{
  "abstract": "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 $\\mathbb{C}^{d}$, 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 $Ω\\subseteq\\mathbb{C}^{d}$ containing the origin, and if $k_{w}^{2}\\in\\mathcal H(k)$ for every $w\\inΩ$, then a bounded linear functional $Λ$ on $\\mathcal H(k)$ satisfying $Λ(1)=1$ is multiplicative if and only if $Λ(k_{w}^{2})=Λ(k_{w})^{2}$ for all $w\\inΩ$. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.21808",
  "author_total": 4,
  "authors": [
    {
      "name": "Tirthankar Bhattacharyya"
    },
    {
      "name": "Jaikishan"
    },
    {
      "name": "Poornendu Kumar"
    },
    {
      "name": "Chaman Kumar Sahu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bx5vb3k8f43t1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.21808",
  "fulltext_url": null,
  "id": "bx5vb3k8f43t1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.21808",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7162077959",
  "pdf_url": "https://arxiv.org/pdf/2605.21808",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bx5vb3k8f43t1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2605.21808",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.21808"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Multiplicative linear functionals on reproducing kernel Hilbert spaces",
  "title_text": "Multiplicative linear functionals on reproducing kernel Hilbert spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/bx5vb3k8f43t1",
  "venue": "arXiv",
  "versions": []
}