Difference of weighted composition operators between Bergman spaces

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].