On the extension of battys theorem on the semigroup asymptotic stability

Abstract

The well-known Batty's theorem states that if a C 0 -semigroup T (t) is bounded and the spectrum of the generator A is contained in the open left-half plane of C, then T (t)A −1 tends to 0. This can be thought of as a particular case of a more general property that, for ω 0 > −∞ and (ω 0 + iR) ∩ σ (A) = / 0 it holds T (t)(A − ω 0 I) −1 / T (t) tends to 0. We show that it is true for T (t) regular enough, however we give examples of unbounded semigroups, with the spectrum of the generator not contained in the open left-half plane of C, with the above property. Moreover we give a more general sufficient condition for this property to hold, thus extending Batty's theorem.