ∈ C . Then dim Kk (f ) = dim Kk (f ∗ ) = γk . We will denote by νk (f ) the number of negative squares of either of the functions deﬁned in (5.21). Theorem 5.13. Let fi , i = 1, 2, be two matrix-J-unitary on TN rational FPSs, with minimal GR-realizations N ' (i) (i) (i) (i) (i) γ (i) γk q = C ,C α = N ; A , B , C , D; C k=1 (i)

(i)

and the associated structured Hermitian matrices H (i) = diag(H1 , . . . , HN ). Assume that the product α = α(1) α(2) is a minimal GR-node. Then, for each k ∈ {1, . . . , N } the matrix (1) (1) (2) (1) (2) Hk 0 Hk = ∈ C(γk +γk )×(γk +γk ) (5.22) (2) 0 Hk is the associated kth Hermitian matrix for α = α(1) α(2) . Corollary 5.14. Let f1 and f2 be two matrix-J-unitary on TN rational FPSs, and assume that the factorization f = f1 f2 is minimal. Then, ν(f1 f2 ) = ν(f1 ) + ν(ff2 ). 5.3. Minimal matrix-J-unitary factorizations In this subsection we consider minimal factorizations of matrix-J-unitary on TN rational FPSs into two factors, both of which are also matrix-J-unitary on TN rational FPSs. Such factorizations will be called minimal matrix-J-unitary factorizations. The following theorem is analogous to its one-variable counterpart [7, Theorem 3.7] and proved in the same way. Theorem 5.15. Let f be a matrix-J-unitary on TN rational FPS and α be its minimal GR-realization of the form (3.11), with the associated structured Hermitian matrix H = diag(H1 , . . . , HN ), and assume that D is invertible. Let (N γ M = k=1 Mk be an A-invariant subspace of C , which is non-degenerate in the associated inner product [ · , · ]H and such that Mk ⊂ Cγk , k = 1, . . . , N . Let Π = diag(Π1 , . . . , ΠN ) be a projection deﬁned by ker Π = M,

and

ran Π = M [⊥] ,

Matrix-J-unitary Rational Formal Power Series

85

that is [⊥]

f or k = 1, . . . , N. ker Πk = Mk , and ran Πk = Mk Then f (z) = f1 (z)ff2 (z), where 2 3 f1 (z) = Iq + C(IIγ − ∆(z)A)−1 ∆(z)(IIγ − Π)BD−1 D1 , 3 2 f2 (z) = D2 Iq + D−1 CΠ(IIγ − ∆(z)A)−1 ∆(z)B ,

(5.23) (5.24)

with D1 = Iq − CH −1 (IIγ − aA∗ )−1 C ∗ J, D = D1 D2 , where a ∈ T belongs to the resolvent set of A1 , and where C1 = C - , A1 = A- , H1 = PM H M

M

M

(with PM being the orthogonal projection onto M in the standard metric of Cγ ), is a minimal matrix-J-unitary factorization of f . Conversely, any minimal matrix-J-unitary factorization of f can be obtained in such a way, and the correspondence between minimal matrix-J-unitary factorizations of f with f1 (a, . . . , a) = Iq and non-degenerate subspaces of A of the form (N M = k=1 Mk , with Mk ⊂ Cγk , k = 1, . . . , N , is one-to-one. Remark 5.16. In the proof of Theorem 5.15, as well as of Theorem 4.10, we make use of Theorem 3.9 and Corollary 3.10. Remark 5.17. Minimal matrix-J-unitary factorizations do not always exist, even in the case N = 1. See [7] for examples in that case. 5.4. Matrix-unitary rational formal power series In this subsection we specialize some of the results in the present section to the case J = Iq . We shall call corresponding rational FPSs matrix-unitary on TN . Theorem 5.18. Let f be a rational FPS and α be its minimal GR-realization of the form (3.11). Then f is matrix-unitary on TN if and only if: (a) There exists an Hermitian matrix H = diag(H1 , . . . , HN ) (with Hk ∈ Cγk ×γk , k = 1, . . . , N ) such that ∗ A B H 0 H 0 A B = . (5.25) C D 0 Iq C D 0 Iq Condition (a) is equivalent to: (a ) There exists an Hermitian matrix G = diag (G1 , . . . , GN ) (with Gk ∈ γk ×γk C , k = 1, . . . , N ) such that ∗ G 0 A B G 0 A B = . (5.26) 0 Iq C D 0 Iq C D Proof. The necessity follows from Theorem 5.1. To prove the suﬃciency, suppose that the Hermitian matrix H = diag(H1 , . . . , HN ) satisﬁes (5.25) and let a ∈ T be such that −a ∈ σ(A). Then, H satisﬁes conditions (4.18) and (4.19) for the GR(N node β = (N ; Aa , Ba , Ca , Da ; Cγ = k=1 Cγk , Cq ) deﬁned by (5.2) (this follows

