Kernel-induced distance and its applications to Composition operators on Large Bergman spaces

TL;DR

This paper characterizes when the difference of two composition operators on weighted Bergman spaces is compact, introduces simple inducing maps, and explores the topology of the space of bounded composition operators.

Contribution

It provides a complete characterization of compact differences of composition operators and analyzes the topological structure of their space on weighted Bergman spaces.

Findings

Characterization of compact differences of composition operators

Introduction of simple inducing maps supporting the main results

Analysis of the topological path connected components in the operator space

Abstract

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