The difference of weighted composition operators on Fock spaces

In this note, we solve the open problem posted by Tien and Khoi (Monatsh Math 188:183–193, 2019). We prove that when \(0<q<p<\infty \) $0<q<p<\infty$ , the difference of two weighted composition operators between Fock spaces \(W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}:\mathcal {F}^p\rightarrow \mathcal {F}^q\) $W_{{\psi }_1,{\phi }_1}-W_{{\psi }_2,{\phi }_2}:{\mathcal{F}}^p\to {\mathcal{F}}^q$ is bounded if and only if both \(W_{\psi _1,\varphi _1}\) $W_{{\psi }_1,{\phi }_1}$ and \(W_{\psi _2,\varphi _2}\) $W_{{\psi }_2,{\phi }_2}$ are bounded. Furthermore, we prove that the same conclusion holds for the differences of a weighted composition operator and a weighted composition-differential operator on \(\mathcal {F}^p\) ${\mathcal{F}}^p$ .