{
  "id": "1302.4117",
  "version": "v2",
  "published": "2013-02-17T20:33:59.000Z",
  "updated": "2014-12-09T08:49:21.000Z",
  "title": "Approximation numbers of composition operators on the $H^2$ space of Dirichlet series",
  "authors": [
    "Hervé Queffélec",
    "Kristian Seip"
  ],
  "comment": "Final version, to appear in Journal of Functional Analysis",
  "categories": [
    "math.FA",
    "math.CV"
  ],
  "abstract": "By a theorem of Gordon and Hedenmalm, $\\varphi$ generates a bounded composition operator on the Hilbert space $\\mathscr{H}^2$ of Dirichlet series $\\sum_n b_n n^{-s}$ with square-summable coefficients $b_n$ if and only if $\\varphi(s)=c_0 s+\\psi(s)$, where $c_0$ is a nonnegative integer and $\\psi$ a Dirichlet series with the following mapping properties: $\\psi$ maps the right half-plane into the half-plane $\\operatorname{Re} s >1/2$ if $c_0=0$ and is either identically zero or maps the right half-plane into itself if $c_0$ is positive. It is shown that the $n$th approximation numbers of bounded composition operators on $\\mathscr{H}^2$ are bounded below by a constant times $r^n$ for some $0<r<1$ when $c_0=0$ and bounded below by a constant times $n^{-A}$ for some $A>0$ when $c_0$ is positive. Both results are best possible. The case when $c_0=0$, $\\psi$ is bounded and smooth up to the boundary of the right half-plane, and $\\sup \\operatorname{Re} \\psi=1/2$, is discussed in depth; it includes examples of non-compact operators as well as operators belonging to all Schatten classes $S_p$. For $\\varphi(s)=c_1+\\sum_{j=1}^d c_{q_j} q_j^{-s}$ with $q_j$ independent integers, it is shown that the $n$th approximation number behaves as $n^{-(d-1)/2}$, possibly up to a factor $(\\log n)^{(d-1)/2}$. Estimates rely mainly on a general Hilbert space method involving finite linear combinations of reproducing kernels. A key role is played by a recently developed interpolation method for $\\mathscr{H}^2$ using estimates of solutions of the $\\bar{\\partial}$ equation. Finally, by a transference principle from $H^2$ of the unit disc, explicit examples of compact composition operators with approximation numbers decaying at essentially any sub-exponential rate can be displayed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-02-17T20:33:59.000Z",
      "abstract": "By a theorem of Gordon and Hedenmalm, $\\varphi$ generates a bounded composition operator on the Hilbert space $\\Ht$ of Dirichlet series $\\sum_n b_n n^{-s}$ with square-summable coefficients $b_n$ if and only if $\\varphi(s)=c_0 s+\\psi(s)$, where $c_0$ is a nonnegative integer and $\\psi$ a Dirichlet series with the following mapping properties: $\\psi$ maps the right half-plane into the half-plane $\\Real s >1/2$ if $c_0=0$ and is either identically zero or maps the right half-plane into itself if $c_0$ is positive. It is shown that the $n$th approximation numbers of bounded composition operators on $\\Ht$ are bounded below by a constant times $r^n$ for some $0<r<1$ when $c_0=0$ and bounded below by a constant times $n^{-A}$ for some $A>0$ when $c_0$ is positive. Both results are best possible. The case when $c_0=0$, $\\psi$ is bounded and smooth up to the boundary of the right half-plane, and $\\sup \\Real \\psi=1/2$, is discussed in depth; it includes examples of non-compact operators as well as operators belonging to all Schatten classes $S_p$. For $\\varphi(s)=c_1+\\sum_{j=1}^d c_{q_j} q_j^{-s}$ with $q_j$ independent integers, it is shown that the $n$th approximation number behaves as $n^{-(d-1)/2}$, possibly up to a factor $(\\log n)^{(d-1)/2}$. Estimates rely mainly on a general Hilbert space method involving finite linear combinations of reproducing kernels. A key role is played by a recently developed interpolation method for $\\Ht$ using estimates of solutions of the $\\bar{\\partial}$ equation. Finally, by a transference principle from $H^2$ of the unit disc, explicit examples of compact composition operators with approximation numbers decaying at essentially any sub-exponential rate can be displayed.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-09T08:49:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B33",
      "30B50",
      "30H10"
    ],
    "keywords": [
      "dirichlet series",
      "right half-plane",
      "bounded composition operator",
      "constant times",
      "general hilbert space method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.4117Q"
    }
  }
}