Li-Yorke chaotic weighted composition operators on Hardy and Bergman spaces over the unit disk

TL;DR

This paper investigates Li-Yorke and mean Li-Yorke chaos for weighted composition operators on Hardy and Bergman spaces, establishing conditions under which these operators exhibit chaotic behavior.

Contribution

It characterizes Li-Yorke chaos for weighted composition operators on classical spaces, linking chaos to non-power-boundedness and non-absolute Cesàro boundedness.

Findings

Li-Yorke chaos occurs if and only if the operator is not power-bounded.

Mean Li-Yorke chaos occurs if and only if the operator is not absolutely Cesàro bounded.

Results apply specifically to Hardy and weighted Bergman spaces.

Abstract

We study Li--Yorke and mean Li--Yorke chaos for weighted composition operators $C_{w,\varphi}$ on Banach spaces of analytic functions on the unit disk $\mathbb{D}$. Under natural conditions on the space, we show that $C_{w,\varphi}$ is (densely) Li--Yorke chaotic if and only if it is not power-bounded, and (densely) mean Li--Yorke chaotic if and only if it is not absolutely Ces\`aro bounded. These results are applied to Hardy spaces $H^p(\mathbb{D})$, $1 \le p \le \infty$, and weighted Bergman spaces $A^p_\beta(\mathbb{D})$, $-1 < \beta < \infty$ and $1 < p < \infty$.