{
  "id": "1804.10693",
  "version": "v1",
  "published": "2018-04-27T21:28:04.000Z",
  "updated": "2018-04-27T21:28:04.000Z",
  "title": "Weak products of complete Pick spaces",
  "authors": [
    "Alexandru Aleman",
    "Michael Hartz",
    "John E. McCarthy",
    "Stefan Richter"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\mathcal H$ be the Drury-Arveson or Dirichlet space of the unit ball of $\\mathbb C^d$. The weak product $\\mathcal H\\odot\\mathcal H$ of $\\mathcal H$ is the collection of all functions $h$ that can be written as $h=\\sum_{n=1}^\\infty f_n g_n$, where $\\sum_{n=1}^\\infty \\|f_n\\|\\|g_n\\|<\\infty$. We show that $\\mathcal H\\odot\\mathcal H$ is contained in the Smirnov class of $\\mathcal H$, i.e. every function in $\\mathcal H\\odot\\mathcal H$ is a quotient of two multipliers of $\\mathcal H$, where the function in the denominator can be chosen to be cyclic in $\\mathcal H$. As a consequence we show that the map $\\mathcal N \\to clos_{\\mathcal H\\odot\\mathcal H} \\mathcal N$ establishes a 1-1 and onto correspondence between the multiplier invariant subspaces of $\\mathcal H$ and of $\\mathcal H\\odot\\mathcal H$. The results hold for many weighted Besov spaces $\\mathcal H$ in the unit ball of $\\mathbb C^d$ provided the reproducing kernel has the complete Pick property. One of our main technical lemmas states that for weighted Besov spaces $\\mathcal H$ that satisfy what we call the multiplier inclusion condition any bounded column multiplication operator $\\mathcal H \\to \\oplus_{n=1}^\\infty \\mathcal H$ induces a bounded row multiplication operator $\\oplus_{n=1}^\\infty \\mathcal H \\to \\mathcal H$. For the Drury-Arveson space $H^2_d$ this leads to an alternate proof of the characterization of interpolating sequences in terms of weak separation and Carleson measure conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-27T21:28:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complete pick spaces",
      "weak product",
      "weighted besov spaces",
      "unit ball",
      "carleson measure conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}