A Gleason-Kahane-\.Zelazko theorem for modules and applications to holomorphic function spaces

TL;DR

This paper extends the Gleason-Kahane-Żelazko theorem to modules and applies it to characterize linear functionals and endomorphisms on Hardy spaces and other holomorphic function spaces, revealing new structural insights without assuming continuity.

Contribution

It generalizes a classical theorem to modules and uses it to characterize linear functionals and endomorphisms on various holomorphic function spaces.

Findings

Linear functionals on Hardy spaces are multiples of point evaluations.

Endomorphisms mapping outer functions to nowhere-zero functions are weighted composition operators.

Extensions to Bergman, Dirichlet, Besov, Bloch, and VMOA spaces.

Abstract

We generalize the Gleason-Kahane-\.Zelazko theorem to modules. As an application, we show that every linear functional on a Hardy space that is non-zero on outer functions is a multiple of a point evaluation. A further consequence is that every linear endomorphism of a Hardy space that maps outer functions to nowhere-zero functions is a weighted composition operator. In neither case is continuity assumed. We also consider some extensions to other function spaces, including the Bergman, Dirichlet and Besov spaces, the little Bloch space and VMOA.