{
  "id": "1911.03236",
  "version": "v1",
  "published": "2019-11-08T13:08:13.000Z",
  "updated": "2019-11-08T13:08:13.000Z",
  "title": "Structure theorems for operators associated with two domains related to $μ$-synthesis",
  "authors": [
    "Bappa Bisai",
    "Sourav Pal"
  ],
  "comment": "30 pages, Bulletin des Sciences Math\\'ematiques, To appear",
  "categories": [
    "math.FA"
  ],
  "abstract": "A commuting tuple of $n$ operators $(S_1, \\dots, S_{n-1}, P)$ defined on a Hilbert space $\\mathcal{H}$, for which the closed symmetrized polydisc \\[ \\Gamma_n = \\left\\{ \\left(\\sum_{i=1}^{n}z_i, \\sum\\limits_{1\\leq i<j\\leq n}z_iz_j, \\dots, \\prod_{i=1}^{n}z_i \\right) : |z_i|\\leq 1, i=1, \\dots, n \\right\\} \\] is a spectral set is called a $\\Gamma_n$-contraction. Also a triple of commuting operators $(A,B,P)$ for which the closed tetrablock $\\overline{\\mathbb E}$ is a spectral set is called an $\\mathbb E$-contraction, where \\[ \\mathbb E = \\{ (x_1,x_2,x_3)\\in\\mathbb C^3\\,:\\, 1-zx_1-wx_2+zwx_3 \\neq 0 \\quad \\forall z, w \\in \\overline{\\mathbb D} \\}. \\] There are several decomposition theorems for contraction operators in the literature due to Sz. Nagy, Foias, Levan, Kubrusly, Foguel and few others which reveal structural information of a contraction. In this article, we obtain analogues of six such major theorems for both $\\Gamma_n$-contractions and $\\mathbb E$-contractions. In each of these decomposition theorems, the underlying Hilbert space admits a unique orthogonal decomposition which is provided by the last component $P$. The central role in determining the structure of a $\\Gamma_n$-contraction or an $\\mathbb E$-contraction is played by positivity of some certain operator pencils and the existence of a unique operator tuple associated with a $\\Gamma_n$-contraction or an $\\mathbb E$-contraction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-08T13:08:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "structure theorems",
      "decomposition theorems",
      "spectral set",
      "unique operator tuple",
      "unique orthogonal decomposition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}