Multiplicativity of linear functionals on function spaces on an open unit disc

TL;DR

This paper generalizes the Gleason-Kahane-Zelazko theorem to various function spaces on the open unit disc, characterizing linear functionals as point evaluations and exploring their properties across different spaces.

Contribution

It extends the GKZ theorem to a broad class of function spaces without topological assumptions, including Hardy, Bergman, and Dirichlet spaces, under continuity conditions.

Findings

Characterization of linear functionals as point evaluations on polynomial spaces

Extension of GKZ theorem to multiple function spaces

Continuity is shown to be necessary for the theorem's conclusions

Abstract

This paper presents a fairly general version of the well-known Gleason-Kahane-$\dot{\text{Z}}$elazko (GKZ) theorem in the spirit of a GKZ type theorem obtained recently by Mashreghi and Ransford for Hardy spaces. In effect, we characterize a class of linear functionals as point evaluations on the vector space of all complex polynomials $\cl P$. We do not make any topological assumptions on $\cl P$. We then apply this characterization to present a version of the GKZ theorem for a vast class of topological spaces of complex-valued functions including the Hardy, Bergman, Dirichlet, and many more well-known function spaces. We obtain this result under the assumption of continuity of the linear functional, which we show, with the help of an example, to be a necessary condition for the desired conclusion. Lastly, we use the GKZ theorem for polynomials to obtain a version of the GKZ theorem for strictly cyclic weighted Hardy spaces.