A spectral characterization of the uniform continuity of strongly continuous groups

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.