{
  "id": "2011.03360",
  "version": "v1",
  "published": "2020-11-04T15:12:41.000Z",
  "updated": "2020-11-04T15:12:41.000Z",
  "title": "A Gleason-Kahane-Żelazko theorem for reproducing kernel Hilbert spaces",
  "authors": [
    "Cheng Chu",
    "Javad Mashreghi",
    "Thomas Ransford"
  ],
  "categories": [
    "math.FA",
    "math.CV"
  ],
  "abstract": "We establish the following Hilbert-space analogue of the Gleason-Kahane-\\.Zelazko theorem. If $\\mathcal{H}$ is a reproducing kernel Hilbert space with a normalized complete Pick kernel, and if $\\Lambda$ is a linear functional on $\\mathcal{H}$ such that $\\Lambda(1)=1$ and $\\Lambda(f)\\ne0$ for all cyclic functions $f\\in\\mathcal{H}$, then $\\Lambda$ is multiplicative, in the sense that $\\Lambda(hf)=\\Lambda(h)\\Lambda(f)$ for all $f\\in\\mathcal{H}$ and for all multipliers $h$ of $\\mathcal{H}$. Continuity of $\\Lambda$ is not assumed. We give an example to show that the theorem fails if the hypothesis of a complete Pick kernel is omitted. We also discuss conditions under which $\\Lambda$ has to be a point evaluation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-04T15:12:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E22"
    ],
    "keywords": [
      "reproducing kernel hilbert space",
      "gleason-kahane-żelazko theorem",
      "normalized complete pick kernel",
      "cyclic functions",
      "hilbert-space analogue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}