La limite de établie par un procédé de démonstration élémentaire
A characterization of polynomially Riesz strongly continuous semigroups
In this paper we characterize the class of polynomially Riesz strongly continuous semigroups on a Banach space . 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 and of with such that the part of the generator in is unbounded with resolvent of Riesz type while its part in is a polynomially Riesz operator.
Some facts from descriptive set theory concerning essential spectra and applications
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...
Representations of Polish groups and continuity
In the first part of the paper, some criteria of continuity of representations of a Polish group in a Banach algebra are given. The second part uses the result of the first part to deduce automatic continuity results of Baire morphisms from Polish groups to locally compact groups or unitary groups. In the final part, the spectrum of an element in the range of a strongly but not norm continuous representation is described.
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