On the extension of battys theorem on the semigroup asymptotic stability

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 $\|T(t)A^{-1}\|$ tends to $0$. This can be thought of as a particular case of a more general property that, for $\omega_0>-\infty$ and $(\omega_0+i\mathbb{R})\cap \sigma(A)=\emptyset$ it holds $\|T(t)(A-\omega_0 I)^{-1}\|/\|T(t)\|$ 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 $\mathbb{C}$, with the above property. Moreover we give a more general sufficient condition for this property to hold, thus extending Batty's theorem.