Invertibility and singularity for bounded linear operators. (English) Zbl 0636.47001

Harte, Robin

Pure and Applied Mathematics, 109. New York-Basel: Marcel Dekker, Inc. xii, 590 p.; $ 99.75 (U.S. & Canada); $ 119.50 (all other countries) (1988).

The book can be characterized as a systematic analysis of all aspects of singularity which prevent a bounded linear operator from being invertible. An unusual attention is paid to the situation where the underlying normed spaces are not complete. This leads to a variety of concepts which usually coincide in the presence of completeness. Basic principles of functional analysis are discussed from the point of view of spectral theory. The analytic tool for the latter is the Liouville theorem. Among the original features of the book one can find various enlargement constructions bridging different kinds of nonsingularity, an algebraic framework for Fredholm theory, and many kinds of spectra including the multiparameter spectral theory.
As far as for these topics the book can be considered as the most competent source of information, to great extent due to the author’s own significant contribution.
Some other areas of spectral theory are neglected: the subharmonic properties of the spectrum, the geometric characteristics of nonsingularity, the deeper variants of the spectral radius formula, extensions of Banach algebras. There is a good bibliography occupying 21 pages but some relevant items could be added: the construction in 6.7.4 appears earlier in B. N. Sadovskiĭ [Usp. Mat. Nauk 27, No. 1, 81–146 (1972; Zbl 0232.47067)], the proof in 18.8.2 goes back to I. Kaplansky [Surveys appl. Math. 4, 1–34 (1958; Zbl 0087.31102)], and also a very recent paper by J. S. Hwang [Bull. Korean Math. Soc. 24, No. 1, 27–29 (1987; Zbl 0658.47003)] is close to the spirit of the book (it deals with 7.11.4.2 in the incomplete situation).
There is an unusual amount of misprints, though not much disturbing: the author has kindly offered to the reviewer two densely typed pages of corrections but it is still possible to detect some more. It seems that the book can serve as a helpful guide to the literature and as an inspiration of new ideas for specialists in spectral theory rather than an introduction for the beginners.

Reviewer: Jaroslav Zemánek (Warszawa)

MSC:

47-02	Research exposition (monographs, survey articles) pertaining to operator theory
46H05	General theory of topological algebras
47A10	Spectrum, resolvent
47A05	General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A65	Structure theory of linear operators

Keywords:

systematic analysis of all aspects of singularity; spectral theory; Liouville theorem; algebraic framework for Fredholm theory; multiparameter spectral theory; subharmonic properties of the spectrum; geometric characteristics of nonsingularity; spectral radius formula; extensions of Banach algebras

Citations:

Zbl 0232.47067; Zbl 0087.31102; Zbl 0658.47003