Series:

Operator Theory: Advances and Applications

14 primary works • 21 total works

Book 8

Cover of Matrices and Indefinite Scalar Products

Book 12

Cover of Topics in Operator Theory Systems and Networks
This volume contains the proceedings of the Workshop on app1ications of linear operator theory to systems and networks, which was held at the Weizmann Institute of Science in the third week of June, 19S3,just be fore the MTNS Conference in Beersheva. For a 10ng time these subjects were studied indepen- dent1y by mathematica1 ana1ysts and e1ectrica1 engineers. Never- the1ess, in spite of the lack of communication, these two groups often deve10ped parallel theories, though in different languages, at different levels of genera1ity and typica11y quite different motivations. In the last severa1 years each side has become aware of the work of the other and there is a seeming1y ever- increasing invo1vement of the abstract theories of factorization, extension and interpolation of operators (and operator/matrix va1ued functions) to the design and analysis of systems and net- works. Moreover, the problems encountered in e1ectrica1 engineering have genera ted new mathematica1 problems, new approaches, and usefu1 new formu1ations.
The papers contained in this volume constitute a more than representative se1ection of the presented talks and dis- cussion at the workshop, and hopefu11y will also serve to give a reasonably accurate picture of the problems which are under active study today and the techniques which are used to deal with them.

Book 49

Cover of Classes of Linear Operators Vol. I
After the book "Basic Operator Theory" by Gohberg-Goldberg was pub lished, we, that is the present authors, intended to continue with another book which would show the readers the large variety of classes of operators and the important role they play in applications. The book was planned to be of modest size, but due to the profusion of results in this area of analysis, the number of topics grew larger than ex pected. Consequently, we decided to divide the material into two volumes - the first volume being presented now. During the past years, courses and seminars were given at our respective in stitutions based on parts of the texts. These were well received by the audience and enabled us to make appropriate choices for the topics and presentation for the two vol umes. We would like to thank G.J. Groenewald, A.B. Kuijper and A.C.M. Ran of the Vrije Universiteit at Amsterdam, who provided us with lists of remarks and corrections. We are now aware that the Basic Operator Theory book should be revised so that it may suitably fit in with our present volumes. This revision is planned to be the last step of an induction and not the first.

Book 53

Cover of One-Dimensional Linear Singular Integral Equations
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.

Book 56

Cover of Time-variant Systems and Interpolation

Book 79

Cover of Partially Specified Matrices and Operators: Classification, Completion, Applications
This book is devoted to a new direction in linear algebra and operator theory that deals with the invariants of partially specified matrices and operators, and with the spectral analysis of their completions. The theory developed centers around two major problems concerning matrices of which part of the entries are given and the others are unspecified. The first is a classification problem and aims at a simplification of the given part with the help of admissible similarities. The results here may be seen as a far reaching generalization of the Jordan canonical form. The second problem is called the eigenvalue completion problem and asks to describe all possible eigenvalues and their multiplicities of the matrices which one obtains by filling in the unspecified entries. Both problems are also considered in an infinite dimensional operator framework. A large part of the book deals with applications to matrix theory and analysis, namely to stabilization problems in mathematical system theory, to problems of Wiener-Hopf factorization and interpolation for matrix polynomials and rational matrix functions, to the Kronecker structure theory of linear pencils, and to non everywhere defined operators. The eigenvalue completion problem has a natural associated inverse, which appears as a restriction problem. The analysis of these two problems is often simpler when a solution of the corresponding classification problem is available.

Book 140

Cover of Orthogonal Systems and Convolution Operators
In this book we study orthogonal polynomials and their generalizations in spaces with weighted inner products. The impetus for our research was a deep theorem due to M.G. Krein along with subsequent results of Krein and H. Langer. Together with our colleagues, we have worked in this area for nearly fifteen years, and the results of our research are presented here in unified form. We are grateful to the Department of mathematics at the University of Maryland in College Park and to Tel-Aviv University for their support and encouragement. The support of the Silver Family Foundation is also highly appreciated. Introduction The starting point ofthis book is a study ofthe orthogonal polynomials {qn In ?: O} obtained by orthogonalizing the power functions I, Z, z2, ... on the unit circle. The orthogonality is with respect to the scalar product defined by where the weight w is a positive integrable function on the unit circle. These ortho gonal polynomials are called the Szego polynomials associated with the weight w.

