Multiplicative linear functionals on reproducing kernel Hilbert spaces

Abstract

The classical Gleason--Kahane--Zelazko theorem characterizes multiplicative linear functionals on a unital Banach algebra through the scalar identity $\Lambda (x^2)=\Lambda (x{)}^2$. We develop analogues of this theorem for bounded linear functionals on reproducing kernel Hilbert spaces of holomorphic functions on domains in ${\mathbb{C}}^d$, replacing conditions on the whole space by tractable conditions involving only kernel functions. Our first main result shows that if $k$ is a diagonal holomorphic kernel on a domain $\Omega \subseteq {\mathbb{C}}^d$ containing the origin, and if $k_w^2\in \mathcal{H}(k)$ for every $w\in \Omega$, then a bounded linear functional $\Lambda$ on $\mathcal{H}(k)$ satisfying $\Lambda (1)=1$ is multiplicative if and only if $\Lambda (k_w^2)=\Lambda (k_w{)}^2$ for all $w\in \Omega$. Kernels satisfying $2$-point Pick property and their powers furnish a natural class of examples. When $k$ arises as a Schur product or a tensor product of complete diagonal Pick kernels, we obtain a further, more algebraic characterization of multiplicativity, expressed in terms of the values of $\Lambda$ on kernel functions and their reciprocals. This framework subsumes the weighted Bergman and Dirichlet-type spaces, as well as the Hardy space on the polydisc. We extend the analysis to Hilbert spaces associated with diagonal Dirichlet series kernels on half-planes, encompassing in particular the Hardy space of Dirichlet series and its Riemann zeta reproducing kernel. Explicit examples demonstrate that the boundedness hypothesis on $\Lambda$ cannot be omitted. Finally, our characterization of multiplicative linear functionals leads to characterizations of weighted composition operators on a reproducing kernel Hilbert space associated with a diagonal holomorphic kernel.

arXiv commentThe manuscript has been thoroughly revised and includes several new results. A new co-author has also been added