86

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

from the proof of Theorem 5.1). Thus, from Theorem 4.13 and Theorem 5.1 we obtain that f is matrix-unitary on TN . Analogously, condition (a ) implies that the FPS f is matrix-unitary on TN . A counterpart of Theorem 4.14 in the present case is the following theorem: Theorem 5.19. Let (C, A) be an observable pair of matrices in the sense that Ok has full column rank for each k = 1, . . . , N . Assume that A ∈ Cr×r is invertible. Then there exists a matrix-unitary on TN rational FPS f with a minimal GR-realization (N α = (N ; A, B, C, D; Cr = k=1 Crk , Cq ) if and only if the Stein equation H − A∗ HA = C ∗ C

(5.27) rk ×rk

, k = has an Hermitian solution H = diag(H1 , . . . , HN ), with Hk ∈ C 1, . . . , N . If such a matrix H exists, it is invertible, and possible choices of D and B are Da and Ba given by (5.16) and (5.17) with J = Iq . Finally, for a given H = diag(H1 , . . . , HN ), all other choices of D and B diﬀer from Da and Ba by a right multiplicative unitary constant. A counterpart of Theorem 4.15 is the following theorem: Theorem 5.20. Let (A, B) be a controllable pair of matrices, in the sense that Ck has full row rank for each k = 1, . . . , N . Assume that A ∈ Cr×r is invertible. Then there exists a matrix-unitary on TN rational FPS f with a minimal GR-realization (N α = (N ; A, B, C, D; Cr = k=1 Crk , Cq ) if and only if the Stein equation G − AGA∗ = BB ∗

(5.28) rk ×rk

has an Hermitian solution G = diag(G1 , . . . , GN ) with Gk ∈ G , k = 1, . . . , N . If such a matrix G exists, it is invertible, and possible choices of D and C are Da and Ca given by (5.18) and (5.19) with H = G−1 and J = Iq . Finally, for a given G = diag(G1 , . . . , GN ), all other choices of D and C diﬀer from Da and Ca by a left multiplicative unitary constant. A counterpart of Theorem 4.16 in the present case is the following: Theorem 5.21. Let f be a matrix-unitary on TN rational FPS and α be its minimal GR-realization of the form (3.11), with the associated structured Hermitian matrix H = diag(H1 , . . . , HN ) and the associated kth inner products [·, ·]Hk , k = 1, . . . , N . Let Pk denote the orthogonal projection in Cγ onto the subspace {0} ⊕ · · · ⊕ {0} ⊕ γ Cγk ⊕ {0} ⊕ · · · ⊕ {0}, and set Ak = AP Pk for k = 1, . . . , N . If x ∈ C is a common eigenvector for A = A1 , . . . , AN corresponding to a common eigenvalue λ = (λ1 , . . . , λN ) ∈ CN , then there exists j ∈ {1, . . . , N } such that |λj | = 1 and [P Pj x, Pj x]Hj = 0. In particular A has no common eigenvalues on TN . The proof of this theorem relies on the equality (1 − |λk |2 )[P Pk x, Pk x]Hk = CP Pk x, CP Pk x,

k = 1, . . . , N,

and follows the same argument as the proof of Theorem 4.16.

Matrix-J-unitary Rational Formal Power Series

87

6. Matrix-J-inner rational formal power series 6.1. A multivariable non-commutative analogue of the half-plane case Let n ∈ N. We deﬁne the matrix open right poly-half-plane as the set 7 N n×n N 6 = Z = (Z1 , . . . , ZN ) ∈ Cn×n : Zk + Zk∗ > 0, k = 1, . . . , N , Π and the matrix closed right poly-half-plane as the set N N = clos Πn×n clos Πn×n 6 7 N = Z = (Z1 , . . . , ZN ) ∈ Cn×n : Zk + Zk∗ ≥ 0, k = 1, . . . , N . We also introduce PN =

0 N Πn×n

and clos PN =

n∈N

It is clear that

0

N clos Πn×n .