Book 162

Cover of Operator Theory in Krein Spaces and Nonlinear Eigenvalue Problems
rd This volume contains papers written by the participants of the 3 Workshop on Operator Theory in Krein spaces and Nonlinear Eigenvalue Problems, held at the Technische Universit. at Berlin, Germany, December 12 to 14, 2003. The workshop covered topics from spectral, perturbation and extension t- ory of linear operators in Krein spaces. They included generalized Nevanlinna functions and related classes of functions, boundary value problems for di?erential operators, spectral problems for matrix polynomials, and perturbation problems forsecondorderevolutionequations.Alltheseproblemsarere?ectedinthepresent volume. The workshop was attended by 46 participants from 12 countries. It is a pleasure to acknowledge the substantial ?nancial support received from the - Research Training Network HPRN-CT-2000-00116 "Analysis and Operators" by the European Community, - DFG-Forschungszentrum MATHEON "Mathematik fur .. Schlussel- .. technologien", - Institute of Mathematics of the Technische Universit. at Berlin. We would also like to thank Petra Grimberger for her great help.
Last but not least, special thanks are due to Jussi Behrndt, Christian Mehl and Carsten Trunk for their excellent workin the organisationof the workshopand the preparationof this volume. Without their assistance the workshop might not have taken place. The Editors Operator Theory: Advances and Applications, Vol. 162, 1-17 c 2005 Birkh. auser Verlag Basel/Switzerland Partial Non-stationary Perturbation Determinants for a Class of J-symmetric Operators Vadim Adamyan, Peter Jonas and Heinz Langer Abstract. We consider the partial non-stationary perturbation determinant (1) itA ?itH ? (t):=det e P e ,t? R.

Book 178

Cover of Factorization of Matrix and Operator Functions: The State Space Method

This book delineates the various types of factorization problems for matrix and operator functions. The problems originate from, or are motivated by, the theory of non-selfadjoint operators, the theory of matrix polynomials, mathematical systems and control theory, the theory of Riccati equations, inversion of convolution operators, and the theory of job scheduling in operations research. The book presents a geometric principle of factorization which has its origins in the state space theory of linear input-output systems and in the theory of characteristic operator functions.

Book 181

Cover of Operator Algebras, Operator Theory and Applications

This book is composed of three survey lecture courses and some twenty invited research papers presented to WOAT 2006 - the International Summer School and Workshop on Operator Algebras, Operator Theory and Applications, held at Lisbon in September 2006. The volume reflects recent developments in the area of operator algebras and their interaction with research fields in complex analysis and operator theory. The book is aimed at postgraduates and researchers in these fields.

Book 192

Cover of Holomorphic Operator Functions of One Variable and Applications
This is a book on holomorphic operator functions of a single variable and their - plications,whichisfocussedontherelationsbetweenlocalandglobaltheories.Itis based on methods and technics of Complex analysis of scalar and matrix functions of several variables. The applications concern: interpolation, holomorphic families of subspaces and frames, spectral theory of polynomials with operator coe?cients, holomorphic equivalence and diagonalization, and Plemelj-Muschelishvili fact- ization. The book also contains a theory of Wiener-Hopf integral equations with operator-valued kernels and a theory of in?nite Toplitz .. matrices with operator entries. We started to work on these topics long ago when one of us was a Ph.D. s- dent of the other in Kishinev (now Cisinau) University. Then our main interests were in problems of factorization of operator-valued functions and singular in- gral operators. Working in this area, we realized from the beginning that di?erent methods and tools from Complex analysis of several variables and their modi?- tions are very useful in obtaining results on factorization for matrix and operator functions. We have in mind di?erent methods and results concerning connections between local and global properties of holomorphic functions.
The ?rst period was very fruitful and during it we obtained the basic results presented in this book.

Book 200

Cover of A State Space Approach to Canonical Factorization with Applications

The present book deals with canonical factorization of matrix and operator functions that appear in state space form or that can be transformed into such a form. A unified geometric approach is used. The main results are all expressed explicitly in terms of matrices or operators, which are parameters of the state space representation. The applications concern different classes of convolution equations. A large part the book deals with rational matrix functions only.

