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Analysis of Toeplitz operators. Lizenzausg. d. Akademie-Verl. Berlin. (English) Zbl 0732.47029

Berlin etc.: Springer-Verlag. 512 p. DM 128.00 (1990).
The theory of Toeplitz operators has been intensively developed lately in various directions. The authors of the book have been taking immediate part in these investigations, and as a result their own scientific interests determined the selection of topics for the book. The main theme is to establish relations between the functional-analytic properties of (mainly one-dimensional) Toeplitz operators and geometric properties of their symbols. With the help of local methods the authors investigate scalar and matrix Toeplitz operators with symbols from various functional classes. Different types of locally sectorial symbols play an important role here. The authors follow some sort of machinery (or “philosophy”) of investigation. The steps of this machinery are the following: algebraization, essentialization, localization and determination of local spectra.
A considerable attention in the book is given also to the studying of a finite section method for Toeplitz operators. New approaches and methods in this direction were worked out by the authors at the beginning of the 1980s. These results exerted essential influence on further development of numerical analysis and gave the possibility to solve several difficult problems. Projectional methods are used in the book for studying the asymptotics of Toeplitz determinants as well.
The book is addressed to a wide audience. Graduate students will find an excellent introduction to the subject, specialists will enjoy the details of proofs and an abundance of factual material. The book should also be of interest to physicists, statisticians and informaticians.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47A53 (Semi-) Fredholm operators; index theories
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
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