On the Dynamics of Weighted Composition Operators II

TL;DR

This paper provides comprehensive characterizations of expansivity for weighted composition operators across various function spaces, extending known results and exploring dynamical properties through conjugation techniques.

Contribution

It introduces complete characterizations of expansivity for weighted composition operators on diverse locally convex spaces and extends results to $L^p$ spaces, including conjugation methods.

Findings

Characterizations of expansivity for weighted composition operators on $C_0(X)$ and $C(X)_c$

Extension of expansivity results to $L^p()$ spaces

Establishment of conjugation between weighted and unweighted operators in dissipative systems

Abstract

We establish complete characterizations of various notions of expansivity for weighted composition operators on a very general class of locally convex spaces of continuous functions. This class includes several classical classes of continuous function spaces, such as the Banach spaces $C_0(X)$ of continuous scalar-valued functions vanishing at infinity on a Hausdorff locally compact space $X$, endowed with the sup norm, and the locally convex spaces $C(X)_c$ of continuous scalar-valued functions on a completely regular space $X$, endowed with the compact-open topology. We also obtain complete characterizations of various notions of expansivity for weighted composition operators on $L^p(\mu)$ spaces, thereby complementing and extending previously known results in the unweighted case. Finally, we establish a conjugation between weighted and unweighted composition operators in the case of dissipative systems on $L^p(\mu)$ spaces and apply it to the study of several dynamical properties.