I. Gohberg – författare

This book provides a comprehensive treatment of the theory of polynomials in a complex variable with matrix coefficients. Basic matrix theory can be viewed as the study of the special case of polynomials of first degree; the theory developed in Matrix Polynomials is a natural extension of this case to polynomials of higher degree. It has applications in many areas, such as differential equations, systems theory, the Wiener–Hopf technique, mechanics and vibrations, and numerical analysis. Although there have been significant advances in some quarters, this work remains the only systematic development of the theory of matrix polynomials.

One of the basic interpolation problems from our point of view is the problem of building a scalar rational function if its poles and zeros with their multiplicities are given. If one assurnes that the function does not have a pole or a zero at infinity, the formula which solves this problem is (1) where Zl , " " Z/ are the given zeros with given multiplicates nl, " " n / and Wb" " W are the given p poles with given multiplicities ml, . . . ,m , and a is an arbitrary nonzero number. p An obvious necessary and sufficient condition for solvability of this simplest Interpolation pr- lern is that Zj :f: wk(1~ j ~ 1, 1~ k~ p) and nl +. . . +n/ = ml +. . . +m '' p The second problem of interpolation in which we are interested is to build a rational matrix function via its zeros which on the imaginary line has modulus 1. In the case the function is scalar, the formula which solves this problem is a Blaschke product, namely z z. )mi n u(z) = all = l~ (2) J ( Z+ Zj where [o] = 1, and the zj''s are the given zeros with given multiplicities mj. Here the necessary and sufficient condition for existence of such u(z) is that zp :f: - Zq for 1~ ]1, q~ n.

One of the basic interpolation problems from our point of view is the problem of building a scalar rational function if its poles and zeros with their multiplicities are given. If one assurnes that the function does not have a pole or a zero at infinity, the formula which solves this problem is (1) where Zl , " " Z/ are the given zeros with given multiplicates nl, " " n / and Wb" " W are the given p poles with given multiplicities ml, . . . ,m , and a is an arbitrary nonzero number. p An obvious necessary and sufficient condition for solvability of this simplest Interpolation pr- lern is that Zj :f: wk(1~ j ~ 1, 1~ k~ p) and nl +. . . +n/ = ml +. . . +m ' p The second problem of interpolation in which we are interested is to build a rational matrix function via its zeros which on the imaginary line has modulus 1. In the case the function is scalar, the formula which solves this problem is a Blaschke product, namely z z. )mi n u(z) = all = l~ (2) J ( Z+ Zj where [o] = 1, and the zj's are the given zeros with given multiplicities mj. Here the necessary and sufficient condition for existence of such u(z) is that zp :f: - Zq for 1~ ]1, q~ n.

This paper is a largely expository account of the theory of p x p matrix polyno­ mials associated with Hermitian block Toeplitz matrices and some related problems of interpolation and extension. Perhaps the main novelty is the use of reproducing kernel Pontryagin spaces to develop parts of the theory in what hopefully the reader will regard as a reasonably lucid way. The topics under discussion are presented in a series of short sections, the headings of which give a pretty good idea of the overall contents of the paper. The theory is a rich one and the present paper in spite of its length is far from complete. The author hopes to fill in some of the gaps in future publications. The story begins with a given sequence h_n" ... , hn of p x p matrices with h-i = hj for j = 0, ... , n. We let k = O, ... ,n, (1.1) denote the Hermitian block Toeplitz matrix based on ho, ... , hk and shall denote its 1 inverse H k by (k)] k [ r = .. k = O, ... ,n, (1.2) k II} . ''-0 '' I- whenever Hk is invertible.

This paper is a largely expository account of the theory of p x p matrix polyno­ mials associated with Hermitian block Toeplitz matrices and some related problems of interpolation and extension. Perhaps the main novelty is the use of reproducing kernel Pontryagin spaces to develop parts of the theory in what hopefully the reader will regard as a reasonably lucid way. The topics under discussion are presented in a series of short sections, the headings of which give a pretty good idea of the overall contents of the paper. The theory is a rich one and the present paper in spite of its length is far from complete. The author hopes to fill in some of the gaps in future publications. The story begins with a given sequence h_n" ... , hn of p x p matrices with h-i = hj for j = 0, ... , n. We let k = O, ... ,n, (1.1) denote the Hermitian block Toeplitz matrix based on ho, ... , hk and shall denote its 1 inverse H k by (k)] k [ r = .. k = O, ... ,n, (1.2) k II} . '-0 ' I- whenever Hk is invertible.

