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 $\mathbb{C}$, then $\Vert T(t)A^{-1}\Vert$ tends to $0$. This can be thought of as a particular case of a more general property that, for ${\omega }_0>-\infty$ and $({\omega }_0+i\mathbb{R})\cap \sigma (A)=\varnothing$ it holds $\Vert T(t)(A-{\omega }_{0}I{)}^{-1}\Vert /\Vert T(t)\Vert$ tends to 0. We show that it is true for $\Vert T(t)\Vert$ regular enough, however we give examples of unbounded semigroups, with the spectrum of the generator not contained in the open left-half plane of $\mathbb{C}$, with the above property. Moreover we give a more general sufficient condition for this property to hold, thus extending Batty's theorem.

14 pages, no figures