{
  "id": "0901.3787",
  "version": "v1",
  "published": "2009-01-23T21:14:59.000Z",
  "updated": "2009-01-23T21:14:59.000Z",
  "title": "Multiplication operators on the Bergman space via analytic continuation",
  "authors": [
    "Ronald G. Douglas",
    "Shunhua Sun",
    "Dechao Zheng"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper, using the group-like property of local inverses of a finite Blaschke product $\\phi$, we will show that the largest $C^*$-algebra in the commutant of the multiplication operator $M_{\\phi}$ by $\\phi$ on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of $\\phi^{-1}\\circ\\phi $ over the unit disk. If the order of the Blaschke product $\\phi$ is less than or equal to eight, then every $C^*$-algebra contained in the commutant of $M_{\\phi}$ is abelian and hence the number of minimal reducing subspaces of $M_{\\phi}$ equals the number of connected components of the Riemann surface of $\\phi^{-1}\\circ\\phi $ over the unit disk.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-01-23T21:14:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B35",
      "46E20"
    ],
    "keywords": [
      "multiplication operator",
      "bergman space",
      "analytic continuation",
      "riemann surface",
      "unit disk"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}