Invertible weighted composition operators

Abstract:

Let $X$ be a set of analytic functions on the open unit disk $\mathbb {D}$, and let $\varphi$ be an analytic function on $\mathbb {D}$ such that $\varphi (\mathbb {D})\subseteq \mathbb {D}$ and $f\mapsto f\circ \varphi$ takes $X$ into itself. We present conditions on $X$ ensuring that if $f\mapsto f\circ \varphi$ is invertible on $X$, then $\varphi$ is an automorphism of $\mathbb {D}$, and we derive a similar result for mappings of the form $f\mapsto \psi \cdot (f\circ \varphi )$, where $\psi$ is some analytic function on $\mathbb {D}$. We obtain as corollaries of this purely function-theoretic work new results concerning invertibility of composition operators and weighted composition operators on Banach spaces of analytic functions such as $S^p$ and the weighted Hardy spaces $H^2(\beta )$.