{
  "id": "1206.1179",
  "version": "v1",
  "published": "2012-06-06T10:58:59.000Z",
  "updated": "2012-06-06T10:58:59.000Z",
  "title": "Estimates for approximation numbers of some classes of composition operators on the Hardy space",
  "authors": [
    "Daniel Li",
    "Hervé Queffélec",
    "Luis Rodriguez-Piazza"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We give estimates for the approximation numbers of composition operators on $H^2$, in terms of some modulus of continuity. For symbols whose image is contained in a polygon, we get that these approximation numbers are dominated by $\\e^{- c \\sqrt n}$. When the symbol is continuous on the closed unit disk and has a domain touching the boundary non-tangentially at a finite number of points, with a good behavior at the boundary around those points, we can improve this upper estimate. A lower estimate is given when this symbol has a good radial behavior at some point. As an application we get that, for the cusp map, the approximation numbers are equivalent, up to constants, to $\\e^{- c \\, n / \\log n}$, very near to the minimal value $\\e^{- c \\, n}$. We also see the limitations of our methods. To finish, we improve a result of O. El-Fallah, K. Kellay, M. Shabankhah and H. Youssfi, in showing that for every compact set $K$ of the unit circle $\\T$ with Lebesgue measure 0, there exists a compact composition operator $C_\\phi \\colon H^2 \\to H^2$, which is in all Schatten classes, and such that $\\phi = 1$ on $K$ and $|\\phi | < 1$ outside $K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-06T10:58:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "approximation numbers",
      "hardy space",
      "compact composition operator",
      "cusp map",
      "closed unit disk"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.1179L"
    }
  }
}