Some facts from descriptive set theory concerning essential spectra and applications
Tom 171 / 2005
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.
