{
  "id": "2008.00981",
  "version": "v1",
  "published": "2020-08-03T16:04:36.000Z",
  "updated": "2020-08-03T16:04:36.000Z",
  "title": "Multiplier tests and subhomogeneity of multiplier algebras",
  "authors": [
    "Alexandru Aleman",
    "Michael Hartz",
    "John E. McCarthy",
    "Stefan Richter"
  ],
  "comment": "39 pages",
  "categories": [
    "math.FA",
    "math.CV",
    "math.OA"
  ],
  "abstract": "Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of $n \\times n$ matrices analogous to the classical Pick matrix. We study for which reproducing kernel Hilbert spaces it suffices to consider matrices of bounded size $n$. We connect this problem to the notion of subhomogeneity of non-selfadjoint operator algebras. Our main results show that multiplier algebras of many Hilbert spaces of analytic functions, such as the Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence one has to test Pick matrices of arbitrarily large matrix size $n$. To treat the Drury-Arveson space, we show that multiplier algebras of certain weighted Dirichlet spaces on the disc embed completely isometrically into the multiplier algebra of the Drury-Arveson space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-03T16:04:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E22",
      "47B32",
      "47L55"
    ],
    "keywords": [
      "multiplier algebra",
      "multiplier tests",
      "reproducing kernel hilbert spaces",
      "drury-arveson space",
      "subhomogeneity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}