A Gleason-Kahane-\.Zelazko theorem for reproducing kernel Hilbert spaces

TL;DR

This paper proves a Hilbert-space analogue of the Gleason-Kahane-Żelazko theorem for reproducing kernel Hilbert spaces with complete Pick kernels, showing such linear functionals are multiplicative and continuous under certain conditions.

Contribution

The paper establishes a new Gleason-Kahane-Żelazko type theorem for RKHS with complete Pick kernels, characterizing linear functionals as multiplicative and continuous.

Findings

Linear functionals are multiplicative under given conditions

Such functionals are automatically continuous

The theorem fails without the complete Pick kernel assumption

Abstract

We establish the following Hilbert-space analogue of the Gleason-Kahane-\.Zelazko theorem. If $\mathcal{H}$ is a reproducing kernel Hilbert space with a normalized complete Pick kernel, and if $\Lambda$ is a linear functional on $\mathcal{H}$ such that $\Lambda(1)=1$ and $\Lambda(f)\ne0$ for all cyclic functions $f\in\mathcal{H}$, then $\Lambda$ is multiplicative, in the sense that $\Lambda(fg)=\Lambda(f)\Lambda(g)$ for all $f,g\in\mathcal{H}$ such that $fg\in\mathcal{H}$. Moreover $\Lambda$ is automatically continuous. We give examples to show that the theorem fails if the hypothesis of a complete Pick kernel is omitted. We also discuss conditions under which $\Lambda$ has to be a point evaluation.