{
  "id": "1607.05815",
  "version": "v1",
  "published": "2016-07-20T04:13:57.000Z",
  "updated": "2016-07-20T04:13:57.000Z",
  "title": "Factorizations of Contractions",
  "authors": [
    "B. Krishna Das",
    "Jaydeb Sarkar",
    "Srijan Sarkar"
  ],
  "comment": "11 pages",
  "categories": [
    "math.FA",
    "math.CV",
    "math.OA"
  ],
  "abstract": "The celebrated theorem of Berger, Coburn and Lebow on pairs of commuting isometries can be formulated as follows: a pure isometry $V$ on a Hilbert space $\\mathcal{H}$ is a product of two commuting isometries $V_1$ and $V_2$ in $\\mathcal{B}(\\mathcal{H})$ if and only if there exists a Hilbert space $\\mathcal{E}$, a unitary $U$ in $\\mathcal{B}(\\mathcal{E})$ and an orthogonal projection $P$ in $\\mathcal{B}(\\mathcal{E})$ such that $(V, V_1, V_2)$ and $(M_z, M_{\\Phi}, M_{\\Psi})$ on $H^2_{\\mathcal{E}}(\\mathbb{D})$ are unitarily equivalent, where \\[ \\Phi(z)=(P+zP^{\\perp})U^*\\;\\text{and}\\; \\Psi(z)=U(P^{\\perp}+zP) \\;;(z \\in \\mathbb{D}). \\] Here we prove a similar factorization result for pure contractions. More particularly, let $T$ be a pure contraction on a Hilbert space $\\mathcal{H}$ and let $P_{\\mathcal{Q}} M_z|_{\\mathcal{Q}}$ be the Sz.-Nagy and Foias representation of $T$ for some canonical $\\mathcal{Q} \\subseteq H^2_{\\mathcal{D}}(\\mathbb{D})$. Then $T = T_1 T_2$, for some commuting contractions $T_1$ and $T_2$ on $\\mathcal{H}$, if and only if there exists $\\mathcal{B}(\\mathcal{D})$-valued polynomials $\\varphi$ and $\\psi$ of degree $ \\leq 1$ such that $\\mathcal{Q}$ is a joint $(M_{\\varphi}^*, M_{\\psi}^*)$-invariant subspace, \\[P_{\\mathcal{Q}} M_z|_{\\mathcal{Q}} = P_{\\mathcal{Q}} M_{\\varphi \\psi}|_{\\mathcal{Q}} = P_{\\mathcal{Q}} M_{\\psi \\varphi}|_{\\mathcal{Q}} \\; \\mbox{and} \\;(T_1, T_2) \\cong (P_{\\mathcal{Q}} M_{\\varphi}|_{\\mathcal{Q}}, P_{\\mathcal{Q}} M_{\\psi}|_{\\mathcal{Q}}).\\]",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-20T04:13:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A13",
      "47A20",
      "47A56",
      "47A68",
      "47B38",
      "46E20",
      "30H10"
    ],
    "keywords": [
      "hilbert space",
      "pure contraction",
      "commuting isometries",
      "similar factorization result",
      "orthogonal projection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}