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Author: L. Boutet de Monvel Publisher: Princeton University Press ISBN: 9780691082790 Category : Mathematics Languages : en Pages : 172
Book Description
The theory of Toeplitz operators has come to resemble more and more in recent years the classical theory of pseudodifferential operators. For instance, Toeplitz operators possess a symbolic calculus analogous to the usual symbolic calculus, and by symbolic means one can construct parametrices for Toeplitz operators and create new Toeplitz operators out of old ones by functional operations. If P is a self-adjoint pseudodifferential operator on a compact manifold with an elliptic symbol that is of order greater than zero, then it has a discrete spectrum. Also, it is well known that the asymptotic behavior of its eigenvalues is closely related to the behavior of the bicharacteristic flow generated by its symbol. It is natural to ask if similar results are true for Toeplitz operators. In the course of answering this question, the authors explore in depth the analogies between Toeplitz operators and pseudodifferential operators and show that both can be viewed as the "quantized" objects associated with functions on compact contact manifolds.
Author: L. Boutet de Monvel Publisher: Princeton University Press ISBN: 9780691082790 Category : Mathematics Languages : en Pages : 172
Book Description
The theory of Toeplitz operators has come to resemble more and more in recent years the classical theory of pseudodifferential operators. For instance, Toeplitz operators possess a symbolic calculus analogous to the usual symbolic calculus, and by symbolic means one can construct parametrices for Toeplitz operators and create new Toeplitz operators out of old ones by functional operations. If P is a self-adjoint pseudodifferential operator on a compact manifold with an elliptic symbol that is of order greater than zero, then it has a discrete spectrum. Also, it is well known that the asymptotic behavior of its eigenvalues is closely related to the behavior of the bicharacteristic flow generated by its symbol. It is natural to ask if similar results are true for Toeplitz operators. In the course of answering this question, the authors explore in depth the analogies between Toeplitz operators and pseudodifferential operators and show that both can be viewed as the "quantized" objects associated with functions on compact contact manifolds.
Author: L. Boutet de Monvel Publisher: Princeton University Press ISBN: 1400881447 Category : Mathematics Languages : en Pages : 166
Book Description
The theory of Toeplitz operators has come to resemble more and more in recent years the classical theory of pseudodifferential operators. For instance, Toeplitz operators possess a symbolic calculus analogous to the usual symbolic calculus, and by symbolic means one can construct parametrices for Toeplitz operators and create new Toeplitz operators out of old ones by functional operations. If P is a self-adjoint pseudodifferential operator on a compact manifold with an elliptic symbol that is of order greater than zero, then it has a discrete spectrum. Also, it is well known that the asymptotic behavior of its eigenvalues is closely related to the behavior of the bicharacteristic flow generated by its symbol. It is natural to ask if similar results are true for Toeplitz operators. In the course of answering this question, the authors explore in depth the analogies between Toeplitz operators and pseudodifferential operators and show that both can be viewed as the "quantized" objects associated with functions on compact contact manifolds.
Author: N. Nikolsky Publisher: Birkhäuser ISBN: 3034855877 Category : Science Languages : en Pages : 420
Book Description
The volume contains selected papers of the Spectral Function Theory seminar, Leningrad Branch of Steklov Mathematical Institute. The papers are mostly devoted to the theory of Toeplitz and model operators. These subjects are considered here from various points of view. Several papers concern the relationships of Toeplitz operators to weighted polynomial approximation. Namely, two papers by B. Solomyak and A. Volberg intensively treat the problem of spectra! multiplicity f~r analytic Toeplitz operators (which are, in fact, multiplication operators) and my paper can serve as an introduction to the problem. This theme of multiplicities is continued in a paper by V. Vasyunin where the multiplicity of the spectrum is computed for Hilbert space contractions with finite defect indices. V. Peller's paper deals with a perturbation theory problem for Toeplitz operators. In a paper by D. Yakubovich a new similarity model for a class of Toeplitz operators is constructed. S. Treil' presents a survey of a part of spectral function theory for vector valued function (Szego-Kolmogorov extreme prob!ems for operator weights, bases of vector rational functions, estimations of Hilbert transform with respect to operator weights, the operator corona problem). As a concluding remark I dare only note that the whole collection convinces us once more without a doubt of the fruitfullness of the natural union of operator theory and complex analysis (if at all the union of these fields is at all different from their intersection).
Author: William Arveson Publisher: Springer Science & Business Media ISBN: 0387953000 Category : Mathematics Languages : en Pages : 140
Book Description
This book presents the basic tools of modern analysis within the context of the fundamental problem of operator theory: to calculate spectra of specific operators on infinite dimensional spaces, especially operators on Hilbert spaces. The tools are diverse, and they provide the basis for more refined methods that allow one to approach problems that go well beyond the computation of spectra: the mathematical foundations of quantum physics, noncommutative K-theory, and the classification of simple C*-algebras being three areas of current research activity which require mastery of the material presented here.
