Representations of linear operators between Banach spaces
The book presents an account of the spectral theory of operators acting in Banach spaces. The basic objects of study are spectral representations for linear compact and in general bounded operators. The spaces are mainly supposed to be reflexive with their duals strictly convex. Chapter 1 presents some of the background material concerning the geometry of Banach spaces, orthogonality, the approximation property, \(p\)-trigonometric functions and other related facts and notions. A brief account of \(s\)-numbers and entropy numbers, including Gelfand numbers and Gelfand widths, are also presented. Chapter 2 is devoted to the spectral decomposition of compact linear operators. The problem posed and solved here is to obtain necessary and sufficient conditions for a compact linear operator to be representable in a series form (similarly as in the classical Hilbert spaces case). The results are applied to (semi-linear) boundary value problems involving the \(p\)-Laplacian. The approach leads, in particular, to a relatively easy proof of the existence of infinitely many eigenvalues of the Dirichlet problems for the \(p\)-Laplacian. The final Chapter 3 is concerned with the problems encountered when trying to extend the results to non-compact operators. Integral representations are described (in terms of the resolution of the identity) for bounded linear operators acting between Banach spaces. Although on the face of it, the topic of the book is rather specialized, the treatment brings out the connexion with the major branches of spectral theory, and emphasizes that all is part of the same subject. The methods developed may give some indication of what is needed to study more general classes of operators. The notes and comments at the end of each chapter give a fairly complete documentation, enabling the reader to trace the material to its sources, pursue the topics further and see them in context. As an authoritative account of a new and rapidly developing branch of spectral theory, this work will be of great interest to research workers and students in the field and related topics.
scientific article; zbMATH DE number 6151540