{
  "id": "2207.03236",
  "version": "v1",
  "published": "2022-07-07T11:44:57.000Z",
  "updated": "2022-07-07T11:44:57.000Z",
  "title": "Functional Models for Commuting Hilbert-space Contractions",
  "authors": [
    "Joseph A. Ball",
    "Haripada Sau"
  ],
  "comment": "Dedicated to the memory of Ron Douglas. It has appeared in Operator Theory: Advances and Applications (Ronald G. Douglas Memorial Volume)",
  "categories": [
    "math.FA"
  ],
  "abstract": "We develop a Sz.-Nagy--Foias-type functional model for a commutative contractive operator tuple $\\underline{T} = (T_1, \\dots, T_d)$ having $T = T_1 \\cdots T_d$ equal to a completely nonunitary contraction. We identify additional invariants ${\\mathbb G}_\\sharp, {\\mathbb W}_\\sharp$ in addition to the Sz.-Nagy--Foias characteristic function $\\Theta_T$ for the product operator $T$ so that the combined triple $({\\mathbb G}_\\sharp, {\\mathbb W}_\\sharp, \\Theta_T)$ becomes a complete unitary invariant for the original operator tuple $\\underline{T}$. For the case $d \\ge 3$ in general there is no commutative isometric lift of $\\underline{T}$; however there is a (not necessarily commutative) isometric lift having some additional structure so that, when compressed to the minimal isometric-lift space for the product operator $T$, generates a special kind of lift of $\\underline{T}$, herein called a {\\em pseudo-commutative contractive lift} of $\\underline{T}$, which in turn leads to the functional model for $\\underline{T}$. This work has many parallels with recently developed model theories for symmetrized-bidisk contractions (commutative operator pairs $(S,P)$ having the symmetrized bidisk $\\Gamma$ as a spectral set) and for tetrablock contractions (commutative operator triples $(A, B, P)$ having the tetrablock domain ${\\mathbb E}$ as a spectral set).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-07T11:44:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "functional model",
      "commuting hilbert-space contractions",
      "product operator",
      "isometric lift",
      "spectral set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}