{
  "id": "1303.1041",
  "version": "v3",
  "published": "2013-03-05T14:14:27.000Z",
  "updated": "2013-04-15T20:29:04.000Z",
  "title": "Jordan Blocks of H^2(D^n)",
  "authors": [
    "Jaydeb Sarkar"
  ],
  "comment": "14 pages. Revised. To appear in the Journal of Operator Theory",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "We develop a several variables analog of the Jordan blocks of the Hardy space $H^2(\\mathbb{D})$. In this consideration, we obtain a complete characterization of the doubly commuting quotient modules of the Hardy module $H^2(\\mathbb{D}^n)$. We prove that a quotient module $\\clq$ of $H^2(\\mathbb{D}^n)$ ($n \\geq 2$) is doubly commuting if and only if \\[\\clq = \\clq_{\\Theta_1} \\otimes \\cdots \\otimes \\clq_{\\Theta_n},\\]where each $\\clq_{\\Theta_i}$ is either a one variable Jordan block $H^2(\\mathbb{D})/\\Theta_i H^2(\\mathbb{D})$ for some inner function $\\Theta_i$ or the Hardy module $H^2(\\mathbb{D})$ on the unit disk for all $i = 1, \\ldots, n$. We say that a submodule $\\cls$ of $H^2(\\mathbb{D}^n)$ is a co-doubly commuting if the quotient module $H^2(\\mathbb{D}^n)/\\cls$ is doubly commuting. We obtain a Beurling like theorem for the class of co-doubly commuting submodules of $H^2(\\mathbb{D}^n)$. We prove that a submodule $\\cls$ of $H^2(\\mathbb{D}^n)$ is co-doubly commuting if and only if \\[\\cls = \\mathop{\\sum}_{i=1}^m \\Theta_i H^2(\\mathbb{D}^n),\\]for some integer $m \\leq n$ and one variable inner functions $\\{\\Theta_i\\}_{i=1}^m$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-04-15T20:29:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A13",
      "47A15",
      "47A20",
      "47A45",
      "47A80",
      "46E20",
      "30H10"
    ],
    "keywords": [
      "hardy module",
      "co-doubly commuting",
      "doubly commuting quotient modules",
      "complete characterization",
      "hardy space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.1041S"
    }
  }
}