{
  "id": "1611.09085",
  "version": "v1",
  "published": "2016-11-28T11:50:37.000Z",
  "updated": "2016-11-28T11:50:37.000Z",
  "title": "Uniform Continuity and Quantization on Bounded Symmetric Domains",
  "authors": [
    "Wolfram Bauer",
    "Raffael Hagger",
    "Nikolai Vasilevski"
  ],
  "comment": "26 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We consider Toeplitz operators $T_f^{\\lambda}$ with symbol $f$ acting on the standard weighted Bergman spaces over a bounded symmetric domain $\\Omega\\subset \\mathbb{C}^n$. Here $\\lambda > genus-1$ is the weight parameter. The classical asymptotic semi-commutator relation $\\lim_{\\lambda \\rightarrow \\infty} \\big{\\|}T_f^{\\lambda} T_g^{\\lambda} -T_{fg}^{\\lambda} \\big{\\|}=0$ with $f,g \\in C(\\overline{\\mathbb{B}^n})$, where $\\Omega=\\mathbb{B}^n$ denotes the complex unit ball, is extended to larger classes of bounded and unbounded operator symbol-functions and to more general domains. We deal with operator symbols that generically are neither continuous inside $\\Omega$ (Section 4) nor admit a continuous extension to the boundary (Section 3 and 4). Let $\\beta$ denote the Bergman metric distance function on $\\Omega$. We prove that the semi-commutator relation remains true for $f$ and $g$ in the space ${\\rm UC}(\\Omega)$ of all $\\beta$-uniformly continuous functions on $\\Omega$. Note that this space contains also unbounded functions. In case of the complex unit ball $\\Omega=\\mathbb{B}^n \\subset \\mathbb{C}^n$ we show that the semi-commutator relation holds true for bounded symbols in ${\\rm VMO}(\\mathbb{B}^n)$, where the vanishing oscillation inside $\\mathbb{B}^n$ is measured with respect to $\\beta$. At the same time the semi-commutator relation fails for generic bounded measurable symbols. We construct a corresponding counterexample using oscillating symbols that are continuous outside of a single point in $\\Omega$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-28T11:50:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B35",
      "81S10",
      "32M15"
    ],
    "keywords": [
      "bounded symmetric domain",
      "uniform continuity",
      "complex unit ball",
      "bergman metric distance function",
      "semi-commutator relation holds true"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}