Generalized Hyperbolicity, Stability and Expansivity for Operators on Locally Convex Spaces

Abstract

We introduce and study the notions of (generalized) hyperbolicity, topological stability and (uniform) topological expansivity for operators on locally convex spaces. We prove that every generalized hyperbolic operator on a locally convex space has the finite shadowing property. Contrary to what happens in the Banach space setting, hyperbolic operators on Fr\'echet spaces may fail to have the shadowing property, but we find additional conditions that ensure the validity of the shadowing property. Assuming that the space is sequentially complete, we prove that generalized hyperbolicity implies the strict periodic shadowing property, but we also show that the hypothesis of sequential completeness is essential. We show that operators with the periodic shadowing property on topological vector spaces have other interesting dynamical behaviors, including the fact that the restriction of such an operator to its chain recurrent set is topologically mixing and Devaney chaotic. We prove that topologically stable operators on locally convex spaces have the finite shadowing property and the strict periodic shadowing property. As a consequence, topologically stable operators on Banach spaces have the shadowing property. Moreover, we prove that generalized hyperbolicity implies topological stability for operators on Banach spaces. We prove that uniformly topologically expansive operators on locally convex spaces are neither Li-Yorke chaotic nor topologically transitive. Finally, we characterize the notion of topological expansivity for weighted shifts on Fr\'echet sequence spaces. Several examples are provided.