Search

This book was originally intended as an extended version of our book'Invertibility and Asymptotics of Toeplitz Matrices', which appeared in 1983. We planned to discuss several topics in more detail, but our main concern was to incorporate a whole series of new results obtained during the last few years. However, we soon realized that the program we had in mind required new thoughts from both the methodological and substantial point of view, and so we decided to attempt writing a completely new book on the analysis of Toeplitz operators. There are at least two reasons for the continuous and increasing interest in Toeplitz operators. On the one hand, Toeplitz operators are of importance in connection with a variety of problems in physics, probability theory, information and control theory, and several other fields. Although we shall not embark on these problems, the selection of the material of this book is to a certain extent determined by such applications. On the other hand, besidesthe differential operators, Toeplitz operators constitute one of the most important classes of non-selfadjoint operators and they are a fascinating example of the fruitful interplay between such topics as operator theory, function theory, and the theory of Banach algebras. One main purpose of this book is to elucidate some of the ideas and methods illustrating just the latter aspect.

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.