Weighted composition semigroups on some Banach spaces

TL;DR

This paper characterizes the strong continuity of weighted composition semigroups on various Banach spaces, including Hardy and Bergman spaces, providing new insights and answering open questions in operator theory.

Contribution

It offers a new characterization of strong continuity for weighted composition semigroups on classical function spaces, improving previous results and addressing open problems.

Findings

Characterization of strong continuity on Lebesgue, Hardy, and Bergman spaces.

Improved results over previous studies by Siskakis and König.

Answering an open question by Siskakis on semigroup continuity.

Abstract

We characterize strong continuity of general operator semigroups on some Lebesgue spaces. In particular, a characterization of strong continuity of weighted composition semigroups on classical Hardy spaces and weighted Bergman spaces with regular weights is given. As applications, our result improves the results of Siskakis, A. G. \cite{AG1} and K\"{o}nig, W. \cite{K} and answers a question of Siskakis, A. G. proposed in \cite{AG4}. We also characterize strongly continuous semigroups of weighted composition operators on weighted Bergman spaces in terms of abelian intertwiners of multiplication operator $M_z$.