Contributions to linear dynamics, sweeping process and regularity of Lipschitz functions

Abstract

This thesis deals with three topics related to linear operators defined on infinite dimensional spaces and two topics of real analysis and variational analysis in finite dimensional spaces. The first chapter contains preliminaries on Banach space theory which will be relevant for the three topics related to linear operators. The second chapter is a characterization of some types of bounded linear operators in terms of the differentiability of Lipschitz functions. Our results include a characterization for the classes of finite rank, compact, limited and weakly compact operators. The third and fourth chapters are inscribed in linear dynamics on infinite dimensional spaces, studying epsilon-hypercyclicity and wild operators respectively. We establish an epsilon-hypercyclicity criterion based on which we can construct epsilon-hypercyclic operators in a large class of separable Banach spaces. With respect to wild operators, we establish results about their spectra and about the norm closure of the set of wild operators in the space of linear bounded operators. In addition, we introduce and explore the concept of asymptotically separated sets to construct linear operators with interesting dynamical properties. The fifth chapter is a generalization of the Kurdyka-Łojasiewicz inequality for multivalued maps which are not necessarily definable in an o-minimal structure. We characterize smooth multivalued functions which satisfy a certain desingularization of the coderivative in terms of the length of the solutions of the related sweeping process as well as the integrability of the oriented talweg. The last chapter is devoted to absolutely minimizing Lipschitz (AML) functions. The main contribution in this subject is a characterization of the regularity of planar AML functions in terms of the regularity of the underlying norm.