{
  "id": "2205.09093",
  "version": "v1",
  "published": "2022-05-18T17:38:22.000Z",
  "updated": "2022-05-18T17:38:22.000Z",
  "title": "Minimal unitary dilations for commuting contractions",
  "authors": [
    "Sourav Pal"
  ],
  "comment": "33 pages. arXiv admin note: text overlap with arXiv:2204.11391",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "For commuting contractions $T_1,\\dots ,T_n$ acting on a Hilbert space $\\mathcal H$ with $T=\\prod_{i=1}^n T_i$, we show that $(T_1, \\dots, T_n)$ dilates to commuting isometries $(V_1, \\dots , V_n)$ on the minimal isometric dilation space of $T$ with $V=\\prod_{i=1}^n V_i$ being the minimal isometric dilation of $T$ if and only if $(T_1^*, \\dots , T_n^*)$ dilates to commuting isometries $(Y_1, \\dots , Y_n)$ on the minimal isometric dilation space of $T^*$ with $Y=\\prod_{i=1}^n Y_i$ being the minimal isometric dilation of $T^*$. Then, we prove an analogue of this result for unitary dilations of $(T_1, \\dots , T_n)$ and its adjoint. We find a necessary and sufficient condition such that $(T_1, \\dots , T_n)$ possesses a unitary dilation $(W_1, \\dots , W_n)$ on the minimal unitary dilation space of $T$ with $W=\\prod_{i=1}^n W_i$ being the minimal unitary dilation of $T$. We show an explicit construction of such a unitary dilation on both Sch$\\ddot{a}$ffer and Sz. Nagy-Foias minimal unitary dilation space of $T$. Also, we show that a relatively weaker hypothesis is necessary and sufficient for the existence of such a unitary dilation when $T$ is a $C._0$ contraction, i.e. when ${T^*}^n \\rightarrow 0$ strongly as $n \\rightarrow \\infty $. We construct a different unitary dilation for $(T_1, \\dots , T_n)$ when $T$ is a $C._0$ contraction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-18T17:38:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "commuting contractions",
      "minimal isometric dilation space",
      "nagy-foias minimal unitary dilation space",
      "commuting isometries"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}