{
  "id": "1708.02609",
  "version": "v1",
  "published": "2017-08-08T18:55:14.000Z",
  "updated": "2017-08-08T18:55:14.000Z",
  "title": "Pairs of commuting isometries - I",
  "authors": [
    "Amit Maji",
    "Jaydeb Sarkar",
    "Sankar T. R"
  ],
  "comment": "25 pages",
  "categories": [
    "math.FA",
    "math.CV",
    "math.OA"
  ],
  "abstract": "We present a sharper version of Berger, Coburn and Lebow's classification result for pure pairs of commuting isometries in the sense of an explicit recipe for constructing pairs of isometric (and bounded holomorphic) multipliers with precise coefficients. One of our main results states that: Let $(V_1, V_2)$ be a pair of commuting isometries on a Hilbert space $\\mathcal{H}$, and let $V = V_1 V_2$ be a pure isometry. Let $\\mathcal{W}_1$, $\\mathcal{W}_2$ and $\\mathcal{W}$ be the wandering subspaces for $V_1$, $V_2$ and $V$, respectively, and let $H^2_{\\mathcal{W}}(\\mathbb{D})$ denote the $\\mathcal{W}$-valued Hardy space over the unit disc $\\mathbb{D}$. Then $(V_1, V_2, V)$ on $\\mathcal{H}$ and $(M_{\\Phi_1}, M_{\\Phi_2}, M_z)$ on $H^2_{\\mathcal{W}}(\\mathbb{D})$ are jointly unitarily equivalent, where \\[ {\\Phi_1}(z) = V_1|_{\\mathcal{W}_2} + V_2^*|_{V_2 \\mathcal{W}_1} z = U^* (P_{\\mathcal{W}_2} + zP_{\\mathcal{W}_2}^{\\perp}), \\] \\[ {\\Phi_2}(z) = V_2|_{\\mathcal{W}_1} + V_1^*|_{V_1 \\mathcal{W}_2} z = (P_{\\mathcal{W}_2}^{\\perp} + zP_{\\mathcal{W}_2}) U, \\] for all $z \\in \\mathbb{D}$, and \\[ U= \\begin{bmatrix} V_2|_{\\mathcal{W}_1} & 0\\\\ 0 & V_1^{*}|_{V_1\\mathcal{W}_2} \\end{bmatrix}, \\] is a unitary operator on $\\mathcal{W} = \\mathcal{W}_1 \\oplus V_1\\mathcal{W}_2 = V_2 \\mathcal{W}_1 \\oplus \\mathcal{W}_2$. As a consequence it follows that the pair $(V_1|_{\\mathcal{W}_2}, V_2^*|_{V_2 \\mathcal{W}_1})$ is a complete set of (joint) unitary invariants for pure pairs of commuting isometries. We also compare the above representation with other natural analytic representations of $(V_1, V_2)$. Finally, we study the defect operators of (not necessarily pure) pairs of commuting isometries. We prove that the defect operator of a pair of commuting isometries is negative if and only if the defect operator is the zero operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-08T18:55:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A05",
      "47A13",
      "47A20",
      "47A45",
      "47A65",
      "46E22",
      "46E40"
    ],
    "keywords": [
      "commuting isometries",
      "defect operator",
      "pure pairs",
      "lebows classification result",
      "natural analytic representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}