{
  "id": "1808.03542",
  "version": "v1",
  "published": "2018-08-10T13:48:17.000Z",
  "updated": "2018-08-10T13:48:17.000Z",
  "title": "Isolated eigenvalues, poles and compact perturbations of Banach space operators",
  "authors": [
    "B. P. Duggal"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Given a Banach space operator $A$, the isolated eigenvalues $E(A)$ and the poles $\\Pi(A)$ (resp., eigenvalues $E^a(A)$ which are isolated points of the approximate point spectrum and the left ploles $\\Pi^a(A)$) of the spectrum of $A$ satisfy $\\Pi(A)\\subseteq E(A)$ (resp., $\\Pi^a(A)\\subseteq E^a(A)$), and the reverse inclusion holds if and only if $E(A)$ (resp., $E^a(A)$) has empty intersection with the B-Weyl spectrum (resp., upper B-Weyl spectrum) of $A$. Evidently $\\Pi(A)\\subseteq E^a(A)$, but no such inclusion exists for $E(A)$ and $\\Pi^a(A)$. The study of identities $E(A)=\\Pi^a(A)$ and $E^a(A)=\\Pi(A)$, and their stability under perturbation by commuting Riesz operators, has been of some interest in the recent past. This paper studies the stability of these identities under perturbation by (non-commuting) compact operators. Examples of analytic Toeplitz operators and operators satisfying the abstract shift condition are considered.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-10T13:48:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A10",
      "47A55",
      "47A53",
      "47B40"
    ],
    "keywords": [
      "banach space operator",
      "isolated eigenvalues",
      "compact perturbations",
      "abstract shift condition",
      "upper b-weyl spectrum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}