Book 235

Cover of Separable Type Representations of Matrices and Fast Algorithms
This two-volume work presents a systematic theoretical and computational study of several types of generalizations of separable matrices. The main attention is paid to fast algorithms (many of linear complexity) for matrices in semiseparable, quasiseparable, band and companion form. The work is focused on algorithms of multiplication, inversion and description of eigenstructure and includes a large number of illustrative examples throughout the different chapters. The first volume consists of four parts. The first part is of a mainly theoretical character introducing and studying the quasiseparable and semiseparable representations of matrices and minimal rank completion problems. Three further completions are treated in the second part. The first applications of the quasiseparable and semiseparable structure are included in the third part where the interplay between the quasiseparable structure and discrete time varying linear systems with boundary conditions play an essential role. The fourth part contains factorization and inversion fast algorithms for matrices via quasiseparable and semiseparable structure. The work is based mostly on results obtained by the authors and their coauthors. Due to its many significant applications and the accessible style the text will be useful to engineers, scientists, numerical analysts, computer scientists and mathematicians alike.

Book 288

Cover of Completeness Theorems and Characteristic Matrix Functions
This monograph presents necessary and sufficient conditions for completeness of the linear span of eigenvectors and generalized eigenvectors of operators that admit a characteristic matrix function in a Banach space setting. Classical conditions for completeness based on the theory of entire functions are further developed for this specific class of operators. The classes of bounded operators that are investigated include trace class and Hilbert-Schmidt operators, finite rank perturbations of Volterra operators, infinite Leslie operators, discrete semi-separable operators, integral operators with semi-separable kernels, and period maps corresponding to delay differential equations. The classes of unbounded operators that are investigated appear in a natural way in the study of infinite dimensional dynamical systems such as mixed type functional differential equations, age-dependent population dynamics, and in the analysis of the Markov semigroup connected to the recently introduced zig-zag process.

Cover of Operator Theory and Copmlex Analysis

v. 49

Cover of Classes of Linear Operators
After the book "Basic Operator Theory" by Gohberg-Goldberg was pub- lished, we, that is the present authors, intended to continue with another book which would show the readers the large variety of classes of operators and the important role they play in applications. The book was planned to be of modest size, but due to the profusion of results in this area of analysis, the number of topics grew larger than ex- pected. Consequently, we decided to divide the material into two volumes - the first volume being presented now. During the past years, courses and seminars were given at our respective in- stitutions based on parts of the texts. These were well received by the audience and enabled us to make appropriate choices for the topics and presentation for the two vol- umes. We would like to thank G.J. Groenewald, A.B. Kuijper and A.C.M. Ran of the Vrije Universiteit at Amsterdam, who provided us with lists of remarks and corrections. We are now aware that the Basic Operator Theory book should be revised so that it may suitably fit in with our present volumes. This revision is planned to be the last step of an induction and not the first.

v. 34

Cover of Orthogonal Matrix-valued Polynomials and Applications
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.

v. 130

Cover of Linear Operators and Matrices
The book is complemented by biographical information. This volume is dedicated to Peter Lancaster, an outstanding expert in operator and matrix theory, numerical analysis and applications, on the occasion of his seventieth birthday. The book contains a selection of recent original research papers in linear algebra and analysis, areas in which Peter Lancaster was very active. The articles are complemented by biographical data and a list of publications. Contributed volume in honor of Peter Lancaster, an outstanding expert in operator theory, matrix theory and numerical analysis. The articles have been carefully selected and refereed and cover topics in linear algebra and analysis where Peter Lancaster was very active.

Cover of Operator Theory in Inner Product Spaces

v. 106

Cover of Contributions to Operator Theory in Spaces with an Indefinite Metric

This volume is dedicated to Heinz Langer, a leading expert in spectral analysis and its applications, in particular to operators in spaces with an indefinite metric, on the occasion of his 60th birthday.
The book begins with his biography and list of publications. It contains a selection of research papers, most of which are devoted to spectral analysis of operators or operator pencils with applications to ordinary and partial differential equations. Other papers deal with time-varying systems, interpolation and factorization problems, and topics from mathematical physics. About half of the papers contain further developments in the theory of operators in spaces with an indefinite metric and treat new applications. The book is of interest to a wide audience of pure and applied mathematicians.