{
  "id": "1706.03570",
  "version": "v1",
  "published": "2017-06-12T11:26:32.000Z",
  "updated": "2017-06-12T11:26:32.000Z",
  "title": "Some examples of composition operators and their approximation numbers on the Hardy space of the bi-disk",
  "authors": [
    "Daniel Li",
    "Hervé Queffélec",
    "Luis Rodríguez-Piazza"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We give examples of composition operators $C\\_\\Phi$ on $H^2 (\\D^2)$ showing that the condition $\\|\\Phi \\|\\_\\infty = 1$ is not sufficient for their approximation numbers $a\\_n (C\\_\\Phi)$ to satisfy $\\lim\\_{n \\to \\infty} [a\\_n (C\\_\\Phi) ]^{1/\\sqrt{n}} = 1$, contrary to the $1$-dimensional case. We also give a situation where this implication holds. We make a link with the Monge-Amp\\`ere capacity of the image of $\\Phi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-12T11:26:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "approximation numbers",
      "composition operators",
      "hardy space",
      "dimensional case",
      "implication holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}