Difference of weighted composition operators between Bergman spaces

We first obtain a simpler proof of the main results in [IEOT, {\bf 93}(2021), 17], which characterized the bounded and compact differences $C_{u,\phi }-C_{v,\psi }$ 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. $C_{u,\phi }-C_{\psi }$ is compact on $H^2(\mathbb{D})$ if and only if both $C_{u^{\mathrm{\prime }},\phi }:H^2(\mathbb{D})\to A_1^2(\mathbb{D})$ and $C_{u{\phi }^{\mathrm{\prime }},\phi }-C_{{\psi }^{\mathrm{\prime }},\psi }:A_1^2(\mathbb{D})\to A_1^2(\mathbb{D})$ are compact, which generalize a result in [JFA, {\bf 278}(2020), 108401].