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.