n∈N

n×n N N iH ⊂ clos Πn×n N

is the essential (or Shilov ) boundary of the matrix poly-half-plane (Πn×n ) (see 1 N [45]) and that JN ⊂ clos PN (recall that JN = n∈N (iHn×n ) ). Let J = J −1 = J ∗ ∈ Cq×q . A matrix-J-unitary on JN rational FPS F is called matrix-J-inner (in PN ) if for each n ∈ N: F (Z)(J ⊗ In )F (Z)∗ ≤ J ⊗ In

(6.1)

N

at those points Z ∈ clos (Πn×n ) where it is deﬁned (the set of such points is N open and dense, in the relative topology, in clos (Πn×n ) since F (Z) is a rational matrix-valued function of the complex variables (Zk )ij , k = 1, . . . , N, i, j = 1, . . . , n). The following theorem is a counterpart of part a) of Theorem 2.16 of [7]. Theorem 6.1. Let F be a matrix-J-unitary on JN rational FPS and α be its minimal GR-realization of the form (3.11). Then F is matrix-J-inner in PN if and only if the associated structured Hermitian matrix H = diag(H1 , . . . , HN ) is strictly positive. Proof. Let n ∈ N. Equality (4.9) can be rewritten as ∗

∗

J ⊗ In − F (Z)(J ⊗ In )F (Z ) = ϕ(Z)∆(Z + Z )(H −1 ⊗ In )ϕ(Z ) where ϕ is a FPS deﬁned by ϕ(z) := C(IIγ − ∆(z)A)−1 ∈ Cq×γ z1 , . . . , zN rat , and (6.2) is well deﬁned at all points Z, Z ∈ (Cn×n )N for which 1 ∈ σ (∆(Z)(A ⊗ In )) ,

1 ∈ σ (∆(Z )(A ⊗ In )) .

∗

(6.2)

88

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

Set ϕk (z) := C(IIγ − ∆(z)A)−1 -Cγk ∈ Cq×γk z1 , . . . , zN rat , k = 1, . . . , N. Then (6.2) becomes: ∗

J ⊗ In − F (Z)(J ⊗ In )F (Z ) =

N

∗

∗

ϕk (Z)(Hk−1 ⊗ (Zk + Zk ))ϕk (Z ) .

(6.3)

k=1

Let X ∈ Hn×n be some positive semideﬁnite matrix, let Y ∈ (Hn×n )N be such that 1 ∈ σ(∆(iY )(A ⊗ In )), and set for k = 1, . . . , N : ek := (0, 0, . . . , 0, 1, 0, . . . , 0) ∈ CN with 1 at the kth place. Then for λ ∈ C set (k)

Y1 , . . . , iY Yk−1 , λX + iY Yk , iY Yk+1 , . . . , iY YN ). ZX,Y (λ) := λX ⊗ ek + iY = (iY Now, (6.3) implies that ∗

ZX,Y (λ))(J ⊗ In )F (Z ZX,Y (λ )) J ⊗ In − F (Z (k)

(k)

∗

= (λ + λ )ϕk (Z ZX,Y (λ))(Hk−1 ⊗ X)ϕk (Z ZX,Y (λ )) . (k)

(k)

(6.4)

(k)

The function h(λ) = F (Z ZX,Y (λ)) is a rational function of λ ∈ C. It is easily seen from (6.4) that h is (J ⊗ In )-inner in the open right half-plane. In particular, it is (J ⊗ In )-contractive in the closed right half-plane (this also follows directly from (6.1)). Therefore (see, e.g., [22]) the function Ψ(λ, λ ) =

∗

ZX,Y (λ))(J ⊗ In )F (Z ZX,Y )(λ ) J ⊗ In − F (Z (k)

(k)

(6.5) λ + λ is a positive semideﬁnite kernel on C: for every choice of r ∈ N, of points λ1 , . . . , λr ∈ C for which the matrices Ψ(λj , λi ) are well deﬁned, and vectors c1 , . . . , cr ∈ Cq ⊗ Cn one has r

c∗j Ψ(λj , λi )ci ≥ 0,

i,j=1 (k)

ZX,Y (0)) = i.e., the matrix (Ψ(λj , λi ))i,j=1,...,r is positive semideﬁnite. Since ϕk (Z ϕk (iY ) is well deﬁned, we obtain from (6.4) that Ψ(0, 0) is also well deﬁned and Ψ(0, 0) = ϕk (iY )(Hk−1 ⊗ X)ϕk (iY )∗ ≥ 0. This inequality holds for every n ∈ N, every positive semideﬁnite X ∈ Hn×n and every Y ∈ (Hn×n )N . Thus, for an arbitrary r ∈ N we can deﬁne n

