Invertible and isometric weighted composition operators

TL;DR

This paper characterizes invertible and isometric weighted composition operators on broad classes of Banach spaces of analytic functions, extending known results to more general settings.

Contribution

It provides a comprehensive description of invertible and isometric weighted composition operators on abstract Banach spaces of analytic functions, generalizing previous specific cases.

Findings

All invertible weighted composition operators are characterized.

Surjective isometric weighted composition operators are described for certain analytic function spaces.

Results extend known characterizations to more general Banach spaces.

Abstract

We consider abstract Banach spaces of analytic functions on general bounded domains that satisfy only a minimum number of axioms. We describe all invertible (equivalently, surjective) weighted composition operators acting on such spaces. For the spaces of analytic functions in the disk whose norm is given in terms of a natural seminorm (such as the typical spaces given in terms of derivatives), we describe all surjective weighted composition operators that are isometric. This generalizes a number of known results.