{
  "id": "0812.2667",
  "version": "v1",
  "published": "2008-12-14T18:39:25.000Z",
  "updated": "2008-12-14T18:39:25.000Z",
  "title": "Upper Triangular Operator Matrices, SVEP and Browder. Weyl Theorems",
  "authors": [
    "B. P. Duggal"
  ],
  "comment": "12 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "A Banach space operator $T\\in B({\\cal X})$ is polaroid if points $\\lambda\\in\\iso\\sigma\\sigma(T)$ are poles of the resolvent of $T$. Let $\\sigma_a(T)$, $\\sigma_w(T)$, $\\sigma_{aw}(T)$, $\\sigma_{SF_+}(T)$ and $\\sigma_{SF_-}(T)$ denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi--Fredholm and lower semi--Fredholm spectrum of $T$. For $A$, $B$ and $C\\in B({\\cal X})$, let $M_C$ denote the operator matrix $(A & C 0 & B)$. If $A$ is polaroid on $\\pi_0(M_C)=\\{\\lambda\\in\\iso\\sigma(M_C) 0<\\dim(M_C-\\lambda)^{-1}(0)<\\infty\\}$, $M_0$ satisfies Weyl's theorem, and $A$ and $B$ satisfy either of the hypotheses (i) $A$ has SVEP at points $\\lambda\\in\\sigma_w(M_0)\\setminus\\sigma_{SF_+}(A)$ and $B$ has SVEP at points $\\mu\\in\\sigma_w(M_0)\\setminus\\sigma_{SF_-}(B)$, or, (ii) both $A$ and $A^*$ have SVEP at points $\\lambda\\in\\sigma_w(M_0)\\setminus\\sigma_{SF_+}(A)$, or, (iii) $A^*$ has SVEP at points $\\lambda\\in\\sigma_w(M_0)\\setminus\\sigma_{SF_+}(A)$ and $B^*$ has SVEP at points $\\mu\\in\\sigma_w(M_0)\\setminus\\sigma_{SF_-}(B)$, then $\\sigma(M_C)\\setminus\\sigma_w(M_C)=\\pi_0(M_C)$. Here the hypothesis that $\\lambda\\in\\pi_0(M_C)$ are poles of the resolvent of $A$ can not be replaced by the hypothesis $\\lambda\\in\\pi_0(A)$ are poles of the resolvent of $A$. For an operator $T\\in B(\\X)$, let $\\pi_0^a(T)=\\{\\lambda:\\lambda\\in\\iso\\sigma_a(T), 0<\\dim(T-\\lambda)^{-1}(0)<\\infty\\}$. We prove that if $A^*$ and $B^*$ have SVEP, $A$ is polaroid on $\\pi_0^a(\\M)$ and $B$ is polaroid on $\\pi_0^a(B)$, then $\\sigma_a(\\M)\\setminus\\sigma_{aw}(\\M)=\\pi_0^a(\\M)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-12-14T18:39:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B47",
      "47A10",
      "47A11"
    ],
    "keywords": [
      "upper triangular operator matrices",
      "weyl theorems",
      "banach space operator",
      "weyl essential approximate",
      "satisfies weyls theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0812.2667D"
    }
  }
}