{
  "id": "2207.03229",
  "version": "v1",
  "published": "2022-07-07T11:25:28.000Z",
  "updated": "2022-07-07T11:25:28.000Z",
  "title": "Dilation theory and functional models for tetrablock contractions",
  "authors": [
    "Joseph A. Ball",
    "Haripada Sau"
  ],
  "comment": "Dedicated to the memory of J\\\"org Eschmeier",
  "categories": [
    "math.FA"
  ],
  "abstract": "A classical result of Sz.-Nagy asserts that a Hilbert space contraction operator $T$ can be dilated to a unitary $\\cU$. A more general multivariable setting for these ideas is the setup where (i) the unit disk is replaced by a domain $\\Omega$ contained in ${\\mathbb C}^d$, (ii) the contraction operator $T$ is replaced by a commuting tuple $\\bfT = (T_1, \\dots, T_d)$ such that $\\| r(T_1, \\dots, T_d) \\|_{\\cL(\\cH)} \\le \\sup_{\\lam \\in \\Omega} | r(\\lam) |$ for all rational functions with no singularities in $\\overline{\\Omega}$ and the unitary operator $\\cU$ is replaced by an $\\Omega$-unitary operator tuple, i.e., a commutative operator $d$-tuple $\\bfU = (U_1, \\dots, U_d)$ of commuting normal operators with joint spectrum contained in the distinguished boundary $b\\Omega$ of $\\Omega$. For a given domain $\\Omega \\subset {\\mathbb C}^d$, the {\\em rational dilation question} asks: given an $\\Omega$-contraction $\\bfT$ on $\\cH$, is it always possible to find an $\\Omega$-unitary $\\bfU$ on a larger Hilbert space $\\cK \\supset \\cH$ so that, for any $d$-variable rational function without singularities in $\\overline{\\Omega}$, one can recover $r(T)$ as $r(T) = P_\\cH r(\\bfU)|_\\cH$. We focus here on the case where $\\Omega $ is the {\\em tetrablock}. (i) We identify a complete set of unitary invariants for a ${\\mathbb E}$-contraction $(A,B,T)$ which can then be used to write down a functional model for $(A,B,T)$, thereby extending earlier results only done for a special case, (ii) we identify the class of {\\em pseudo-commutative ${\\mathbb E}$-isometries} (a priori slightly larger than the class of ${\\mathbb E}$-isometries) to which any ${\\mathbb E}$-contraction can be lifted, and (iii) we use our functional model to recover an earlier result on the existence and uniqueness of a ${\\mathbb E}$-isometric lift $(V_1, V_2, V_3)$ of a special type for a ${\\mathbb E}$-contraction $(A,B,T)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-07T11:25:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "functional model",
      "tetrablock contractions",
      "dilation theory",
      "hilbert space contraction operator",
      "rational function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}