On the uniform ergodic theorem. II

Lin, Michael

doi:10.1090/S0002-9939-1974-0417822-8

On the uniform ergodic theorem. II
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by Michael Lin

Proc. Amer. Math. Soc. 46 (1974), 217-225

DOI: https://doi.org/10.1090/S0002-9939-1974-0417822-8

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Abstract:

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.

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