Bücher der Reihe Operator Theory: Advances and Applications

This is the first of two volumes containing peer-reviewed research and survey papers based on talks at the International Conference on Modern Analysis and Applications. The papers describe the contemporary development of subjects influenced by Mark Krein.

This is the second of two volumes containing peer-reviewed research and survey papers based on talks at the International Conference on Modern Analysis and Applications. The papers describe the contemporary development of subjects influenced by Mark Krein.

The specific character of the theory of series manifests itself when one considers rearrangements (permutations) of the terms of a series, which brings combinatorial considerations into the problems studied.

Bridges the gap between nonlinear analysis, nonlinear operator equations and the theory of holomorphic mappings on Banach spaces. This book concludes with a brief exposition of the theory of spaces with indefinite metrics, and some relevant applications of the holomorphic mappings theory in this setting.

This text is dedicated to the memory of Israel Glazman, who was the author of many papers and books on operator theory and its applications. The book opens with an essay on Glazman's life and scientific achievements, and it then focuses on the areas of his unusually wide interests,

Periodic differential operators have a rich mathematical theory as well as important physical applications. This book lays out the theoretical foundations and then moves on to give a coherent account of more recent results, relating in particular to the eigenvalue and spectral theory of the Hill and Dirac equations.

This research monograph contains a study of discrete-time nodes, the discrete counterpart of the theory elaborated by Bart, Gohberg and Kaashoek for the continuous case, discrete-time Lyapunov and Riccati equations, discrete-time Hamiltonian systems in connection with input-output operators and associated Hankel and Toeplitz operators.

These 35 refereed articles report on recent and original results in various areas of operator theory and connected fields, many of them strongly related to contributions of Sz.-Nagy. The scientific part of the book is preceeded by fifty pages of biographical material, including several photos.

The notions of positive functions and of reproducing kernel Hilbert spaces play an important role in various fields of mathematics, such as stochastic processes, linear systems theory, operator theory, and the theory of analytic functions.

This book explores new difference schemes for approximating the solutions of regular and singular perturbation boundary-value problems for PDEs.

The specific character of the theory of series manifests itself when one considers rearrangements (permutations) of the terms of a series, which brings combinatorial considerations into the problems studied.

This volume contains the proceedings of the International Workshop on Operator Theory and Applications held at the University of Algarve in Faro, Portugal, September 12-15, in the year 2000. The main topics of the conference were !> Factorization Theory; !> Operator Theoretical Methods in Diffraction Theory;

This volume is dedicated to Bernd Silbermann on the oc­ casion of his sixtieth birthday. It consists of selected papers devoted to the inexhaustible and ever-young fields of Toeplitz matrices and singular integral equations, and thus to areas Bernd Silbermann has been enriching by fundamental con­ tributions for the last three decades. Most authors of this volume participated in the conference organized and sponsored by the Department of Mathematics (under the deans Dieter Happel and Jiirgen vom Scheidt) of the Chemnitz University of Technology in honor of Bernd Silbermann in Pobershau, April 8-12, 2001. The majority of the papers presented here are based on the talks given on that conference. We thank all contributors for their enthusiasm when preparing the articles for this volume. BERND SILBERMANN Operator Theory: Advances and Applications, Vol. 135, 1-12 © 2002 Birkhiiuser Verlag Basel/Switzerland Essay on Bernd Silbermann Albrecht Bottcher Bernd Silbermann was born on 6 April 1941 in Langhennersdorf, a village in Sax­ ony. His parents were farmers. To this day, he is proud of his ability to drive a tractor. He went to school in Langhennersdorffrom 1947 to 1955, and in the sub­ sequent two years he apprenticed to a grocer (and really sold fish). From 1958 to 1962 he attended a school in Chemnitz, and from 1962 to 1967, in the heyday of Soviet mathematics, he was a student of mathematics at the Lomonosov University in Moscow. His lecturers included such eminent mathematicians as P. S.

We offer the reader of this book some specimens of "infinity" that we seized from the "mathematical jungle" and trapped within the solid cage of analysis The creation of the theory of singular integral equations in the mid 20th century is associated with the names of N.1. Muskhelishvili, F.D. Gakhov, N.P. Vekua and their numerous students and followers and is marked by the fact that it relied principally on methods of complex analysis. In the early 1960s, the development of this theory received a powerful impulse from the ideas and methods of functional analysis that were then brought into the picture. Its modern architecture is due to a constellation of brilliant mathemati­ cians and the scientific collectives that they produced (S.G. Mikhlin, M.G. Krein, B.V. Khvedelidze, 1. Gohberg, LB. Simonenko, A. Devinatz, H. Widom, R.G. Dou­ glas, D. Sarason, A.P. Calderon, S. Prossdorf, B. Silbermann, and others). In the ensuing period, the Fredholm theory of singular integral operators with a finite index was completed in its main aspects in wide classes of Banach and Frechet spaces.

