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Banach algebra techniques in operator theory. 2nd ed. (English) Zbl 0920.47001
Graduate Texts in Mathematics. 179. New York NY: Springer. xvi, 194 p. (1998).
This book is meant as an introduction to Banach algebras and their application to linear operator theory, with a particular emphasis on Toeplitz operators. The contents is described by the author in the preface as follows.
The book begins with a chapter presenting the basic results in the theory of Banach spaces along with many relevant examples. The second chapter concerns the elementary theory of commutative Banach algebras since these techniques are essential for the approach to operator theory presented in the later chapters. Then after a short chapter on the geometry of Hilbert space, the study of operator theory begins in earnest. In the fourth chapter operators on Hilbert space are studied and a rather sophisticated version of the spectral theorem is obtained. The notion of a $$C^*$$-algebra is introduced and used throughout the last half of this chapter. The study of compact operators and Fredholm operators is taken up in the fifth chapter along with certain ancillary results concerning ideals in $$C^*$$-algebras. The approach here is a bit unorthodox but is suggested by modern developments.
The last two chapters are of a slightly different character and present a systematic development including recent research of the theory of Toeplitz operators. This latter class of operators has attracted the attention of several mathematicians recently and occurs in several rather diverse contexts.
In the sixth chapter certain topics from the theory of Hardy spaces are developed. The selection by needs of the last chapter and proofs are based on the techniques obtained earlier in the book. The study of Toeplitz operators is taken up in the seventh chapter. Most of what is known in the scalar case is presented including Widom’s result on the connectedness of the spectrum.
At the end of each chapter, the author adds a detailed list of exercises, ranging from elementary to unsolvable. The book closes with a list of 117 references and a subject index.
This monograph is certainly a useful contribution to the vast market of books on functional analysis and operator theory; indeed, there is no comparable work in book form. Without any doubt, this has become, and will continue to be, a standard reference in the field which is indispensable for anyone interested in both the theory and applications of linear operators.

MSC:
 47-02 Research exposition (monographs, survey articles) pertaining to operator theory 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 46L10 General theory of von Neumann algebras 46L05 General theory of $$C^*$$-algebras 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 47A53 (Semi-) Fredholm operators; index theories 47B07 Linear operators defined by compactness properties 46J05 General theory of commutative topological algebras 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 47B48 Linear operators on Banach algebras