The associated weight and the essential norm of weighted composition operators

Abstract

For an almost radial and typical weight $v$, we characterize the continuity and compactness of the weighted composition operator $u C_{\varphi}$ acting on the weighted Banach spaces of analytic functions $H_{v}^{\infty}$ in terms of the $n$th power of the symbol $\varphi$, where $u: \mathbb{D} \to \mathbb{C}$ is an analytic function and $\varphi: \mathbb{D} \to \mathbb{D}$ is analytic. More precisely, we extend the results of Hyv\"arinen, Kemppainen, Lindstr\"om, Rautio and Saukko relating to the continuity and essential norm calculus of the operator $u C_{\varphi}$.