{
  "id": "1409.3017",
  "version": "v1",
  "published": "2014-09-10T10:49:42.000Z",
  "updated": "2014-09-10T10:49:42.000Z",
  "title": "Composition Operators on Bohr-Bergman Spaces of Dirichlet Series",
  "authors": [
    "Maxime Bailleul",
    "Ole Fredrik Brevig"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "For $\\alpha \\in \\mathbb{R}$, let $\\mathscr{D}_\\alpha$ denote the scale of Hilbert spaces consisting of Dirichlet series $f(s) = \\sum_{n=1}^\\infty a_n n^{-s}$ that satisfy $\\sum_{n=1}^\\infty |a_n|^2/[d(n)]^\\alpha < \\infty$. The Gordon--Hedenmalm Theorem on composition operators for $\\mathscr{H}^2=\\mathscr{D}_0$ is extended to the Bergman case $\\alpha>0$. These composition operators are generated by functions of the form $\\Phi(s) = c_0 s + \\varphi(s)$, where $c_0$ is a nonnegative integer and $\\varphi(s)$ is a Dirichlet series with certain convergence and mapping properties. For the operators with $c_0=0$ a new phenomenon is discovered: If $0 < \\alpha < 1$, the space $\\mathscr{D}_\\alpha$ is mapped by the composition operator into a smaller space in the same scale. When $\\alpha > 1$, the space $\\mathscr{D}_\\alpha$ is mapped into a larger space in the same scale. Moreover, a partial description of the composition operators on the Dirichlet--Bergman spaces $\\mathscr{A}^p$ for $1 \\leq p < \\infty$ are obtained, in addition to new partial results for composition operators on the Dirichlet--Hardy spaces $\\mathscr{H}^p$ when $p$ is an odd integer.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-10T10:49:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B33",
      "30B50"
    ],
    "keywords": [
      "composition operator",
      "dirichlet series",
      "bohr-bergman spaces",
      "dirichlet-bergman spaces",
      "partial description"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.3017B"
    }
  }
}