Difference of Composition operators over Bergman spaces with exponential weights

TL;DR

This paper characterizes when the difference of two composition operators on weighted Bergman spaces is compact, using a pseudodistance related to the weights, and explores the topology of the space of bounded composition operators.

Contribution

It provides a complete characterization of compact differences of composition operators on weighted Bergman spaces with exponential weights, and analyzes the topological structure of these operators.

Findings

Characterization of compact differences via $ ext{eta}$-derived pseudodistance.

Identification of simple inducing maps supporting the main results.

Analysis of the topological path components in the space of bounded composition operators.

Abstract

In this paper, we obtain a complete characterization for the compact difference of two composition operators acting on Bergman spaces with weight $\omega=e^{-\eta}$, $\Delta\eta>0$ in terms of the $\eta$-derived pseudodistance of two analytic self maps. In addition, we provide simple inducing maps which support our main result. We also study the topological path component of the space of all bounded composition operators on $A^2(\omega)$ endowed with the Hilbert-Schmidt norm topology.