These papers are followed by a leading article entitled Sonja Kovalevsky: Her life and professorship in Stockholm, written especially for this volume by Jan-Erik Bjork in preparation for his major address to the Symposium.

We have considered writing the present book for a long time, since the lack of a sufficiently complete textbook about complex analysis in infinite dimensional spaces was apparent. There are, however, some separate topics on this subject covered in the mathematical literature. For instance, the elementary theory of holomorphic vector­ functions.and mappings on Banach spaces is presented in the monographs of E. Hille and R. Phillips [1] and L. Schwartz [1], whereas some results on Banach algebras of holomorphic functions and holomorphic operator-functions are discussed in the books of W. Rudin [1] and T. Kato [1]. Apparently, the need to study holomorphic mappings in infinite dimensional spaces arose for the first time in connection with the development of nonlinear anal­ ysis. A systematic study of integral equations with an analytic nonlinear part was started at the end ofthe 19th and the beginning ofthe 20th centuries by A. Liapunov, E. Schmidt, A. Nekrasov and others. Their research work was directed towards the theory of nonlinear waves and used mainly the undetermined coefficients and the majorant power series methods. The most complete presentation of these methods comes from N. Nazarov. In the forties and fifties the interest in Liapunov's and Schmidt's analytic methods diminished temporarily due to the appearence of variational calculus meth­ ods (M. Golomb, A. Hammerstein and others) and also to the rapid development of the mapping degree theory (J. Leray, J. Schauder, G. Birkhoff, O. Kellog and others).

The last decades have demonstrated that quantum mechanics is an inexhaustible source of inspiration for contemporary mathematical physics.

That interfusion turned out to have great advan tages and gave rise to a vast number of significant results, of which we want to mention especially the classical results on the theory of Fourier series in L2 ( -7r, 7r) and their continual analog - Plancherel's theorem on the Fourier transform in L2 ( -00, +00).

Uncertainty principles for time-frequency operators.- 1. Introduction.- 2. Sampling results for time-frequency transformations.- 3. Uncertainty principles for exact Gabor and wavelet frames.- References.- Distribution of zeros of matrix-valued continuous analogues of orthogonal polynomials.- 1. Preliminary results.- 1.1. Matrix-valued Krein functions of the first and second kinds.- 1.2. Partitioned integral operators.- 2. Orthogonal operator-valued polynomials.- 2.1. Stein equations for operators.- 2.2. Zeros of orthogonal polynomials.- 2.3. On Toeplitz matrices with operator entries.- 3. Zeros of mat rix-valued Krein functions.- 3.1 On Wiener-Hopf operators.- 3.2. Proof of the main theorem.- References.- The band extension of the real line as a limit of discrete band extensions, II. The entropy principle.- 0. Introduction.- I. Preliminaries.- II. Main results.- References.- Weakly positive matrix measures, generalized Toeplitz forms, and their applications to Hankel and Hilbert transform operators.- 1. Lifting properties of generalized Toeplitz forms and weakly positive matrix measures.- 2. The GBT and the theorems of Helson-Szeg¿ and Nehari.- 3. GNS construction, Wold decomposition and abstract lifting theorems.- 4. Multiparameter and n-conditional lifting theorems, the A-A-K theorem and applications in several variables.- References.- Reduction of the abstract four block problem to a Nehari problem.- 0. Introduction.- 1. Main theorems.- 2. Proofs of the main theorems.- References.- The state space method for integro-differential equations of Wiener-Hopf type with rational matrix symbols.- 1. Introduction and main theorems.- 2. Preliminaries on matrix pencils.- 3. Singular differential equations on the full-line.- 4. Singular differential equations on the half-line.- 5. Preliminaries on realizations.- 6. Proof of theorem 1.1.- 7. Proofs of theorems 1.2 and 1.3.- 8. An example.- References.- Symbols and asymptotic expansions.- 0. Introduction.- I. Smooth symbols on Rn.- II. Piecewise smooth symbols on T.- III. Piecewise smooth symbols on Rn.- IV. Symbols discontinuous across a hyperplane in Rn ¿Rn.- References.- Program of Workshop.