= nr, Y = (1) (r) (j) n

× n N (Y 1 , . . . , Y N ) ∈ (H ) , where Y k = diag(Y Yk , . . . , Yk ) and Yk ∈ Hn×n , k = 1, . . . , N, j = 1, . . . , r, such that ϕk (iY ) is well deﬁned, ⎛ ⎞ In · · · In . .. ⎟ ∈ Cn×n ⊗ Cr×r ∼ Cn × n

=⎜ X = ⎝ .. .⎠ In

···

In

Matrix-J-unitary Rational Formal Power Series

89

and get

k (iY )∗ 0 ≤ ϕk (iY )(Hk−1 ⊗ X)ϕ = diag(ϕk (iY (1) ), . . . , ϕk (iY (r) ))× ⎛ ⎞ ⎛ ⎞ In ⎟ ⎜ ⎜ ⎟ × ⎝Hk−1 ⊗ ⎝ ... ⎠ In · · · In ⎠ diag(ϕk (iY (1) )∗ , . . . , ϕk (iY (r) )∗ ) In ⎛ ⎞ ϕk (iY (1) ) ⎜ ⎟ −1 .. =⎝ ⎠ (Hk ⊗ In ) ϕk (iY (1) )∗ . ϕk (iY (r) ) = ϕk (iY (µ) )(Hk−1 ⊗ In )ϕk (iY (ν) )∗

ϕk (iY (r) )∗

···

.

µ,ν=1,...,r

Therefore, the function Kk (iY, iY ) = ϕk (iY )(Hk−1 ⊗ In )ϕk (iY )∗ is a positive semideﬁnite kernel on any subset of (iHn×n )N where it is deﬁned, and in particular in some neighborhood of the origin. One can extend this function to Kk (Z, Z ) = ϕk (Z)(Hk−1 ⊗ In )ϕk (Z )∗

(6.6)

at those points Z, Z ∈ (C ) × (C ) where ϕk is deﬁned. Thus, on some neighborhood Γ of the origin in (Cn×n )N × (Cn×n )N , the function Kk (Z, Z ) is holomorphic in Z and anti-holomorphic in Z . On the other hand, it is well known (see, e.g., [9]) that one can construct a reproducing kernel Hilbert space (which we will denote by H(Kk )) with reproducing kernel Kk (iY, iY ), which is obtained as the completion of H0 = span Kk (·, iY )x ; iY ∈ (iHn×n )N ∩ Γ, x ∈ Cq ⊗ Cn n×n N

n×n N

with respect to the inner product 8 r 9 (µ) (ν) Kk (·, iY )xµ , Kk (·, iY )xν µ=1

=

ν=0 r :

Kk (iY (ν) , iY (µ) )xµ , xν

µ=1 ν=1

H0

; Cq ⊗Cn

.

The reproducing kernel property reads: f (·), Kk (·, iY )xH(Kk ) = f (iY ), xCq ⊗Cn , ∗

and thus Kk (iY, iY ) = Φ(iY )Φ(iY ) where Φ(iY ) : f (·) → f (iY ) is the evaluation map. In view of (6.6), the kernel Kk (·, ·) is extendable on Γ × Γ to the function K(Z, Z ) which is holomorphic in Z and antiholomorphic in Z ,

90

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

all the elements of H(Kk ) have holomorphic continuations to Γ, and so has the function Φ(·). Thus, Kk (Z, Z ) = Φ(Z)Φ(Z )∗ and so Kk (Z, Z ) is a positive semideﬁnite kernel on Γ. (We could also use [3, Theorem 1.1.4, p.10] to obtain this conclusion.) Therefore, for any choice of ∈ N and Z (1) , . . . , Z () ∈ Γ the matrix ϕk (Z (µ) )(Hk−1 ⊗ In )ϕk (Z (ν) )∗ µ,ν=1,..., ⎞ ⎛ ϕk (Z (1) ) (6.7) ⎟ ⎜ .. −1 (1) ∗ () ∗ =⎝ ⊗ I ) · · (H ϕ (Z ) · · · ϕ (Z ) ⎠ n k k k . ϕk (Z () ) is positive semideﬁnite. Since the coeﬃcients of the FPS ϕk are (ϕk )w = (CA)wgk , w ∈ FN , and since α is an observable GR-node, we have ) ker(CA)wgk = {0} . w∈F FN

Hence, by Theorem 2.1 we can chose n, ∈ N and Z (1) , . . . , Z () ∈ Γ such that )

ker ϕk (Z (j) ) = {0} .

j=1

Thus the matrix colj=1,..., ϕk (Z (j) ) has full column rank. (We could also use Theorem 3.7.) From (6.7) it then follows that Hk−1 > 0. Since this holds for all k ∈ {1, . . . , N }, we get H > 0. Conversely, if H > 0 then it follows from (6.2) that for every n ∈ N and N Z ∈ (Πn×n ) for which 1 ∈ σ(∆(Z)(A ⊗ In )), one has J ⊗ In − F (Z)(J ⊗ In )F (Z)∗ ≥ 0. Therefore F is matrix-J-inner in PN , and the proof is complete.

