{
  "id": "2208.04737",
  "version": "v1",
  "published": "2022-08-05T19:41:39.000Z",
  "updated": "2022-08-05T19:41:39.000Z",
  "title": "On decomposition for pairs of contractions",
  "authors": [
    "Satyabrata Majee",
    "Amit Maji"
  ],
  "comment": "Preliminary version, 21 pages",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "This paper presents Wold-type decomposition for various pairs of commuting contractions on Hilbert spaces. As a consequence, we obtain a new and simple proof of S\\l{}o\\'{c}inski's theorem for pairs of doubly commuting isometries. We also achieve an explicit decomposition for pairs of commuting contractions such that the c.n.u. parts of the contractions are in $C_{00}$. It is also shown that if a pair $(T, V)$ of commuting operators with $T$ as a contraction and $V$ as an isometry satisfying $T^*V=VT^*$, then there exists a unique pair of doubly commuting isometries on the minimal isometric dilation space of $T$. As an application, we provide a new proof for pairs of commuting operators consisting of an isometry and a co-isometry are doubly commuting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-05T19:41:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A45",
      "47A20",
      "47A15",
      "47A13",
      "47A05"
    ],
    "keywords": [
      "minimal isometric dilation space",
      "doubly commuting isometries",
      "commuting contractions",
      "commuting operators",
      "explicit decomposition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}