International audienc

Abstract. 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.

Let L(H) be the algebra of all bounded operators in a Hilbert space H, θ : G → L(H) denotes a strongly continuous unitary representation of a locally compact and second countable group G in H, σ(θ(g)) the spectrum of θ(g) and Conv(S) the convex hull of any subset S in a vector space. We prove here that θ is uniformly continuous if and only if {g ∈ G/0 / ∈ Conv(σ(θ(g)))} is non-meager.Mathematics Subject Classification (2010). 47A10, 47D03.

International audienceLet X be a Banach space and (T(t))t Î \mathbbR(T(t))tR a strongly continuous group of linear operators on X. Set $\sigma^1(T(t)) := \{ \frac{\lambda}{\mid \lambda \mid}\, : \,\lambda\, \in\, \sigma(T(t)) \}$1(T(t)):=:(T(t)) and c(T) : = {t Î \mathbbR : s1(T(t)) ¹ \mathbbT}(T):=tR:1(T(t))=T where \mathbbTT is the unit circle and s(T(t))(T(t)) denotes the spectrum of T(t). The main result of this paper is: (T(t))t Î \mathbbR(T(t))tR is uniformly continuous if and only if c(T)(T) is non-meager. Similar characterizations in terms of the approximate point spectrum and essential spectra are also derived