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Archiv der Mathematik

, Volume 90, Issue 5, pp 420–428 | Cite as

A spectral characterization of the uniform continuity of strongly continuous groups

  • Khalid Latrach
  • J. Martin Paoli
  • Pierre Simonnet
Article

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.

Mathematics Subject Classification (2000).

47A10 47D03 54E52 

Keywords.

Strongly continuous groups uniformly continuous groups Baire’s property Baire measurable functions essential spectra 

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Copyright information

© Birkhaeuser 2008

Authors and Affiliations

  • Khalid Latrach
    • 1
  • J. Martin Paoli
    • 2
  • Pierre Simonnet
    • 2
  1. 1.Département de MathématiquesUniversité Blaise Pascal (Clermont II)AubièreFrance
  2. 2.Département de MathématiquesUniversité de CorseCorteFrance

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