{
  "id": "2210.10617",
  "version": "v1",
  "published": "2022-10-19T15:00:56.000Z",
  "updated": "2022-10-19T15:00:56.000Z",
  "title": "A generalization of Ando's dilation, and isometric dilations for a class of tuples of $q$-commuting contractions",
  "authors": [
    "Sibaprasad Barik",
    "Bappa Bisai"
  ],
  "comment": "23 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Given a bounded operator $Q$ on a Hilbert space $\\mathcal{H}$, a pair of bounded operators $(T_1, T_2)$ on $\\mathcal{H}$ is said to be $Q$-commuting if one of the following holds: \\[ T_1T_2=QT_2T_1 \\text{ or }T_1T_2=T_2QT_1 \\text{ or }T_1T_2=T_2T_1Q. \\] We give an explicit construction of isometric dilations for pairs of $Q$-commuting contractions for unitary $Q$, which generalizes the isometric dilation of Ando [2] for pairs of commuting contractions. In particular, for $Q=qI_{\\mathcal{H}}$, where $q$ is a complex number of modulus $1$, this gives, as a corollary, an explicit construction of isometric dilations for pairs of $q$-commuting contractions which are well studied. There is an extended notion of $q$-commutativity for general tuples of operators and it is known that isometric dilation does not hold, in general, for an $n$-tuple of $q$-commuting contractions, where $n\\geq 3$. Generalizing the class of commuting contractions considered by Brehmer [8], we construct a class of $n$-tuples of $q$-commuting contractions and find isometric dilations explicitly for the class.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-19T15:00:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "commuting contractions",
      "isometric dilation",
      "andos dilation",
      "generalization",
      "explicit construction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}