{
  "id": "2102.01062",
  "version": "v1",
  "published": "2021-02-01T18:51:45.000Z",
  "updated": "2021-02-01T18:51:45.000Z",
  "title": "Partially isometric Toeplitz operators on the polydisc",
  "authors": [
    "Deepak K. D",
    "Deepak Pradhan",
    "Jaydeb Sarkar"
  ],
  "comment": "12 pages",
  "categories": [
    "math.FA",
    "math.CV",
    "math.OA"
  ],
  "abstract": "A Toeplitz operator $T_\\varphi$, $\\varphi \\in L^\\infty(\\mathbb{T}^n)$, is a partial isometry if and only if there exist inner functions $\\varphi_1, \\varphi_2 \\in H^\\infty(\\mathbb{D}^n)$ such that $\\varphi_1$ and $\\varphi_2$ depends on different variables and $\\varphi = \\bar{\\varphi}_1 \\varphi_2$. In particular, for $n=1$, along with new proof, this recovers a classical theorem of Brown and Douglas. \\noindent We also prove that a partially isometric Toeplitz operator is hyponormal if and only if the corresponding symbol is an inner function in $H^\\infty(\\mathbb{D}^n)$. Moreover, partially isometric Toeplitz operators are always power partial isometry (following Halmos and Wallen), and hence, up to unitary equivalence, a partially isometric Toeplitz operator with symbol in $L^\\infty(\\mathbb{T}^n)$, $n > 1$, is either a shift, or a co-shift, or a direct sum of truncated shifts. Along the way, we prove that $T_\\varphi$ is a shift whenever $\\varphi$ is inner in $H^\\infty(\\mathbb{D}^n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-01T18:51:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B35",
      "47B20",
      "30J05",
      "30H10",
      "15B05",
      "46L99"
    ],
    "keywords": [
      "partially isometric toeplitz operator",
      "inner function",
      "power partial isometry",
      "direct sum",
      "unitary equivalence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}