{
  "id": "1511.01408",
  "version": "v1",
  "published": "2015-11-04T17:35:52.000Z",
  "updated": "2015-11-04T17:35:52.000Z",
  "title": "Compact perturbations and consequent hereditarily polaroid operators",
  "authors": [
    "B. P. Duggal"
  ],
  "comment": "10 Pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "A Banach space operator $A\\in B({\\cal{X}})$ is polaroid, $A\\in {\\cal{P}}$, if the isolated points of the spectrum $\\sigma(A)$ are poles of the operator; $A$ is hereditarily polaroid, $A\\in{\\cal{HP}}$, if every restriction of $A$ to a closed invariant subspace is polaroid. Operators $A\\in{\\cal{HP}}$ have SVEP on $\\Phi_{sf}(A)=\\{\\lambda: A-\\lambda$ is semi Fredholm $\\}$: This, in answer to a question posed by Li and Zhou (Studia Math. 221(2014), 175-192), proves the necessity of the condition $\\Phi_{sf}^+(A)=\\emptyset$. A sufficient condition for $A\\in B({\\cal{X}})$ to have SVEP on $\\Phi_{sf}(A)$ is that its component $\\Omega_a(A)=\\{\\lambda\\in\\Phi_{sf}(A): \\rm{ind}(A-\\lambda)\\leq 0\\}$ is connected. We prove: If $A\\in B({\\cal{H}})$ is a Hilbert space operator, then a necessary and sufficient condition for there to exist a compact operator $K$ such that $A+K\\in{\\cal{HP}}$ is that $\\Omega_a(A)$ is connected.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-04T17:35:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A10",
      "47A55",
      "47A53",
      "47B40"
    ],
    "keywords": [
      "consequent hereditarily polaroid operators",
      "compact perturbations",
      "sufficient condition",
      "hilbert space operator",
      "banach space operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}