Weighted composition operators on extended analytic Lipschitz algebras

Abstract

In this paper, we study weighted composition operators on extended analytic Lipschitz algebras ${\rm Lip}_{A}(X,K,\alpha)$ where $X$ is a compact plane set, $K$ is a closed subset of $X$ with nonempty interior and $0 < \alpha \leq 1$. We first give necessary conditions and sufficient conditions on a function $u \in \mathbb{C} ^{X}$ and self-map $\varphi$ of $X$ for which $T=uc_{\varphi}$ to be a weighted composition operator on ${\rm Lip}_{A}(X,K,\alpha)$. We next give the necessary conditions for these operators to be compact and provide some sufficient conditions for the compactness of such operators.