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 Λ(x2)=Λ(x)2. We develop analogues of this theorem for bounded linear functionals on reproducing kernel Hilbert spaces of holomorphic functions on domains in ℂd, 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 Ω⊆ℂd containing the origin, and if kw2∈ℋ(k) for every w∈Ω, then a bounded linear functional Λ on ℋ(k) satisfying Λ(1)=1 is multiplicative if and only if Λ(kw2)=Λ(kw)2 for all w∈Ω. 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 Λ 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 Λ 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.
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BibTeX
@misc{bhattacharyya2026multiplicative,
title = {{Multiplicative linear functionals on reproducing kernel Hilbert spaces}},
author = {Tirthankar Bhattacharyya and Jaikishan and Poornendu Kumar and Chaman Kumar Sahu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.21808},
eprint = {2605.21808},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.21808},
}
RIS
TY - UNPB
TI - Multiplicative linear functionals on reproducing kernel Hilbert spaces
AU - Tirthankar Bhattacharyya
AU - Jaikishan
AU - Poornendu Kumar
AU - Chaman Kumar Sahu
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2605.21808
UR - https://arxiv.org/abs/2605.21808
AB - The classical Gleason--Kahane--Zelazko theorem characterizes multiplicative linear functionals on a unital Banach algebra through the scalar identity Λ(x^2)=Λ(x)^2. We develop analogues of this theorem for bounded linear functionals on reproducing kernel Hilbert spaces of holomorphic functions on domains in 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 Ω⊆ℂ^d containing the origin, and if k_w^2∈ℋ(k) for every w∈Ω, then a bounded linear functional Λ on ℋ(k) satisfying Λ(1)=1 is multiplicative if and only if Λ(k_w^2)=Λ(k_w)^2 for all w∈Ω. 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 Λ 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 Λ 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.
ER -