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

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

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