A Gleason-Kahane-Żelazko theorem for reproducing kernel Hilbert spaces

Abstract

We establish the following Hilbert-space analogue of the Gleason-Kahane-Å»elazko theorem. If $\mathcal{H}$ is a reproducing kernel Hilbert space with a normalized complete Pick kernel, and if $\hat{I}›$ is a linear functional on $\mathcal{H}$ such that $\hat{I}›(1)=1$ and $\hat{I}›(f)\ne 0$ for all cyclic functions $f\in \mathcal{H}$, then $\hat{I}›$ is multiplicative, in the sense that $\hat{I}›(fg)=\hat{I}›(f)\hat{I}›(g)$ for all $f,g\in \mathcal{H}$ such that $fg\in \mathcal{H}$. Moreover $\hat{I}›$ 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 $\hat{I}›$ has to be a point evaluation.