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

We establish the following Hilbert-space analog of the Gleason-Kahane-Zelazko theorem. If H${\mathcal {H}}$ is a reproducing kernel Hilbert space with a normalized complete Pick kernel, and if ?$\Lambda$ is a linear functional on H${\mathcal {H}}$ such that ?(1)=1$\Lambda (1)=1$ and ?(f)not equal 0$\Lambda (f)\ne 0$ for all cyclic functions f is an element of H$f\in {\mathcal {H}}$, then ?$\Lambda$ is multiplicative, in the sense that ?(fg)=?(f)?(g)$\Lambda (fg)=\Lambda (f)\Lambda (g)$ for all f,g is an element of H$f,g\in {\mathcal {H}}$ such that fg is an element of H$fg\in {\mathcal {H}}$. 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.