Toeplitz Matrices and Operators Cambridge Studies in Advanced Mathematics Series

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.

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.

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.