Isometries of the product of composition operators on weighted Bergman space

TL;DR

This paper characterizes when the product of composition operators on weighted Bergman spaces is an isometry, revealing differences from classical Hardy spaces and generalizing known results on operator multiplicativity.

Contribution

It provides necessary and sufficient conditions for isometric products of composition operators on weighted Bergman spaces and extends classical results to new function spaces.

Findings

Product of composition operators isometry conditions derived.

Counterexample shows composition operators need not be norm one on certain spaces.

Generalization of Schwartz's result on almost multiplicative operators.

Abstract

In this paper the necessary and sufficient conditions for the product of composition operators to be isometry are obtained on weighted Bergman space. With the help of a counter example we also proved that unlike on $\mathcal{H}^2(\mathbb{D})$ and $\mathcal{A}_{\alpha}^2(\mathbb{D}),$ the composition operator on $\mathcal{S}^2(\mathbb{D})$ induced by an analytic self map on $\mathbb{D}$ with fixed origin need not be of norm one. We have generalized the Schwartz's well known result on $\mathcal{A}_{\alpha}^2(\mathbb{D})$ which characterizes the almost multiplicative operator on $\mathcal{H}^2(\mathbb{D}).$