Theorem 6.2. Let F ∈ C z1 , . . . , zN rat be matrix-J-inner in PN . Then F has a minimal GR-realization of the form (3.11) with the associated structured Hermitian matrix H = Iγ . This realization is unique up to a unitary similarity. q×q

Proof. Let α◦ = (N ; A◦ , B ◦ , C ◦ , D; Cγ =

N '

Cγ k , Cq )

k=1

be a minimal GR-realization of F , with the associated structured Hermitian ma◦ trix H ◦ = diag(H1◦ , . . . , HN ). By Theorem 6.1 the matrix H ◦ is strictly positive. ◦ 1/2 ◦ 1/2 Therefore, (H ) = diag((H1◦ )1/2 , . . . , (H HN ) ) is well deﬁned and strictly positive, and N ' Cγk , Cq ), α = (N ; A, B, C, D; Cγ = k=1

Matrix-J-unitary Rational Formal Power Series

91

where A = (H ◦ )1/2 A◦ (H ◦ )−1/2 ,

B = (H ◦ )1/2 B ◦ ,

C = C ◦ (H ◦ )−1/2 ,

(6.8)

is a minimal GR-realization of F satisfying A∗ + A =

−C ∗ JC,

(6.9)

=

∗

−C JD,

(6.10)

A∗ + A = C =

−BJB ∗ , −DJB ∗ ,

(6.11) (6.12)

B or equivalently,

and thus having the associated structured Hermitian matrix H = Iγ . Since in this case the inner product [ · , · ]H coincides with the standard inner product · , · of Cγ , by Remark 4.6 this minimal GR-realization with the property H = Iγ is unique up to unitary similarity. We remark that a one-variable counterpart of the latter result is essentially contained in [20], [38] (see also [10, Section 4.2]). 6.2. A multivariable non-commutative analogue of the disk case Let n ∈ N. We deﬁne the matrix open unit polydisk as 7 N n×n N 6 D = W = (W W1 , . . . , WN ) ∈ Cn×n : Wk Wk∗ < In , k = 1, . . . , N , and the matrix closed unit polydisk as N N clos Dn×n = clos Dn×n 6 7 N = W = (W W1 , . . . , WN ) ∈ Cn×n : Wk Wk∗ ≤ In , k = 1, . . . , N . N

The matrix unit torus (Tn×n ) is the essential (or Shilov) boundary of (Dn×n ) (see [45]). In our setting, the set 0 0 n×n N n×n N DN = D resp., clos DN = clos D n∈N

N

n∈N

is a multivariable non-commutative counterpart of the open (resp., closed) unit disk. Let J = J −1 = J ∗ ∈ Cq×q . A rational FPS f which is matrix-J-unitary on TN is called matrix-J-inner in DN if for every n ∈ N: f (W )(J ⊗ In )f (W )∗ ≤ J ⊗ In N

(6.13)

at those points W ∈ clos (Dn×n ) where it is deﬁned. We note that the set of N such points is open and dense (in the relative topology) in clos (Dn×n ) since f (W ) is a rational matrix-valued function of the complex variables (W Wk )ij , k = 1, . . . , N, i, j = 1, . . . , n.