VI A. S. MARKUS, A. A. SEMENCUL und 1. B. SIMONENKO für die Diskussionen über verschiedene Fragen und für ihre wertvollen Bemerkungen. Die Autoren bringen ihre Dankbarkeit dem Redakteur des Buches, F. V. SIROKOV, zum Ausdruck. Seine Hilfe trug maßgeblich zur einfachen und exakten Darlegung bei. Kisinev, am 18. Februar 1970 VORWORT ZUR DEUTSCHEN AUSGABE Die vorliegende Ausgabe dieses Buches unterscheidet sich nur in einem Teil wesentlich von dem russischen Original. Es handelt sich dabei um den Schluß des dritten Kapitels, wo Verfahren zur Umkehrung endlicher TOEPLITz-Matrizen und ihrer stetigen Analoga dargelegt werden. Die beiden letzten Paragraphen von KapitelIII (§ 6 und § 7) der russischen Ausgabe sind durch drei neue Paragraphen (§ 6, § 7, § 8) ersetzt worden. Die neue Darlegung ist vollständiger und zeichnet sich auch durch größere Allgemeinheit und Einfachheit aus. Darüber hinaus sind die Literaturhinweise sowie das Literaturverzeichnis er­ weitert worden. Es wurden einige unbedeutende Druckfehler berichtigt. Die Autoren danken aufrichtig Herrn Prof. Dr. S. PRÖSSDORF, der der Initiator dieser übersetzung ist, sowie dem Akademie-Verlag und den beiden übersetzern, Herrn Dr. J. LEITERER und Herrn Dr. R. LEHMANN. Kisinev Die Autoren 1. Mai 1972 INHALTSVERZEICHNIS Einführung ................................................................ 1 Kapitel I. Allgemeine Sätze über WIENER-HoPF-Gleichungen ...................... 9 § 1. Polynome von einseitig umkehrbaren Operatoren ......................... 9 1. Einige Hilfssätze. ................................................. 9 2. Einseitig umkehrbare Operatoren. . . . . . . . . . . . . . . . . . . . . . . . . . 11 . . . . . . . . . • 3. Umkehrung von Polynomen von einseitig umkehrbaren Operatoren....... 16 § 2. Stetige Funktionen von einseitig umkehrbaren Operatoren. . . . .. . . . . . . . . . 18 .

VI A. S. MARKUS, A. A. SEMENCUL und 1. B. SIMONENKO für die Diskussionen über verschiedene Fragen und für ihre wertvollen Bemerkungen. Die Autoren bringen ihre Dankbarkeit dem Redakteur des Buches, F. V. SIROKOV, zum Ausdruck. Seine Hilfe trug maßgeblich zur einfachen und exakten Darlegung bei. Kisinev, am 18. Februar 1970 VORWORT ZUR DEUTSCHEN AUSGABE Die vorliegende Ausgabe dieses Buches unterscheidet sich nur in einem Teil wesentlich von dem russischen Original. Es handelt sich dabei um den Schluß des dritten Kapitels, wo Verfahren zur Umkehrung endlicher TOEPLITz-Matrizen und ihrer stetigen Analoga dargelegt werden. Die beiden letzten Paragraphen von KapitelIII (§ 6 und § 7) der russischen Ausgabe sind durch drei neue Paragraphen (§ 6, § 7, § 8) ersetzt worden. Die neue Darlegung ist vollständiger und zeichnet sich auch durch größere Allgemeinheit und Einfachheit aus. Darüber hinaus sind die Literaturhinweise sowie das Literaturverzeichnis er­ weitert worden. Es wurden einige unbedeutende Druckfehler berichtigt. Die Autoren danken aufrichtig Herrn Prof. Dr. S. PRÖSSDORF, der der Initiator dieser übersetzung ist, sowie dem Akademie-Verlag und den beiden übersetzern, Herrn Dr. J. LEITERER und Herrn Dr. R. LEHMANN. Kisinev Die Autoren 1. Mai 1972 INHALTSVERZEICHNIS Einführung ................................................................ 1 Kapitel I. Allgemeine Sätze über WIENER-HoPF-Gleichungen ...................... 9 § 1. Polynome von einseitig umkehrbaren Operatoren ......................... 9 1. Einige Hilfssätze. ................................................. 9 2. Einseitig umkehrbare Operatoren. . . . . . . . . . . . . . . . . . . . . . . . . . 11 . . . . . . . . . • 3. Umkehrung von Polynomen von einseitig umkehrbaren Operatoren....... 16 § 2. Stetige Funktionen von einseitig umkehrbaren Operatoren. . . . .. . . . . . . . . . 18 .

