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 $$E\hookrightarrow (C(X),\tau _p)$$ with X being a topological Hausdorff space containing at least two different points and such that the evaluations $$\{\delta _x: x\in X\}$$ are linearly independent in $$E'$$ . We prove, when X is compact and E is a Banach space containing a nowhere vanishing function, that a weighted composition operator $$C_{w,\varphi }$$ is never weakly supercyclic on E. We also prove that if the symbol $$\varphi $$ lies in the unit ball of $$A(\mathbb {D})$$ , then every weighted composition operator can never be $$\tau _p$$ -supercyclic neither on $$C(\mathbb {D})$$ nor on the disc algebra $$A(\mathbb {D})$$ . 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.