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An introduction to local spectral theory. (English) Zbl 0957.47004
London Mathematical Society Monographs. New Series. 20. Oxford: Clarendon Press. xii, 591 p. (2000).
A bounded linear operator \(T\) on a complex Banach space is called decomposable if, given any pair of open sets \(U,V\subset\mathbb{C}\) covering \(\mathbb{C}\), one can find \(T\)-invariant closed subspaces \(Y,Z\subset X\) such that \(X= Y+ Z\), \(\sigma(T|_Y)\subset U\), and \(\sigma(T|_Z)\subset V\). This idea of “localizing by decomposing” is the Leitmotiv of this book.
The book consists of six chapters. The first chapter deals with decomposable operators in the sense just mentioned. In the second chapter, the authors discuss duality theory and invariant subspaces, while the third chapter is concerned with spectral inclusions and spectral subspaces, the heart of what they call “local spectral theory”. A typical application refers to multipliers which are discussed in detail in the fourth chapter. A topic which seems to reflect the authors’ particular field of interest is automatic continuity which is treated in the fifth chapter. Finally, the sixth chapter is devoted to some open problems. The book closes with a detailed list of references, a subject and name index, and a list of symbols which is helpful for not getting “drowned” in the very technical details of the presentation.
This monograph may be reviewed as an essential update and extension of the books by I. Colojoară and C. Foiaş [“Theory of generalized spectral operators”, New York (1968; Zbl 0189.44201)] and F.-H. Vasilescu [“Analytic functional calculus and spectral decompositions”, Dordrecht (1982; Zbl 0495.47013)]. It will be of interest to specialists in the field.

MSC:
47A11 Local spectral properties of linear operators
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47A15 Invariant subspaces of linear operators
46H40 Automatic continuity
47A25 Spectral sets of linear operators
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