Author: N. Nikolsky Publisher: Birkhäuser ISBN: 9783764323448 Category : Science Languages : en Pages : 0
Book Description
The volume contains selected papers of the Spectral Function Theory seminar, Leningrad Branch of Steklov Mathematical Institute. The papers are mostly devoted to the theory of Toeplitz and model operators. These subjects are considered here from various points of view. Several papers concern the relationships of Toeplitz operators to weighted polynomial approximation. Namely, two papers by B. Solomyak and A. Volberg intensively treat the problem of spectra! multiplicity f~r analytic Toeplitz operators (which are, in fact, multiplication operators) and my paper can serve as an introduction to the problem. This theme of multiplicities is continued in a paper by V. Vasyunin where the multiplicity of the spectrum is computed for Hilbert space contractions with finite defect indices. V. Peller's paper deals with a perturbation theory problem for Toeplitz operators. In a paper by D. Yakubovich a new similarity model for a class of Toeplitz operators is constructed. S. Treil' presents a survey of a part of spectral function theory for vector valued function (Szego-Kolmogorov extreme prob!ems for operator weights, bases of vector rational functions, estimations of Hilbert transform with respect to operator weights, the operator corona problem). As a concluding remark I dare only note that the whole collection convinces us once more without a doubt of the fruitfullness of the natural union of operator theory and complex analysis (if at all the union of these fields is at all different from their intersection).
Author: Albrecht Boettcher Publisher: SIAM ISBN: 0898715997 Category : Mathematics Languages : en Pages : 410
Book Description
“This is a wonderful book, full of the latest material on Toeplitz matrices and operators, including norms, spectra, pseudospectra, fields of values, and polynomial hulls. The notes at the end of the chapters are especially interesting and the exercises are challenging. The writing is careful and precise but also entertaining.” --Anne Greenbaum, Professor of Mathematics, University of Washington.“This book is a tremendous resource for all aspects of the spectral theory of banded Toeplitz matrices. It will be the first place I turn when looking for many results in this field, and given this book's amazing breadth and depth, I expect to find just what I need.” -- Mark Embree, Assistant Professor of Computational and Applied Mathematics, Rice University.This self-contained introduction to the behavior of several spectral characteristics of large Toeplitz band matrices is the first systematic presentation of a relatively large body of knowledge. Covering everything from classic results to the most recent developments, Spectral Properties of Banded Toeplitz Matrices is an important resource. The spectral characteristics include determinants, eigenvalues and eigenvectors, pseudospectra and pseudomodes, singular values, norms, and condition numbers. Toeplitz matrices emerge in many applications and the literature on them is immense. They remain an active field of research with many facets, and the material on banded ones until now has primarily been found in research papers. The book may serve both as a text for introducing the material and as a reference. The approach is based on the know-how and experience of the authors in combining functional analytical methods with hard analysis and in applying operator theoretical methods to matrix theory, which reveals the essence of several phenomena and leads to significant improvements in existing results. All basic results presented in the book are precisely stated as theorems and accompanied by full proofs.Audience This book is written for applied mathematicians, engineers, and scientists who encounter Toeplitz matrices in their research. It also will be of interest to mathematicians in the fields of operator theory, numerical analysis, structured matrices, or random matrix theory, and physicists, chemists, biologists, and economists who deal with stationary statistical and stochastic problems. Parts of the book are suitable for use as a graduate-level text on Toeplitz matrices or analysis.Contents Preface; Chapter 1: Infinite Matrices; Chapter 2: Determinants; Chapter 3: Stability; Chapter 4: Instability; Chapter 5: Norms; Chapter 6: Condition Numbers; Chapter 7: Substitutes for the Spectrum; Chapter 8: Transient Behavior; Chapter 9: Singular Values; Chapter 10: Extreme Eigenvalues; Chapter 11: Eigenvalue Distribution; Chapter 12: Eigenvectors and Pseudomodes; Chapter 13: Structured Perturbations; Chapter 14: Impurities; Bibliography; Index.
Author: L. Boutet de Monvel
Author: L. Boutet de Monvel
Author: L. Boutet de Monvel
Author: Anthony Bak
Author: Edward Kevan Billowitz
Author: William Wilbur Johnston
Author: N. Nikolsky
Author: William Arveson
Author: N. Nikolsky
Author: Albrecht Boettcher
Author: Abraham Daniel Bergsagel Isgur