This book aims to present the theory of interpolation for rational matrix functions as a recently matured independent mathematical subject with its own problems, methods and applications. The authors decided to start working on this book during the regional CBMS conference in Lincoln, Nebraska organized by F. Gilfeather and D. Larson. The principal lecturer, J. William Helton, presented ten lectures on operator and systems theory and the interplay between them. The conference was very stimulating and helped us to decide that the time was ripe for a book on interpolation for matrix valued functions (both rational and non-rational). When the work started and the first partial draft of the book was ready it became clear that the topic is vast and that the rational case by itself with its applications is already enough material for an interesting book. In the process of writing the book, methods for the rational case were developed and refined. As a result we are now able to present the rational case as an independent theory. After two years a major part of the first draft was prepared. Then a long period of revising the original draft and introducing recently acquired results and methods followed. There followed a period of polishing and of 25 chapters and the appendix commuting at various times somewhere between Williamsburg, Blacksburg, Tel Aviv, College Park and Amsterdam (sometimes with one or two of the authors).

This book aims to present the theory of interpolation for rational matrix functions as a recently matured independent mathematical subject with its own problems, methods and applications. The authors decided to start working on this book during the regional CBMS conference in Lincoln, Nebraska organized by F. Gilfeather and D. Larson. The principal lecturer, J. William Helton, presented ten lectures on operator and systems theory and the interplay between them. The conference was very stimulating and helped us to decide that the time was ripe for a book on interpolation for matrix valued functions (both rational and non-rational). When the work started and the first partial draft of the book was ready it became clear that the topic is vast and that the rational case by itself with its applications is already enough material for an interesting book. In the process of writing the book, methods for the rational case were developed and refined. As a result we are now able to present the rational case as an independent theory. After two years a major part of the first draft was prepared. Then a long period of revising the original draft and introducing recently acquired results and methods followed. There followed a period of polishing and of 25 chapters and the appendix commuting at various times somewhere between Williamsburg, Blacksburg, Tel Aviv, College Park and Amsterdam (sometimes with one or two of the authors).

On November 12-14, 1997 a workshop was held at the Vrije Universiteit Amsterdam on the occasion of the sixtieth birthday ofM. A. Kaashoek. The present volume contains the proceedings of this workshop. The workshop was attended by 44 participants from all over the world: partici­ pants came from Austria, Belgium, Canada, Germany, Ireland, Israel, Italy, The Netherlands, South Africa, Switzerland, Ukraine and the USA. The atmosphere at the workshop was very warm and friendly. There where 21 plenary lectures, and each lecture was followed by a lively discussion. The workshop was supported by: the Vakgroep Wiskunde of the Vrije Univer­ siteit, the department of Mathematics and Computer Science of the Vrije Univer­ siteit, the Stichting VU Computer Science & Mathematics Research Centre, the Thomas Stieltjes Institute for Mathematics, and the department of Economics of the Erasmus University Rotterdam. The organizers would like to take this opportunity to express their gratitude for the support. Without it the workshop would not have been so successful as it was. Table of Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Photograph of M. A. Kaashoek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Curriculum Vitae of M. A. Kaashoek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv List of Publications of M. A. Kaashoek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix l. Gohberg Opening Address . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi H. Bart, A. C. M. Ran and H. I. Woerdeman Personal Reminiscences . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv V. Adamyan and R. Mennicken On the Separation of Certain Spectral Components of Selfadjoint Operator Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Conditions for the Separation of Spectral Components . . . . . . . 4 3. Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Our goal is to find Grabner bases for polynomials in four different sets of expressions: 1 x- , (1 - x)-1 (RESOL) X, 1 x- (1 - xy)-1 (EB) X, , y-1, (1-yx)-1 y, (1_y)-1 (1-x)-1 (preNF) (EB) plus and (1 - xy)1/2 (1 - yx )1/2 (NF) (preNF) plus and Most formulas in the theory of the Nagy-Foias operator model [NF] are polynomials in these expressions where x = T and y = T*. Complicated polynomials can often be simplified by applying "replacement rules". For example, the polynomial (1 - xy)-2 - 2xy(1-xy)-2 + xy2 (1 - xy)-2 -1 simplifies to O. This can be seen by three applications of the replacement rule (1-xy) -1 xy -t (1 - xy)-1 -1 which is true because of the definition of (1-xy)-1. A replacement rule consists of a left hand side (LHS) and a right hand side (RHS). The LHS will always be a monomial. The RHS will be a polynomial whose terms are "simpler" (in a sense to be made precise) than the LHS. An expression is reduced by repeatedly replacing any occurrence of a LHS by the corresponding RHS. The monomials will be well-ordered, so the reduction procedure will terminate after finitely many steps. Our aim is to provide a list of substitution rules for the classes of expressions above. These rules, when implemented on a computer, provide an efficient automatic simplification process. We discuss and define the ordering on monomials later.

