Let X be a separable Banach space and denote by 𝓛(X) (resp. 𝒦(ℂ)) the set of all bounded linear operators on X (resp. the set of all compact subsets of ℂ). We show that the maps from 𝓛(X) into 𝒦(ℂ) which assign to each element of 𝓛(X) its spectrum, approximate point spectrum, essential spectrum, Weyl essential spectrum, Browder essential spectrum, respectively, are Borel maps, where 𝓛(X) (resp. 𝒦(ℂ)) is endowed with the strong operator topology (resp. Hausdorff topology). This enables us to derive the topological complexity of some subsets of 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-3-1,
author = {Khalid Latrach and J. Martin Paoli and Pierre Simonnet},
title = {Some facts from descriptive set theory concerning essential spectra and applications},
journal = {Studia Mathematica},
volume = {166},
year = {2005},
pages = {207-225},
zbl = {1158.54316},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-3-1}
}
Khalid Latrach; J. Martin Paoli; Pierre Simonnet. Some facts from descriptive set theory concerning essential spectra and applications. Studia Mathematica, Tome 166 (2005) pp. 207-225. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-3-1/