Bøger i Operator Theory: Advances and Applications serien

This volume collects a selected number of papers presented at the International Workshop on Operator Theory and its Applications (IWOTA) held in July 2014 at Vrije Universiteit in Amsterdam.

This book defines and examines the counterpart of Schur functions and Schur analysis in the slice hyperholomorphic setting. It is organized into three parts: the first introduces readers to classical Schur analysis, while the second offers background material on quaternions, slice hyperholomorphic functions, and quaternionic functional analysis.

This book presents a collection of expository and research papers on various topics in matrix and operator theory, contributed by several experts on the occasion of Albrecht Boettcher's 60th birthday.

This book presents novel results by participants of the conference "Control theory of infinite-dimensional systems" that took place in January 2018 at the FernUniversitat in Hagen.

A collection of articles emphasizing modern interpolation theory, a topic which has seen much progress in recent years. These ideas and problems in operator theory, often arising from systems and control theories, bring the reader to the forefront of current research in this area.

Convolutions by integrable kernels have continuous symbols and the Cauchy singular integral operator is the most prominent example of a convolution operator with a piecewise continuous symbol.

Linear differential equations with periodic coefficients constitute a well developed part of the theory of ordinary differential equations [17, 94, 156, 177, 178, 272, 389].

Incomplete second order linear differential equations in Banach spaces as well as first order equations have become a classical part of functional analysis. Special emphasis is placed on new surprising effects arising for complete second order equations which do not take place for first order and incomplete second order equations.

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.

Capturing the state of the art of the interplay between partial differential equations, functional analysis, maximal regularity, and probability theory, this volume was initiated at the Delft conference on the occasion of the retirement of Philippe Clement. It will be of interest to researchers in PDEs and functional analysis.

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.

This volume contains a selection of papers on the topics of Clifford analysis and wavelets and multiscale analysis, the latter being understood in a very wide sense. Most of the articles have been written on invitation and they provide a unique collection of material, particularly relating to Clifford analysis and the theory of wavelets.

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.