{
  "id": "1808.01788",
  "version": "v1",
  "published": "2018-08-06T09:13:58.000Z",
  "updated": "2018-08-06T09:13:58.000Z",
  "title": "Toeplitz operators with piecewise continuous symbols on the Hardy space $H^1$",
  "authors": [
    "Santeri Miihkinen",
    "Jani A. Virtanen"
  ],
  "comment": "6 pages, 1 figure",
  "categories": [
    "math.FA",
    "math.CV",
    "math.SP"
  ],
  "abstract": "The geometric descriptions of the (essential) spectra of Toeplitz operators with piecewise continuous symbols are among the most beautiful results about Toeplitz operators on Hardy spaces $H^p$ with $1<p<\\infty$. In the Hardy space $H^1$, the essential spectra of Toeplitz operators are known for continuous symbols and symbols in the Douglas algebra $C+H^\\infty$. It is natural to ask whether the theory for piecewise continuous symbols can also be extended to $H^1$. We answer this question in negative and show in particular that the Toeplitz operator is never bounded on $H^1$ if its symbol has a jump discontinuity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-06T09:13:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B35",
      "30H10"
    ],
    "keywords": [
      "toeplitz operator",
      "piecewise continuous symbols",
      "hardy space",
      "essential spectra",
      "douglas algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}