{
  "id": "2108.10635",
  "version": "v1",
  "published": "2021-08-24T10:38:32.000Z",
  "updated": "2021-08-24T10:38:32.000Z",
  "title": "Necessary conditions for existence of $Γ_n$-contractions and examples of $Γ_3$-contractions",
  "authors": [
    "Shubhankar Mandal",
    "Avijit Pal"
  ],
  "comment": "12 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "The fundamental result of B. Sz. Nazy states that every contraction has a coisometric extension and a unitary dilation. The isometric dilation of a contraction on a Hilbert space motivated whether this theory can be extended sensibly to families of operators. It is natural to ask whether this idea can be generalized, where the contraction $T$ is substituted by a commuting $n$-tuples of operators $(S_1,\\cdots, S_n)$ acting on some Hilbert space having $\\Gamma_n$ as a spectral set. We derive the necessary conditions for the existence of a $\\Gamma_n$-isometric dilation for $\\Gamma_n$-contractions. Also we discuss an example of a $\\Gamma_3$-contraction $(S_1, S_2, S_3)$ acting on some Hilbert space $\\mathcal H,$ which has a $\\Gamma_3$-isometric dilation, but it fails to satisfy the following condition: $$E_1^*E_1-E_1E_1^*= E_2^*E_2-E_2E_2^*,$$ where $E_1$ and $E_2$ are the fundamental operators of $(S_1, S_2, S_3),$ $(S_1,S_2)$ is a pair of commuting contractions and $S_3$ is a partial isometry. Thus, the set of sufficient conditions for the existence of a $\\Gamma_3$-isometric dilation breaks down, in general, to be necessary, even when the $\\Gamma_3$-contraction $(S_1, S_2, S_3)$ has the special structure as described above.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-24T10:38:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "contraction",
      "necessary conditions",
      "hilbert space",
      "isometric dilation breaks",
      "fundamental operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}