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Some facts from descriptive set theory concerning essential spectra and applications

Khalid Latrach;  J. Martin Paoli;  Pierre Simonnet — 2005

Studia Mathematica

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...

A characterization of polynomially Riesz strongly continuous semigroups

Khalid Latrach;  Martin J. Paoli;  Mohamed Aziz Taoudi — 2006

Commentationes Mathematicae Universitatis Carolinae

In this paper we characterize the class of polynomially Riesz strongly continuous semigroups on a Banach space X . Our main results assert, in particular, that the generators of such semigroups are either polynomially Riesz (then bounded) or there exist two closed infinite dimensional invariant subspaces X 0 and X 1 of X with X = X 0 ⊕ X 1 such that the part of the generator in X 0 is unbounded with resolvent of Riesz type while its part in X 1 is a polynomially Riesz operator.

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