{
  "id": "math/0009074",
  "version": "v2",
  "published": "2000-09-07T14:34:54.000Z",
  "updated": "2000-12-26T18:13:24.000Z",
  "title": "Multipliers of the Hardy space H^1 and power bounded operators",
  "authors": [
    "Gilles Pisier"
  ],
  "comment": "Submitted to Colloquium Math",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "We study the space of functions $\\phi\\colon \\NN\\to \\CC$ such that there is a Hilbert space $H$, a power bounded operator $T$ in $B(H)$ and vectors $\\xi,\\eta$ in $H$ such that $$\\phi(n) = < T^n\\xi,\\eta>.$$ This implies that the matrix $(\\phi(i+j))_{i,j\\ge 0}$ is a Schur multiplier of $B(\\ell_2)$ or equivalently is in the space $(\\ell_1 \\buildrel {\\vee}\\over {\\otimes} \\ell_1)^*$. We show that the converse does not hold, which answers a question raised by Peller [Pe]. Our approach makes use of a new class of Fourier multipliers of $H^1$ which we call ``shift-bounded''. We show that there is a $\\phi$ which is a ``completely bounded'' multiplier of $H^1$, or equivalently for which $(\\phi(i+j))_{i,j\\ge 0}$ is a bounded Schur multiplier of $B(\\ell_2)$, but which is not ``shift-bounded'' on $H^1$. We also give a characterization of ``completely shift-bounded'' multipliers on $H^1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2000-12-26T18:13:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B15",
      "47D03"
    ],
    "keywords": [
      "power bounded operator",
      "hardy space",
      "fourier multipliers",
      "hilbert space",
      "bounded schur multiplier"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......9074P"
    }
  }
}