This volume contains the proceedings of the international conference on " Operator Calculus and Spectral Theory" in Lambrecht, Germany, Decem­ ber 9. - 14. , 1991, sponsored by the Deutsche Forschungsgemeinschaft, and by the Karl-Weierstrass-Institute of Mathematics, Berlin. The idea was to bring together specialists from different areas of modern analysis, geometry and mathematical physics, in a similar spirit as in the earlier series of conferences of the Karl-Weierstrass- Institute (Ludwigsfelde, 1976; Reinhardsbrunn, 1985; Holzhau, 1988; Breitenbrunn,1990). Berlin, Mainz M. Demuth B. Gramsch B. -W. Schulze List of talks given in La. brecht. 8. - 14. December 91 Name Title S. Albeverio Some recent deveiopments in Dirichlet forms and associated processes U. Bunke On the spectral flow of Dirac operators with negative definite functions as symbols - applications in probability theory and mathematical physics M. Combescure Recurrent versus diffusive behaviour for time-dependent quantum Hamiltonian E. B. Davies Analysis on graphs and noncommutative geometry M. Demuth On stochastic spectral analysis and asymptotics for Schrodinger operators P. Duclos A global approach to the location of quantum resonances v. v. Egorov On negative spectrum of elliptic operators V. EnB Non-threshold states of long-range N-body problems B. Gramsch -algebras and microlocal analysis P. B. Gilkey On the index of geometrical operators for Riemannian manifolds with boundary B. Helffer Spectral problems in statistical mechanics R. Hempe I Second order pertubations of elliptic operators with periodic principal part T. Ichinose On the Weyl quantized relativistic Hamiltonian V. V.

This volume, addressed to researchers and postgraduate students, compiles up-to-date research and expository papers on different aspects of complex analysis, including relations to operator theory and hypercomplex analysis. Subjects include the Schroedinger equation, subelliptic operators, Lie algebras and superalgebras, among others.

Theorems of factorising matrix functions and the operator identity method play an essential role in this book in constructing the spectral theory (direct and inverse problems) of canonical differential systems. Includes many varied applications of the general theory.

th This volume is dedicated to Carl Pearcy on his 60 birthday. It collects recent contributions to operator theory, nonselfadjoint operator algebras, measure the­ ory, and the theory of moments by several of the lead­ ing specialists in those areas. Many of the contributors are collaborators or former students of Carl Pearcy, and the variety of the topics bears witness to the wide range of his work and interests. The editors were helped by many in the compi­ lation of this volume. Srdjan Petrovic helped com­ pile Carl's list of publications, while Arlen Brown and George Exner helped in writing the biographical and mathematical sketch. The work of many referees, who must remain anonymous, was very valuable. Israel Gohberg suggested that we publish this volume in the distinguished series Operator Theory: Advances and Applications. The whole volume was expertly typeset by Elena Fraboschi. We wish to extend to all of these people our heartfelt thanks. CARL M. PEARCY Carl M. Pearcy: A Biographical Sketch H. BERCOVICI fj C. FOIAS Carl Mark Pearcy, Jr. was born on August 23, 1935 in Beaumont, Texas. He was the eldest of two sons of Carl Mark Pearcy, Sr., and Carrie Edith (Tilbury) Pearcy.

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.

Pseudo-differential operators belong to the most powerful tools in the analysis of partial differential equations. Basic achievements in the early sixties have initiated a completely new understanding of many old and important problems in analy­ sis and mathematical physics. The standard calculus of pseudo-differential and Fourier integral operators may today be considered as classical. The development has been continuous since the early days of the first essential applications to ellip­ ticity, index theory, parametrices and propagation of singularities for non-elliptic operators, boundary-value problems, and spectral theory. The basic ideas of the calculus go back to Giraud, Calderon, Zygmund, Mikhlin, Agranovich, Dynin, Vishik, Eskin, and Maslov. Subsequent progress was greatly stimulated by the classical works of Kohn, Nirenberg and Hormander. In recent years there developed a new vital interest in the ideas of micro­ local analysis in connection with analogous fields of applications over spaces with singularities, e.g. conical points, edges, corners, and higher singularities. The index theory for manifolds with singularities became an enormous challenge for analysists to invent an adequate concept of ellipticity, based on corresponding symbolic structures. Note that index theory was another source of ideas for the later development of the theory of pseudo-differential operators. Let us mention, in particular, the fundamental contributions by Gelfand, Atiyah, Singer, and Bott.

Generalized Schur functions are scalar- or operator-valued holomorphic functions such that certain associated kernels have a finite number of negative squares.

The following tract is divided into three parts: Hilbert spaces and their (bounded and unbounded) self-adjoint operators, linear Hamiltonian systemsand their scalar counterparts and their application to orthogonal polynomials.

PREFACE The classic statement of the Riemann boundary problem consists in finding a function (z) which is analytic and bounded in two domains D+ and D-, with a common boundary - a smooth closed contour L admitting a continuous extension onto L both from D+ and D- and satisfying on L the boundary condition +(t) = G(t)-(t) + g(t).

A collection of papers on different aspects of operator theory and complex analysis, covering the recent achievements of the Odessa-Kharkov school, where Potapov was very active. The book appeals to a wide group of mathematicians and engineers, and much of the material can be used for advanced courses and seminars.

Kaashoek, "Bitangential interpola tion for input-output operators of time varying systems: the discrete time case" contains solutions of time varying interpolation problems.