Eigenvalues of set-valued operators in Banach spaces
The paper under review is a valuable mathematical contribution, especially, but not only, to the spectral theory, generalizing herewith so many areas like linear algebra, but also optimization theory (non-smoothness allowed now), dynamical systems and control theory. In fact, this work deals with the spectral analysis of set-valued operators from a Banach space \(X\) into its dual space \(X^*\). This article is very carefully organized; is is rounded in contents and written in a sound way. The authors look for results in their generality, but they also pay attention to the detail, interesting special cases, examples and observations. There are three sections: 1. Introduction, providing the necessary terminology, helpful settings and, very importantly, motivating examples from differential inclusions and their generalizations; 2. Eigenvalues and Eigenvectors, dealing, e.g., with duality map, various convergence and (sub)sequence properties related with smoothness of the Banach space, its characterization by Gâteaux differentiability of the norm, the case of reflexivity and Fréchet differentiability the spectral terminology is provided. Referring to the spectral objects, the main part of the paper is 3. Semicontinuity Properties of the Eigenset Operator and the Spectrum, with a wealth of results in terms of upper and lower semicontinuity, but also on graphical convergence, Kuratowski convergence, pseudo-similarity transformations, subdifferentials, etc. These convergence or continuity results may in the future serve for the study of inverse problems, the numerical solution of problems with, e.g., differential inclusions and their controllability, and the corresponding stability theory.
scientific article; zbMATH DE number 2176208