Weighted composition operators: isometries and asymptotic behaviour

TL;DR

This paper investigates the iterates of weighted composition operators on Hardy spaces, focusing on their convergence properties and conditions under which they act as isometries, providing new insights into their asymptotic behavior.

Contribution

It offers new characterizations of when weighted composition operators are isometries and analyzes their asymptotic behavior on Hardy and Bergman spaces.

Findings

Resolved convergence questions for operator iterates

Identified conditions for isometric weighted composition operators

Provided new results for Hardy and Bergman spaces

Abstract

This paper studies the behaviour of iterates of weighted composition operators acting on spaces of analytic functions, with particular emphasis on the Hardy space $H^2$. Questions relating to uniform, strong and weak convergence are resolved in many cases. Connected to this is the question when a weighted composition operators is an isometry, and new results are given in the case of the Hardy and Bergman spaces.