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

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.