Porosity and Supercyclic Operators on 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 Morrey spaces. Also, we characterize supercyclic weighted composition operators on certain Segal algebras of functions and on non-unital commutative C*-algebras. Moreover, we introduce the concept of Cesáro p-hyper-transitivity for a positive rational number p and we characterize Cesáro p-hyper-transitive weighted composition operators on all these spaces. We illustrate our results with concrete examples and we give in addition an example of Cesáro hypercyclic weighted composition operator which is not Cesáro p-hyper-transitive for any positive rational p different from 1. 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 characterize a class of weighted composition operators on this space whose the set of non-hypercyclic vectors is non-porous.