{
  "id": "1405.1388",
  "version": "v2",
  "published": "2014-05-06T18:21:38.000Z",
  "updated": "2014-10-22T21:05:44.000Z",
  "title": "Rudin's Submodules of $H^2(\\mathbb{D}^2)$",
  "authors": [
    "B. K. Das",
    "Jaydeb sarkar"
  ],
  "comment": "6 pages. Revised. To appear in C. R. Acad. Sci. Paris",
  "categories": [
    "math.FA",
    "math.CV",
    "math.OA"
  ],
  "abstract": "Let $\\{\\alpha_n\\}_{n\\geq 0}$ be a sequence of scalars in the open unit disc of $\\mathbb{C}$, and let $\\{l_n\\}_{n\\geq 0}$ be a sequence of natural numbers satisfying $\\sum_{n=0}^\\infty (1 - l_n|\\alpha_n|) <\\infty$. Then the joint $(M_{z_1}, M_{z_2})$ invariant subspace \\[\\mathcal{S}_{\\Phi} = \\vee_{n=0}^\\infty \\Big( z_1^n \\prod_{k=n}^\\infty \\left(\\frac{-\\bar{\\alpha}_k}{|\\alpha_k|} \\frac{z_2 - \\alpha_k}{1 - \\bar{\\alpha}_k z_2}\\right)^{l_k} H^2(\\mathbb{D}^2)\\Big),\\] is called a Rudin submodule. In this paper we analyze the class of Rudin submodules and prove that \\[ \\text{dim} (\\mathcal{S}_{\\Phi}\\ominus (z_1 \\mathcal{S}_{\\Phi}+ z_2\\mathcal{S}_{\\Phi}))= 1+\\#\\{n\\ge 0: \\alpha_n=0\\}<\\infty. \\]In particular, this answer a question earlier raised by Douglas and Yang (2000).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-06T18:21:38.000Z",
      "abstract": "Let $\\{\\alpha_n\\}_{n\\ge 0}$ be a sequence of scalars in the open unit disc of $\\mathbb{C}$, and let $\\{l_n\\}_{n\\ge 0}$ be a sequence of natural numbers satisfying $\\sum_{n=0}^\\infty (1 - l_n|\\alpha_n|) <\\infty$. Then the joint $(M_{z_1}, M_{z_2})$ invariant subspace \\[\\mathcal{S}_{\\Phi} = \\vee_{n=0}^\\infty \\Big(z_1^n \\prod_{k=n}^\\infty \\left(\\frac{-\\bar{\\alpha}_k}{|\\alpha_k|} \\frac{z_2 - \\alpha_k}{1 - \\bar{\\alpha}_k z_2}\\right)^{l_k} H^2(\\mathbb{D}^2)\\Big),\\] is called a Rudin submodule. In this paper we analyze the class of Rudin submodules and prove that \\[ \\text{dim} (\\mathcal{S}_{\\Phi}\\ominus (z_1 \\cls_{\\Phi}+ z_2\\cls_{\\Phi}))= 1+#\\{n\\ge 0: \\alpha_n=0\\}<\\infty. \\]In particular, this answer a question earlier raised by Douglas and Yang.",
      "comment": "6 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-22T21:05:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A13",
      "47A15",
      "47B38",
      "46E20",
      "30H10"
    ],
    "keywords": [
      "rudins submodules",
      "rudin submodule",
      "open unit disc",
      "question earlier",
      "invariant subspace"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.1388D"
    }
  }
}