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<infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<q<p<\infty $$\end{document}, the difference of two weighted composition operators between Fock spaces W psi 1,phi 1-W psi 2,phi 2:Fp -> Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{\psi _1,\varphi _1}-W_{\psi _2,\varphi _2}:\mathcal {F}<^>p\rightarrow \mathcal {F}<^>q$$\end{document} is bounded if and only if both W psi 1,phi 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{\psi _1,\varphi _1}$$\end{document} and W psi 2,phi 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{\psi _2,\varphi _2}$$\end{document} 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 Fp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}<^>p$$\end{document}.