This volume is dedicated to Harold Widom, a distinguished mathematician and renowned expert in the area of Toeplitz, Wiener-Hopf and pseudodifferential operators, on the occasion of his sixtieth birthday. The book opens with biographical material and a list of the mathematician''s publications, this being followed by two papers based on Toeplitz lectures which he delivered at Tel Aviv University in March, 1993. The rest of the book consists of a selection of papers containing some recent achievements in the following areas: Szegö-Widom asymptotic formulas for determinants of finite sections of Toeplitz matrices and their generalizations, the Fisher-Hartwig conjecture, random matrices, analysis of kernels of Toeplitz matrices, projectional methods and eigenvalue distribution for Toeplitz matrices, the Fredholm theory for convolution type operators, the Nehari interpolation problem with generalizations and applications, and Toeplitz-Hausdorff type theorems. The book will appeal to a wide audience of pure and applied mathematicians.

This volume consists of eight papers containing recent advances in interpolation theory for matrix functions and completion theory for matrices and operators. In the first paper, D. Alpay and P. Loubaton, "The tangential trigonometric moment problem on an interval and related topics" a trigonometric moment problem on an interval for matrix valued functions is studied. The realization approach plays an important role in solving this problem. The second paper, M. Bakonyi, V.G. Kaftal, G. Weiss and H.J. Woerdeman, "Max­ imum entropy and joint norm bounds for operator extensions" is dedicated to a matrix completion problem. In it is considered the problem when only the lower triangular part of the operator entries of a matrix is identified. Completions which have simultaneously a small usual norm and a small Hilbert-Schmidt norm are considered. Bounds for these norms are obtained. The analysis of the maximum entropy extension plays a special role. The paper contains applications to nest algebras and integral operators. The third paper, J .A. Ball, I. Gohberg and M.A. Kaashoek, "Bitangential interpola­ tion for input-output operators of time varying systems: the discrete time case" contains solutions of time varying interpolation problems. The main attention is focused on the time varying analog of the Nevanlinna-Pick tangential problem in the case where the inter­ polation conditions appear from two sides. The state space theory of time varying systems play an important role.

This volume is dedicated to Tsuyoshi Ando, a foremost expert in operator theory, matrix theory, complex analysis, and their applications, on the occasion of his 60th birthday. The book opens with his biography and list of publications. It contains a selection of papers covering a broad spectrum of topics ranging from abstract operator theory to various concrete problems and applications. The majority of the papers deal with topics in modern operator theory and its applications. This volume also contains papers on interpolation and completion problems, factorization problems and problems connected with complex analysis. The book will appeal to a wide audience of pure and applied mathematicians.

