Toeplitz Matrices and Operators

by Nikolaï Nikolski, Danièle Gibbons (Translator), Greg Gibbons (Translator)

Overview

The theory of Toeplitz matrices and operators is a vital part of modern analysis, with applications to moment problems, orthogonal polynomials, approximation theory, integral equations, bounded- and vanishing-mean oscillations, and asymptotic methods for large structured determinants, among others. This friendly introduction to Toeplitz theory covers the classical spectral theory of Toeplitz forms and Wiener–Hopf integral operators and their manifestations throughout modern functional analysis. Numerous solved exercises illustrate the results of the main text and introduce subsidiary topics, including recent developments. Each chapter ends with a survey of the present state of the theory, making this a valuable work for the beginning graduate student and established researcher alike. With biographies of the principal creators of the theory and historical context also woven into the text, this book is a complete source on Toeplitz theory.

Product Details

ISBN-13:	9781107198500
Publisher:	Cambridge University Press
Publication date:	01/02/2020
Series:	Cambridge Studies in Advanced Mathematics , #182
Pages:	450
Product dimensions:	6.14(w) x 9.25(h) x 1.14(d)

About the Author

Nikolaï Nikolski is Professor Emeritus at the Université de Bordeaux, working primarily in analysis and operator theory. He has been co-editor of four international journals, editor of more than fifteen books, and published numerous articles and research monographs. He has also supervised twenty-six Ph.D. students, including three Salem Prize winners. Professor Nikolski was elected Fellow of the American Mathematical Society (AMS) in 2013 and received the Prix Ampère of the French Academy of Sciences in 2010.

Table of Contents

1. Why Toeplitz–Hankel? Motivations and panorama; 2. Hankel and Toeplitz – brother operators on the space H2; 3. H2 theory of Toeplitz operators; 4. Applications: Riemann–Hilbert, Wiener–Hopf, singular integral operators (SIO); 5. Toeplitz matrices: moments, spectra, asymptotics; Appendix A. Key notions of Banach spaces; Appendix B. Key notions of Hilbert spaces; Appendix C. An overview of Banach algebras; Appendix D. Linear operators; Appendix E. Fredholm operators and the Noether index; Appendix F. A brief overview of Hardy spaces; References; Notation; Index.

Editorial Reviews

‘The author's love for all facets of the topic and his distinctive sense for culture and history resulted in a book of unique charm. The reader will enjoy a captivating journey from the historical roots up to the latest achievements of a fascinating field of concrete operator theory. An excellent source and a true treat for both beginners and professionals!' Albrecht Böttcher, Technische Universität Chemnitz, Germany

'This is a detailed and clearly-written treatment of Toeplitz operators and Toeplitz matrices. The subject is strongly motivated by applications, and an extensive historical background is provided. The book should be of great interest to both students and experts in the subject. It contains several useful appendices, giving the necessary background material on functional analysis and complex analysis.' Jonathan Partington, University of Leeds

'The monograph provides a concise and comprehensive treatment of the beautiful theory of Toeplitz operators and their deep connections with other topics in Analysis, as well as applications. Written by a world's leading expert, it is precise and at the same time accessible. Appendices make it self-contained, and selected biographical sketches are read as a breathtaking historical novel. A real treat for everyone interested in Operator Theory and its developments, both classical and recent.' Ilya Spitkovsky, New York University Abu Dhabi

‘This book covers the history of Toeplitz forms and Wiener-Hopf integral operators and their applications in harmonic and functional analysis, interpolated with personal biographies of the important mathematical theorists who contributed to their development … this book will surely appeal to all who care for the ‘dramaturgy of mathematics'.’ J. A. Bakal, Choice

‘This is an extremely thorough introduction to the theory of Toeplitz matrices (and their siblings the Hankel matrices; known collectively by the portmanteau Ha-plitz). It provides most of the needed background, and probably could be used for very advanced undergraduates but seems more like a second-year graduate text.’ Allen Stenger, MAA Reviews

‘I strongly recommend it for both students and experts. The unique style of the book will captivate the reader.’ Maria T. Nowak, MathSciNet

From the Publisher