{
  "id": "2311.06764",
  "version": "v1",
  "published": "2023-11-12T07:53:19.000Z",
  "updated": "2023-11-12T07:53:19.000Z",
  "title": "The $2 \\times 2$ block matrices associated with an annulus",
  "authors": [
    "Sourav Pal",
    "Nitin Tomar"
  ],
  "comment": "This is a short paper, submitted to journal",
  "categories": [
    "math.FA",
    "math.CV"
  ],
  "abstract": "A bounded Hilbert space operator $T$ for which the closure of the annulus \\[ \\mathbb{A}_r=\\{z \\ : \\ r<|z|<1\\} \\subseteq \\mathbb{C}, \\qquad (0<r<1) \\] is a spectral set is called an $\\mathbb A_r$-contraction. A celebrated theorem due to Douglas, Muhly and Pearcy gives a necessary and sufficient condition such that a $2 \\times 2$ block matrix of operators $ \\begin{bmatrix} T_1 & X 0 & T_2 \\end{bmatrix} $ is a contraction. We seek an answer to the same question in the setting of annulus, i.e., under what conditions $\\widetilde{T}_Y=\\begin{bmatrix} T_1 & Y 0 & T_2 \\end{bmatrix} $ becomes an $\\mathbb A_r$-contraction. For a pair of $\\mathbb A_r$-contractions $T_1,T_2$ and an operator $X$ that commutes with $T_1,T_2$, here we find a necessary and sufficient condition such that each of the block matrices \\[ T_X= \\begin{bmatrix} T & X 0 & T \\end{bmatrix} \\,, \\quad \\widehat{T}_X=\\begin{bmatrix} T_1 & X(T_1-T_2) 0 & T_2 \\end{bmatrix} \\] becomes an $\\mathbb A_r$-contraction. Thus, the general block matrix $\\widetilde{T}_Y$ is an $\\mathbb A_r$-contraction if $T_1-T_2$ (as in $\\widehat{T}_X$) is invertible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-12T07:53:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A08",
      "47A25"
    ],
    "keywords": [
      "contraction",
      "sufficient condition",
      "general block matrix",
      "bounded hilbert space operator",
      "spectral set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}