{
  "id": "1701.07476",
  "version": "v1",
  "published": "2017-01-25T20:33:58.000Z",
  "updated": "2017-01-25T20:33:58.000Z",
  "title": "The Smirnov class for spaces with the complete Pick property",
  "authors": [
    "Alexandru Aleman",
    "Michael Hartz",
    "John E. McCarthy",
    "Stefan Richter"
  ],
  "comment": "18 pages",
  "categories": [
    "math.FA",
    "math.CV"
  ],
  "abstract": "We show that every function in a reproducing kernel Hilbert space with a normalized complete Pick kernel is the quotient of a multiplier and a cyclic multiplier. This extends a theorem of Alpay, Bolotnikov and Kaptano\\u{g}lu. We explore various consequences of this result regarding zero sets, spaces on compact sets and Gleason parts. In particular, using a construction of Salas, we exhibit a rotationally invariant complete Pick space of analytic functions on the unit disc for which the corona theorem fails.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-25T20:33:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E22",
      "30H15",
      "30H80"
    ],
    "keywords": [
      "complete pick property",
      "smirnov class",
      "rotationally invariant complete pick space",
      "normalized complete pick kernel",
      "corona theorem fails"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}