On the uniform ergodic for α−times integrated semigroups

Resumen

Let $A$ be a generator of an $\alpha-$times integrated semigroup
$(S(t))_{t\geq 0}$. We study the uniform ergodicity of $(S(t))_{t\geq 0}$ and we show that the range of $A$ is closed if and only if $\lambda R(\lambda,A)$ is uniformly ergodic.
Moreover, we obtain that $(S(t))_{t\geq 0}$ is uniformly ergodic if and only if $\alpha=0$. Finally, we get that $\frac{1}{t^{\alpha+1}}\int_{0}^{t}S(s)ds$ converge uniformly for all $\alpha\geq 0$.