{
  "id": "1312.0322",
  "version": "v2",
  "published": "2013-12-02T04:11:27.000Z",
  "updated": "2014-11-12T06:34:00.000Z",
  "title": "A Note on Tetrablock Contractions",
  "authors": [
    "Haripada Sau"
  ],
  "comment": "19 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "A commuting triple of operators $(A,B,P)$ on a Hilbert space $\\mathcal{H}$ is called a tetrablock contraction if the closure of the set $$ E = \\{\\underline{x}=(x_1,x_2,x_3)\\in \\mathbb{C}^3: 1-x_1z-x_2w+x_3zw \\neq 0 \\text{whenever}|z| \\leq 1\\text{and}|w| \\leq 1 \\} $$ is a spectral set. In this paper, we have constructed a functional model and produced a complete unitary invariant for a pure tetrablock contraction. In this construction, the fundamental operators, which are the unique solutions of the operator equations $$ A-B^*P = D_PX_1D_P \\text{and} B-A^*P=D_PX_2D_P, \\text{where $X_1,X_2 \\in \\mathcal{B}(\\mathcal{D}_P)$}, $$ play a big role. As a corollary to the functional model, we show that every pure tetrablock isometry $(A,B,P)$ on a Hilbert space $\\mathcal{H}$ is unitarily equivalent to $(M_{G_1^*+G_2z}, M_{G_2^*+G_1z},M_z)$ on $H^2_{\\mathcal{D}_{P^*}}(\\mathbb{D})$, where $G_1$ and $G_2$ are the fundamental operators of $(A^*,B^*,P^*)$. We prove a Beurling-Lax-Halmos type theorem for a triple of operators $(M_{F_1^*+F_2z},M_{F_2^*+F_1z},M_z)$, where $\\mathcal{E}$ is a Hilbert space and $F_1,F_2 \\in \\mathcal{B}(\\mathcal{E})$. We deal with a natural example of tetrablock contraction on functions space to find out its fundamental operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-02T04:11:27.000Z",
      "abstract": "A commuting triple of operators $(A,B,P)$ on a Hilbert space $\\mathcal{H}$ is called a tetrablock contraction if the closure of the set $$ E = \\{\\underline{x}=(x_1,x_2,x_3)\\in \\mathbb{C}^3: 1-x_1z-x_2w+x_3zw \\neq 0 \\text{ whenever }|z| < 1\\text{ and }|w| < 1 \\} $$ is a spectral set. In this paper, we have constructed a functional model and produced a complete unitary invariant for a pure tetrablock contraction. In this construction, the fundamental operators, which are the unique solutions of the operator equations $$ A-B^*P = D_PX_1D_P \\text{ and } B-A^*P=D_PX_2D_P, \\text{ where $X_1,X_2 \\in \\mathcal{B}(\\mathcal{D}_P)$}, $$ play a big role. As a corollary to the functional model, we show that every pure tetrablock isometry $(A,B,P)$ on a Hilbert space $\\mathcal{H}$ is unitarily equivalent to $(M_{G_1^*+G_2z}, M_{G_2^*+G_1z},M_z)$ on $H^2(\\mathcal{D}_{P^*})$, where $G_1$ and $G_2$ are fundamental operators of $(A^*,B^*,P^*)$. In \\cite{sir and me}, a tetrablock unitary dilation of a tetrablock contraction has been explicitly constructed with the assumptions that the fundamental operators $F_1,F_2$ and $G_1,G_2$ of $(A,B,P)$ and $(A^*,B^*,P^*)$ respectively satisfy $[X_1,X_2]=0$ and $[X_1,X_1^*]=[X_2X_2^*]$, in place of $X_1$ and $X_2$ respectively. Here it is shown that the conditions can be relaxed. We prove a Beurling-Lax-Halmos type theorem for a triple of operators $(M_{F_1^*+F_2z},M_{F_2^*+F_1z},M_z)$, where $F_1,F_2 \\in \\mathcal{B}(\\mathcal{E})$ are two operators and $\\mathcal{E}$ is a Hilbert space. We deal with a natural example of tetrablock contraction on functions space to find out its fundamental operators.",
      "comment": "16 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-12T06:34:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A15",
      "47A20",
      "47A25",
      "47A45"
    ],
    "keywords": [
      "fundamental operators",
      "hilbert space",
      "functional model",
      "pure tetrablock contraction",
      "tetrablock unitary dilation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.0322S"
    }
  }
}