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A spectral characterization of the uniform continuity of strongly continuous groups

Authors:	Khalid Latrach J Martin Paoli Pierre Simonnet

Institution:	(1) Département de Mathématiques, Université Blaise Pascal (Clermont II), CNRS (UMR 6620) 24 avenue des Landais, 63170 Aubière, France;(2) Département de Mathématiques, Université de Corse, Quartier Grossetti, BP 52, F-20250 Corte, France

Abstract:	Let X be a Banach space and $(T(t))_{t \in {\mathbb{R}}}$ a strongly continuous group of linear operators on X. Set $\sigma^1(T(t)) := \{ \frac{\lambda}{\mid \lambda \mid}\, : \,\lambda\, \in\, \sigma(T(t)) \}$ and $\chi(T) := \{t \in {\mathbb{R}} : \sigma^1(T(t)) \neq {\mathbb{T}}\}$ where ${\mathbb{T}}$ is the unit circle and $\sigma(T(t))$ denotes the spectrum of T(t). The main result of this paper is: $(T(t))_{t \in {\mathbb{R}}}$ is uniformly continuous if and only if $\chi(T)$ is non-meager. Similar characterizations in terms of the approximate point spectrum and essential spectra are also derived. Received: 14 June 2006, Revised: 27 September 2007

Keywords:	Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 47A10 47D03 54E52
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