Porosity and supercyclic operators on solid Banach function spaces

Abstract

In this paper, we characterize supercyclic weighted composition operators on a large class of solid Banach function spaces, in particular on Lebesgue, Orlicz and Morrey spaces. Also, we characterize supercyclic weighted composition operators on certain Segal algebras of functions and nonunital commutative C*-algebras. Moreover, we introduce the concept of Ces\'aro hyper-transitivity and we characterize Ces\'aro hyper-transitive weighted composition operators on all these spaces. We illustrate our results with concrete examples and we give in addition an example of a hypercyclic weighted composition operator which is not Ces\'aro hyper-transitive. Next, we introduce a class of non-porous subsets of the space of continuous functions vanishing at infinity on the real line. As an application, we consider weighted composition operator on this space and we give sufficient conditions that induce that the set of non-hypercyclic vectors for this operator is non-porous.