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Multivalued linear operators. (English) Zbl 0911.47002
Pure and Applied Mathematics, Marcel Dekker. 213. New York NY: Marcel Dekker. vi, 335 p. (1998).
The book deals with the abstract theory of multivalued linear operators in normed spaces. Although some applications are mentioned, the main advantage of this theory appears to be a unification (and better understanding) of the classical theory of linear operators. The first third of the book (Chapter I-III) is introductory and requires only basic knowledge in functional analysis: Highlights here are a unified version of the closed graph/open mapping and the closed range theorem which may replace the classical variants in an introductory course on functional analysis.
Multivalued linear operators occur naturally in connection with Fredholm operators (consider e.g. inverses), and the largest part of the book is devoted to this field (Chapter V) and related topics like the essential spectrum (Chapter VII). In the remaining chapters, operational quantities (related to e.g. Fredholm operators) and the second adjoint are considered. In the second part of the book there are many technical results; the reader should have experience in linear functional analysis to put them in the right place.

MSC:
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47A53 (Semi-) Fredholm operators; index theories
47A10 Spectrum, resolvent
47H04 Set-valued operators
47A57 Linear operator methods in interpolation, moment and extension problems
47A55 Perturbation theory of linear operators
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