{
  "id": "1203.1134",
  "version": "v1",
  "published": "2012-03-06T08:49:06.000Z",
  "updated": "2012-03-06T08:49:06.000Z",
  "title": "A note on property (gb) and perturbations",
  "authors": [
    "Qingping Zeng",
    "Huaijie Zhong"
  ],
  "comment": "10 pages",
  "journal": "Abstract and Applied Analysis Volume 2012 (2012), Article ID 523986, 10 pages",
  "doi": "10.1155/2012/523986",
  "categories": [
    "math.FA"
  ],
  "abstract": "An operator $T \\in \\mathcal{B}(X)$ defined on a Banach space $X$ satisfies property $(gb)$ if the complement in the approximate point spectrum $\\sigma_{a}(T)$ of the upper semi-B-Weyl spectrum $\\sigma_{SBF_{+}^{-}}(T)$ coincides with the set $\\Pi(T)$ of all poles of the resolvent of $T$. In this note we continue to study property $(gb)$ and the stability of it, for a bounded linear operator $T$ acting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, by quasi-nilpotent operators commuting with $T$. Two counterexamples show that property $(gb)$ in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-03-06T08:49:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A10",
      "47A11",
      "47A53",
      "47A55"
    ],
    "keywords": [
      "banach space",
      "approximate point spectrum",
      "upper semi-b-weyl spectrum",
      "finite rank operators",
      "commuting finite rank perturbations"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Hindawi",
      "journal": "Adv. High Energ. Phys."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.1134Z"
    }
  }
}