92

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

Theorem 6.3. Let f be a rational FPS which is matrix-J-unitary on TN , and let α be its minimal GR-realization of the form (3.11). Then f is matrix-J-inner in DN if and only if the associated structured Hermitian matrix H = diag(H1 , . . . , HN ) is strictly positive. Proof. The statement of this theorem follows from Theorem 6.1 and Theorem 5.1, since the Cayley transform deﬁned in Theorem 5.1 maps each open matrix unit polydisk (Dn×n )N onto the open right matrix poly-half-plane (Πn×n )N , and the inequality (6.13) turns into (6.1) for the function F deﬁned in (5.1). The following theorem is an analogue of Theorem 6.2. Theorem 6.4. Let f be a rational FPS which is matrix-J-inner in DN . Then there exists its minimal GR-realization α of the form (3.11), with the associated structured Hermitian matrix H = Iγ . Such a realization is unique up to a unitary similarity. In the special case of Theorem 6.4 where J = Iq the FPS f is called matrixinner, and the GR-node α satisﬁes ∗ A B A B = Iγ +q , C D C D i.e., α is a unitary GR-node, which has been considered ﬁrst by J. Agler in [1]. In what follows we will show that Theorem 6.4 for J = Iq is a special case of the theorem of J. A. Ball, G. Groenewald and T. Malakorn on unitary GR-realizations of FPSs from the non-commutative Schur–Agler class [12], which becomes in several aspects stronger in this special case. Let U and Y be Hilbert spaces. Denote by L(U, Y) the Banach space of bounded linear operators from U into Y. A GR-node in the general setting of Hilbert spaces is α = (N ; A, B, C, D; X =

N '

Xk , U, Y),

k=1

i.e., a collection of Hilbert spaces X , X1 , . . . , XN , U, Y and operators A ∈ L(X ) = L(X , X ), B ∈ L(U, X ), C ∈ L(X , Y), and D ∈ L(U, Y). Such a GR-node α is called unitary if ∗ ∗ A B A B A B A B = IX ⊕Y , = IX ⊕U , C D C D C D C D

A B i.e., C is a unitary operator from X ⊕ U onto X ⊕ Y. The non-commutative D transfer function of α is

Tαnc (z) = D + C(I − ∆(z)A)−1 ∆(z)B,

(6.14)

Matrix-J-unitary Rational Formal Power Series

93

where the expression (6.14) is understood as a FPS from L(U, Y) z1 , . . . , zN given by ∞ w k (CAB) z w = D + C (∆(z)A) ∆(z)B. (6.15) Tαnc (z) = D + w∈F FN \{∅}

k=0

The non-commutative Schur–Agler class SAnc N (U, Y) consists of all FPSs f ∈ L(U, Y) z1 , . . . , zN such that for any separable Hilbert space K and any N tuple δ = (δ1 , . . . , δN ) of strict contractions in K the limit in the operator norm topology fw ⊗ δ w f (δ) = lim m→∞

w∈F FN : |w|≤m

exists and deﬁnes a contractive operator f (δ) ∈ L(U ⊗ K, Y ⊗ K). We note that the non-commutative Schur–Agler class was deﬁned in [12] also for a more general class of operator N -tuples δ. ). Consider another set of non-commuting indeterminates z = (z1 , . . . , zN For f (z) ∈ L(V, Y) z1 , . . . , zN and f (z ) ∈ L(V, U) z1 , . . . , zN we deﬁne a FPS ∗ f (z)f (z ) ∈ L(U, Y) z1 , . . . , zN , z1 , . . . , zN by w T ∗ ∗ f (z)f (z ) = fw (ffw ) z w z . (6.16) w,w ∈F FN

In [12] the class

SAnc N (U, Y)

was characterized as follows:

Theorem 6.5. Let f ∈ L(U, Y) z1 , . . . , zN . The following statements are equivalent: (1) f ∈ SAnc N (U, Y); (2) there exist auxiliary Hilbert spaces H, H1 , . . . , HN which are related by H = (N k=1 Hk , and a FPS ϕ ∈ L(H, Y) z1 , . . . , zN such that ∗

∗

IY − f (z)f (z ) = ϕ(z)(IIH − ∆(z)∆(z )∗ )ϕ(z ) ; (6.17) (N (3) there exists a unitary GR-node α = (N ; A, B, C, D; X = k=1 Xk , U, Y) such that f = Tαnc. We now give another characterization of the Schur–Agler class SAnc N (U, Y). Theorem 6.6. A FPS f belongs to SAnc N (U, Y) if and only if for every n ∈ N and W ∈ (Dn×n )N the limit in the operator norm topology f (W ) = lim fw ⊗ W w (6.18) m→∞

w∈F FN : |w|≤m

exists and f (W ) ≤ 1. Proof. The necessity is clear. We prove the suﬃciency. We set fk (z) = fw z w , k = 0, 1, . . . . w∈F FN : |w|=k

94

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

Then for every n ∈ N and W ∈ (Dn×n )N , (6.18) becomes f (W ) = lim

m→∞

m

fk (W ),

(6.19)

k=0

where the limit is taken in the operator norm topology. Let r ∈ (0, 1) and choose τ > 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 W x λW x = λ fk 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

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close