On the Uniform Ergodic Theorem. II
Michael Lin · Proceedings of the American Mathematical Society · 1974
Let $\{ {T_t}\}$ be a strongly continuous semigroup of bounded linear operators on a Banach space $X$, satisfying ${\lim _{t \to \infty }}||{T_t}||/t = 0$. We prove the equivalence of the following conditions: (1) ${t^{ - 1}}\int _0^t {{T_r}dr}$ converges uniformly as $t \to \infty$. (2) The infinitesimal generator $A$ has closed range. (3) ${\lim _{\lambda \to {0^ + }}}\lambda {R_\lambda }$ exists in the uniform operator topology.