Supercyclicity of weighted composition operators on spaces of continuous functions

Abstract

Our study is focused on the dynamics of weighted composition operators defined on a locally convex space E ,→ (C(X), τp) with X being a topological Hausdorff space containing at least two different points and such that the evaluations {δx : x ∈ X} are linearly independent in E0. We prove, when X is compact and E is a Banach space containing a nowhere vanishing function, that a weighted composition operator Cw,ϕ is never weakly supercyclic on E. We also prove that if the symbol ϕ lies in the unit ball of A(D), then every weighted composition operator can never be τp-supercyclic neither on C(D) nor on the disc algebra A(D). Finally, we obtain Ansari-Bourdon type results and conditions on the spectrum for arbitrary weakly supercyclic operators, and we provide necessary conditions for a composition operator to be weakly supercyclic on the space of holomorphic functions defined in non necessarily simply connected planar domains. As a consequence, we show that no composition operator can be weakly supercyclic neither on the space of holomorphic functions on the punctured disc nor in the punctured plane.