On the Dynamics of Weighted Composition Operators

TL;DR

This paper investigates various dynamical properties of weighted composition operators on different function and sequence spaces, providing new insights into their behavior and applications to classical operators like weighted shifts and translations.

Contribution

It offers a comprehensive analysis of power-boundedness, chaos, and boundedness properties of weighted composition operators, with new results and applications to classical sequence and function spaces.

Findings

Characterization of chaos and boundedness for weighted composition operators

Applications to weighted shifts on classical sequence spaces

Applications to weighted translation operators on classical function spaces

Abstract

We study the properties of power-boundedness, Li-Yorke chaos, distributional chaos, absolutely Ces\`aro boundedness and mean Li-Yorke chaos for weighted composition operators on $L^p(\mu)$ spaces and on $C_0(\Omega)$ spaces. We illustrate the general results by presenting several applications to weighted shifts on the classical sequence spaces $c_0(\mathbb{N})$, $c_0(\mathbb{Z})$, $\ell^p(\mathbb{N})$ and $\ell^p(\mathbb{Z})$ ($1 \leq p < \infty$) and to weighted translation operators on the classical function spaces $C_0[1,\infty)$, $C_0(\mathbb{R})$, $L^p[1,\infty)$ and $L^p(\mathbb{R})$ ($1 \leq p < \infty$).