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Some facts from descriptive set theory concerning essential spectra and applications

Tom 171 / 2005

Khalid Latrach, J. Martin Paoli, Pierre Simonnet Studia Mathematica 171 (2005), 207-225 MSC: 54H05, 47A10, 47D03. DOI: 10.4064/sm171-3-1

Streszczenie

Let $X$ be a separable Banach space and denote by $\mathcal{L}(X)$ (resp. $\mathcal{K}(\mathbb{C})$) the set of all bounded linear operators on $X$ (resp. the set of all compact subsets of $\mathbb{C}$). We show that the maps from $\mathcal{L}(X)$ into $\mathcal{K}(\mathbb{C})$ which assign to each element of $\mathcal{L}(X)$ its spectrum, approximate point spectrum, essential spectrum, Weyl essential spectrum, Browder essential spectrum, respectively, are Borel maps, where $\mathcal{L}(X)$ (resp. $\mathcal{K}(\mathbb{C})$) is endowed with the strong operator topology (resp. Hausdorff topology). This enables us to derive the topological complexity of some subsets of $\mathcal{L}(X)$ and to discuss the properties of a class of strongly continuous semigroups. We close the paper by giving a characterization of strongly continuous semigroups on hereditarily indecomposable Banach spaces.

Autorzy

  • Khalid LatrachLaboratoire de Mathématiques
    CNRS (UMR 6620)
    Université Blaise Pascal
    24 avenue des Landais
    63117 Aubière, France
    e-mail
  • J. Martin PaoliDépartement de Mathématiques
    Université de Corse
    Quartier Grossetti, BP 52
    20250 Corte, France
  • Pierre SimonnetDépartement de Mathématiques
    Université de Corse
    Quartier Grossetti, BP 52
    20250 Corte, France

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