{
  "id": "2008.12322",
  "version": "v1",
  "published": "2020-08-27T18:23:27.000Z",
  "updated": "2020-08-27T18:23:27.000Z",
  "title": "Pairs of projections and commuting isometries",
  "authors": [
    "Sandipan De",
    "Shankar. P",
    "Jaydeb Sarkar",
    "Sankar T. R"
  ],
  "comment": "26 pages",
  "categories": [
    "math.FA",
    "math.CV",
    "math.OA"
  ],
  "abstract": "Given a separable Hilbert space $\\mathcal{E}$, let $H^2_{\\mathcal{E}}(\\mathbb{D})$ denote the $\\mathcal{E}$-valued Hardy space on the open unit disc $\\mathbb{D}$ and let $M_z$ denote the shift operator on $H^2_{\\mathcal{E}}(\\mathbb{D})$. It is known that a commuting pair of isometries $(V_1, V_2)$ on $H^2_{\\mathcal{E}}(\\mathbb{D})$ with $V_1 V_2 = M_z$ is associated to an orthogonal projection $P$ and a unitary $U$ on $\\mathcal{E}$ (and vice versa). In this case, the ``defect operator'' of $(V_1, V_2)$, say $T$, is given by the difference of orthogonal projections on $\\mathcal{E}$: \\[ T = UPU^* - P. \\] This paper is an attempt to determine whether irreducible commuting pairs of isometries $(V_1, V_2)$ can be built up from compact operators $T$ on $\\mathcal{E}$ such that $T$ is a difference of two orthogonal projections. The answer to this question is sometimes in the affirmative and sometimes in the negative. \\noindent The range of constructions of $(V_1, V_2)$ presented here also yields examples of a number of concrete pairs of commuting isometries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-27T18:23:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A13",
      "47A65",
      "47B47",
      "15A15"
    ],
    "keywords": [
      "commuting isometries",
      "orthogonal projection",
      "open unit disc",
      "commuting pair",
      "vice versa"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}