This monograph is the second volume of a graduate text book on the modern theory of linear one-dimensional singular integral equations. Both volumes may be regarded as unique graduate text books. Singular integral equations attract more and more attention since this class of equations appears in numerous applications, and also because they form one of the few classes of equations which can be solved explicitly. The present book is to a great extent based upon material contained in the second part of the authors'' monograph [6] which appeared in 1973 in Russian, and in 1979 in German translation. The present text includes a large number of additions and complementary material, essentially changing the character, structure and contents of the book, and making it accessible to a wider audience. Our main subject in the first volume was the case of closed curves and continuous coeffi­ cients. Here, in the second volume, we turn to general curves and discontinuous coefficients. We are deeply grateful to the editor Professor G. Heinig, to the translator Dr. S. Roeh, and to the typist Mr. G. Lillack, for their patient work. The authors Ramat-Aviv, Ramat-Gan, May 26, 1991 11 Introduction This book is the second volume of an introduction to the theory of linear one-dimensional singular integral operators. The main topics of both parts of the book are the invertibility and Fredholmness of these operators. Special attention is paid to inversion methods.

This book is an introduction to the theory of linear one-dimensional singular integral equations. It is essentually a graduate textbook. Singular integral equations have attracted more and more attention, because, on one hand, this class of equations appears in many applications and, on the other, it is one of a few classes of equations which can be solved in explicit form. In this book material of the monograph [2] of the authors on one-dimensional singular integral operators is widely used. This monograph appeared in 1973 in Russian and later in German translation [3]. In the final text version the authors included many addenda and changes which have in essence changed character, structure and contents of the book and have, in our opinion, made it more suitable for a wider range of readers. Only the case of singular integral operators with continuous coefficients on a closed contour is considered herein. The case of discontinuous coefficients and more general contours will be considered in the second volume. We are grateful to the editor Professor G. Heinig of the volume and to the translators Dr. B. Luderer and Dr. S. Roch, and to G. Lillack, who did the typing of the manuscript, for the work they have done on this volume.

This book presents a unified approach for solving both stationary and nonstationary interpolation problems, in finite or infinite dimensions, based on the commutant lifting theorem from operator theory and the state space method from mathematical system theory. Initially the authors planned a number of papers treating nonstationary interpolation problems of Nevanlinna-Pick and Nehari type by reducing these nonstationary problems to stationary ones for operator-valued functions with operator arguments and using classical commutant lifting techniques. This reduction method required us to review and further develop the classical results for the stationary problems in this more general framework. Here the system theory turned out to be very useful for setting up the problems and for providing natural state space formulas for describing the solutions. In this way our work involved us in a much wider program than original planned. The final results of our efforts are presented here. The financial support in 1994 from the "NWO-stimulansprogramma" for the Thomas Stieltjes Institute for Mathematics in the Netherlands enabled us to start the research which lead to the present book. We also gratefully acknowledge the support from our home institutions: Indiana University at Bloomington, Purdue University at West Lafayette, Tel-Aviv University, and the Vrije Universiteit at Amsterdam. We warmly thank Dr. A.L. Sakhnovich for his carefully reading of a large part of the manuscript. Finally, Sharon Wise prepared very efficiently and with great care the troff file of this manuscript; we are grateful for her excellent typing.

The present volume contains the proceedings of the International Conference on Ap­ plications of Operator Theory held in Winnipeg, Canada (October 2nd to 6th, 1994), which was organized by the Institute of Industrial Mathematical Sciences (IIMS) of the University of Manitoba. At this conference 92 participants representing 15 countries par­ ticipated, and 64 papers were presented. This meeting was the second of a linked pair. The first was a program of advanced instruction held at the Fields Institute, Ontario, followed by a research conference. The first of these events gave rise to the volume "Lectures on Operator Theory and its Applications", published by the American Mathematical Society for the Fields Institute in 1995. These two events were the creation of the following Program Committee: M. A. Dahleh (M. I. T. ) P. A. Fillmore (Dalhousie) B. A. Francis (Toronto) F. Ghahramani (Manitoba) K. Glover (Cambridge) I. Gohberg (Tel Aviv) T. Kailath (Stanford) P. Lancaster (Calgary), Chair H. Langer (Vienna) P. N. Shivakumar (Manitoba) A. A. Shkalikov (Moscow) B. Simon (Cal. Tech. ) H. Widom (Santa Cruz) Both events focused on the following main topics: Infinite matrices and projection methods, linear operators on indefinite scalar product spaces, differential operators and mathematical systems theory and control. This volume contains a selection of papers in modern operator theory and its appli­ cations. They are dedicated to recent achievements and many are written by leaders in the mentioned fields.