A Gleason-Kahane-Żelazko theorem for modules and applications to holomorphic function spaces

2015/10/27 by Javad Mashreghi, Thomas Ransford, Mashreghi, Javad +1 · 1 citation
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #math.CV #math.FA

paper · pdf · doi:10.48550/arxiv.1510.08128

arxiv created 2015/10/27 · arxiv updated 2015/10/29

Abstract

We generalize the Gleason-Kahane-Żelazko 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.

Cited by

• Gleason–Kahane–Zelazko theorems in function spaces

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