Decomposability of Weighted Composition Operators on $L^p$ of Atomic Measure Space

Abstract

In this paper, we discuss the decomposability of weighted composition operator $uC_\phi$ on $L^p(X)(1\leq p<\infty)$ of a $\sigma$-finite atomic measure space $(X,\mathcal{S},\mu)$ with the assumption that $u\in L^\infty(X)$ and $|u|$ has positive ess inf. We prove that if the analytic core of $uC_\phi$ is zero and $uC_\phi$ is not quasinilpotent, then it is not decomposable. We also show that if $\phi$ is either injective almost everywhere or surjective almost everywhere but not both, then $uC_\phi$ is not decomposable. Finally, we give a necessary condition for decomposability of $uC_\phi$.