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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the uniform ergodic theorem. II
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by Michael Lin PDF
Proc. Amer. Math. Soc. 46 (1974), 217-225 Request permission

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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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 46 (1974), 217-225
  • MSC: Primary 47A35; Secondary 47D05
  • DOI: https://doi.org/10.1090/S0002-9939-1974-0417822-8
  • MathSciNet review: 0417822