Invertible weighted composition operators

TL;DR

This paper investigates conditions under which invertible weighted composition operators on certain spaces of analytic functions imply that the symbol functions are automorphisms of the disk, providing new insights into their invertibility properties.

Contribution

It establishes criteria linking invertibility of weighted composition operators to automorphisms of the disk, extending results to various Banach spaces of analytic functions.

Findings

Invertibility of composition operators implies the symbol is an automorphism.

Results apply to Banach spaces like S^p and weighted Hardy spaces H^2(beta).

Provides new criteria for invertibility of weighted composition operators.

Abstract

Let X be a set of analytic functions on the open unit disk D, and let phi be an analytic function on D such that phi(D) is contained in D and f |-> f o phi takes X into itself. We present conditions on X ensuring that if f |-> f o phi is invertible on X, then phi is an automorphism of D, and we derive a similar result for mappings of the form f |-> psi.(f o phi), where psi is some analytic function on 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).