Some facts from descriptive set theory concerning essential spectra and applications

Khalid Latrach¹ ; J. Martin Paoli² ; Pierre Simonnet²

¹ Laboratoire de Mathématiques CNRS (UMR 6620) Université Blaise Pascal 24 avenue des Landais 63117 Aubière, France
² Département de Mathématiques Université de Corse Quartier Grossetti, BP 52 20250 Corte, France

Studia Mathematica, Tome 171 (2005) no. 3, pp. 207-225

Résumé

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.

Détail

DOI : 10.4064/sm171-3-1

Keywords: separable banach space denote mathcal resp mathcal mathbb set bounded linear operators resp set compact subsets mathbb maps mathcal mathcal mathbb which assign each element mathcal its spectrum approximate point spectrum essential spectrum weyl essential spectrum browder essential spectrum respectively borel maps where mathcal resp mathcal mathbb endowed strong operator topology resp hausdorff topology enables derive topological complexity subsets mathcal discuss properties class strongly continuous semigroups close paper giving characterization strongly continuous semigroups hereditarily indecomposable banach spaces