Difference of weighted composition operators between Bergman spaces

TL;DR

This paper simplifies proofs and extends characterizations of the boundedness, compactness, and Schatten class membership of differences of weighted composition operators between Bergman spaces and Hardy spaces, revealing new rigidity results.

Contribution

It provides a simpler proof of existing results and introduces new characterizations for the difference of weighted composition operators, including their Schatten class membership and compactness criteria.

Findings

Simplified proof of boundedness and compactness characterizations.

Characterizations for Schatten class membership.

Rigidity result for compact differences on Hardy space.

Abstract

We first obtain a simpler proof of the main results in [IEOT, {\bf 93}(2021), 17], which characterized the bounded and compact differences $C_{u,\varphi}-C_{v,\psi}$ of two weighted composition operators acting from $A_{\alpha}^p(\mathbb{D})$ to $A_{\alpha}^q(\mathbb{D})$. Then we get some characterizations for the difference of weighted composition operator belonging to Schatten class. Moreover, compact difference of a weighted composition operator and a unweighted composition operator on Hardy space is also studied. It shows a rigidity, i.e. $C_{u,\varphi}-C_{\psi}$ is compact on $H^{2}(\mathbb{D})$ if and only if both $C_{u',\varphi}:H^2(\mathbb{D})\to A_{1}^2(\mathbb{D})$ and $C_{u\varphi',\varphi}-C_{\psi',\psi}:A_{1}^2(\mathbb{D})\to A_{1}^2(\mathbb{D})$ are compact, which generalize a result in [JFA, {\bf 278}(2020), 108401].