# Dynamics of weighted composition operators on spaces of continuous functions

## TL;DR

This paper investigates the dynamics of weighted composition operators on various spaces of continuous and holomorphic functions, establishing conditions under which these operators are not supercyclic and exploring their spectral properties.

## Contribution

It provides new results on the non-supercyclicity of weighted composition operators in specific functional spaces and extends the analysis to spectral conditions and supercyclicity in complex domains.

## Key findings

- Weighted composition operators are never weakly supercyclic on certain Banach spaces.
- Operators with symbols in the unit ball of A(𝔻) are not τ_p-supercyclic on C(𝔻) or A(𝔻).
- No composition operator is weakly supercyclic on spaces of holomorphic functions on punctured domains.

## 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 $C_{\varphi,\omega}$ 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.