{
  "id": "1307.0924",
  "version": "v1",
  "published": "2013-07-03T06:48:59.000Z",
  "updated": "2013-07-03T06:48:59.000Z",
  "title": "Beurling's Theorem And Invariant Subspaces For The Shift On Hardy Spaces",
  "authors": [
    "Zhijian Qiu"
  ],
  "journal": "Science in China, Series A: Math, Jan 2008, Vol. 51, No. 1, 131-142",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $G$ be a bounded open subset in the complex plane and let $H^{2}(G)$ denote the Hardy space on $G$. We call a bounded simply connected domain $W$ perfectly connected if the boundary value function of the inverse of the Riemann map from $W$ onto the unit disk $D$ is almost 1-1 rwith respect to the Lebesgure on $\\partial D$ and if the Riemann map belongs to the weak-star closure of the polynomials in $H^{\\infty}(W)$. Our main theorem states: In order that for each $M\\in Lat(M_{z})$, there exist $u\\in H^{\\infty}(G)$ such that $ M = \\vee\\{u H^{2}(G)\\}$, it is necessary and sufficient that the following hold: 1) Each component of $G$ is a perfectly connected domain. 2) The harmonic measures of the components of $G$ are mutually singular. 3) % $P^{\\infty}(\\omega) The set of polynomials is weak-star dense in $ H^{\\infty}(G)$. \\noindent Moreover, if $G$ satisfies these conditions, then every $M\\in Lat(M_{z})$ is of the form $u H^{2}(G)$, where %$u\\in H^{\\infty}(G)$ and the restriction of $u$ to each of the components of $G$ is either an inner function or zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-03T06:48:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30H05",
      "30E10",
      "46E15",
      "47B20"
    ],
    "keywords": [
      "hardy space",
      "invariant subspaces",
      "beurlings theorem",
      "connected domain",
      "boundary value function"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}