Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the uniform ergodic theorem. II


Author: Michael Lin
Journal: Proc. Amer. Math. Soc. 46 (1974), 217-225
MSC: Primary 47A35; Secondary 47D05
MathSciNet review: 0417822
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \{ {T_t}\} $ be a strongly continuous semigroup of bounded linear operators on a Banach space $ X$, satisfying $ {\lim _{t \to \infty }}\vert\vert{T_t}\vert\vert/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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A35, 47D05

Retrieve articles in all journals with MSC: 47A35, 47D05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0417822-8
Keywords: Ergodic theorem, ergodicity of semigroups, Abel ergodicity
Article copyright: © Copyright 1974 American Mathematical Society