{
  "id": "2207.01278",
  "version": "v1",
  "published": "2022-07-04T09:16:54.000Z",
  "updated": "2022-07-04T09:16:54.000Z",
  "title": "Models for q-commutative tuples of isometries",
  "authors": [
    "Joseph A. Ball",
    "Haripada Sau"
  ],
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "A pair of Hilbert space linear operators $(V_1,V_2)$ is said to be $q$-commutative, for a unimodular complex number $q$, if $V_1V_2=qV_2V_1$. A concrete functional model for $q$-commutative pairs of isometries is obtained. The functional model is parametrized by a collection of Hilbert spaces and operators acting on them. As a consequence, the collection serves as a complete unitary invariance for $q$-commutative pairs of isometries. A $q$-commutative operator pair $(V_1,V_2)$ is said to be doubly $q$-commutative, if in addition, it satisfies $V_2V_1^*=qV_1^*V_2$. Doubly $q$-commutative pairs of isometries are also characterized. Special attention is given to doubly $q$-commutative pairs of shift operators. The notion of $q$-commutativity is then naturally extended to the case of general tuples of operators to obtain a similar model for tuples of $q$-commutative isometries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-04T09:16:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "isometries",
      "q-commutative tuples",
      "commutative pairs",
      "hilbert space linear operators",
      "concrete functional model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}