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

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