论文标题
GLEASON-KAHANE-孙-Melazko定理，用于复制内核希尔伯特空间
A Gleason-Kahane-Żelazko theorem for reproducing kernel Hilbert spaces
论文作者
论文摘要
我们建立了Gleason-Kahane-Zelazko定理的以下希尔伯特空间类似物。如果$ \ Mathcal {h} $是具有归一化的完整挑选内核的复制品Hilbert空间，并且如果$λ$是在$ \ MATHCAL {H} $上的线性功能，则$λ（1）= 1 $ and $λ（f）$ f）$ flipitions $ flieliates $ flacitions $ flipions $ f \ in Mather placitions $ f \ in Matire placitions \ in Mruptim placitions placitions placitions placitions placitions placitions placitions placitions placitions。从某种意义上说，$λ（fg）=λ（f）λ（g）$ for All $ f，g \ in \ Mathcal {H} $，使得$ fg \ in \ Mathcal {h h} $。而且，$λ$自动连续。我们举例说明，如果省略了完整的挑选内核的假设，则定理将失败。我们还讨论了$λ$必须作为点评估的条件。
We establish the following Hilbert-space analogue of the Gleason-Kahane-Żelazko theorem. If $\mathcal{H}$ is a reproducing kernel Hilbert space with a normalized complete Pick kernel, and if $Λ$ is a linear functional on $\mathcal{H}$ such that $Λ(1)=1$ and $Λ(f)\ne0$ for all cyclic functions $f\in\mathcal{H}$, then $Λ$ is multiplicative, in the sense that $Λ(fg)=Λ(f)Λ(g)$ for all $f,g\in\mathcal{H}$ such that $fg\in\mathcal{H}$. Moreover $Λ$ 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 $Λ$ has to be a point evaluation.
