Synopses & Reviews
Toeplitz operators arise in plenty of applications. They constitute one of the most important classes of non-selfadjoint operators, and the ideas and methods prevailing in the field of Toeplitz operators are a fascinating illustration of the fruitful interplay between operator theory, complex analysis, and Banach algebra techniques. This book is a systematic introduction to the advanced analysis of block Toeplitz operators and includes both classical results and recent developments. Its first edition has been a standard reference for fifteen years. The present second edition is enriched by several results obtained only in the last decade. The topics treated range from the analysis of locally sectorial matrix functions through Toeplitz and Wiener-Hopf operators on Banach spaces, projection methods, and quarter-plane operators up to Toeplitz and Wiener-Hopf determinants. The book is addressed to both graduate students approaching the subject for the first time and specialists in the theory of Toeplitz operators, but should also be of interest to physicists, probabilists, and computer scientists.
From the reviews of the second edition: "The first edition of this book was published in 1990 and became a standard reference ... . The area of mathematics that includes the analysis of Toeplitz operators has been particularly active over the last two decades and the authors of this book have been actively taking part in this development. ... new material was incorporated with recent developments in topics ... . this new edition is certainly welcome and will find a wide audience." (Amarino Lebre, Zentralblatt MATH, Vol. 1098 (24), 2006) "This edition, like the first, is a clear, detailed, and self-contained development of the field. The scope of this book is immense. It is 661 pages long. ... For those of us that work in the area of Toeplitz operators, and for those who only occasionally think about Toeplitz operators, this book is an invaluable resource. We are grateful that we have a complete, elegant, and current account of an interesting and intriguing subject." (Estelle L. Basor, Mathematical Reviews, Issue 2007 k)
Since the late 1980s, Toeplitz operators and matrices have remained a ?eld of extensive research and the development during the last nearly twenty years is impressive. One encounters Toeplitz matrices in plenty of applications on the one hand, and Toeplitz operators con?rmed their role as the basic elementary building blocks of more complicated operators on the other. Several monographs on Toeplitz and Hankel operators were written d- ing the last decade. These include Peller s grandiose book on Hankel ope- tors and their applications and Nikolski s beautiful easy reading on operators, functions, and systems, with emphasis on topics connected with the names of Hardy, Hankel, and Toeplitz. They also include books by the authors together withHagen, Roch, Yu.Karlovich, Spitkovsky, Grudsky, andRabinovich.Thus, results, techniques, and developments in the ?eld of Toeplitz operators are now well presented in the monographic literature. Despite these competitive works, we felt that large parts of the ?rst edition of the present monograp- whichismeanwhileoutofstock-havenotlosttheirfascinationandrelevance. Moreover, the ?rst edition has received a warm reception by many colleagues and became a standard reference. This encouraged us to venture on thinking about a second edition, and we are grateful to the Springer Publishing House for showing an interest in this.
A revised introduction to the advanced analysis of block Toeplitz operators including recent research. This book builds on the success of the first edition which has been used as a standard reference for fifteen years. Topics range from the analysis of locally sectorial matrix functions to Toeplitz and Wiener-Hopf determinants. This will appeal to both graduate students and specialists in the theory of Toeplitz operators.
Table of Contents
Auxiliary material.- Basic theory.- Symbol analysis.- Toeplitz operators on H2.- Toeplitz operators on Hp.- Toeplitz operators on Lp.- Finite section method.- Toeplitz operators over the quarter-plane.- Wiener-Hopf integral operators.- Toeplitz determinants.