On the Uniform Ergodic for α-times Integrated Semigroups

Let A be the generator of an alpha-times integrated semigroup (S(t))(t >= 0). We study the uniform ergodicity of (S(t)(t >= 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 >= 0) is uniformly ergodic if and only if alpha = 0. Finally, we get that 1/t(alpha+1) integral(t)(0) S(s)ds converge uniformly for all alpha > 0.