{
  "id": "1806.10757",
  "version": "v1",
  "published": "2018-06-28T03:36:53.000Z",
  "updated": "2018-06-28T03:36:53.000Z",
  "title": "Reducing subspaces for multiplication operators on the Dirichlet space through local inverses and Riemann surface",
  "authors": [
    "Caixing Gu",
    "Shuaibing Luo",
    "Jie Xiao"
  ],
  "categories": [
    "math.FA",
    "math.CV",
    "math.OA"
  ],
  "abstract": "This paper is devoted to the study of reducing subspaces for multiplication operator $M_\\phi$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_\\phi$ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surface to study the reducing subspaces of $M_\\phi$ on the Bergman space, and we discover a new way to study the Riemann surface for $\\phi^{-1}\\circ\\phi$. By this means, we determine the reducing subspaces of $M_\\phi$ on the Dirichlet space when the order of $\\phi$ is $5$; $6$; $7$ and answer some questions of Douglas-Putinar-Wang \\cite{DPW12}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-28T03:36:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "reducing subspaces",
      "dirichlet space",
      "riemann surface",
      "multiplication operator",
      "local inverses"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}