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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Stability of individual elements under one-parameter semigroups
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by Charles J. K. Batty and Qu么c Ph贸ng V霉 PDF
Trans. Amer. Math. Soc. 322 (1990), 805-818 Request permission

Abstract:

Let $\{ T(t):t \geqslant 0\}$ be a ${C_0}$-semigroup on a Banach space $X$ with generator $A$, and let $x \in X$. If $\sigma (A) \cap i{\mathbf {R}}$ is empty and $t \mapsto T(t)x$ is uniformly continuous, then $||T(t)x|| \to 0$ as $t \to \infty$. If the semigroup is sun-reflexive, $\sigma (A) \cap i{\mathbf {R}}$ is countable, $P\sigma (A) \cap i{\mathbf {R}}$ is empty, and $t \mapsto T(t)x$ is uniformly weakly continuous, then $T(t)x \to 0$ weakly as $t \to \infty$. Questions of almost periodicity and of stabilization of contraction semigroups on Hilbert space are also discussed.
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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 322 (1990), 805-818
  • MSC: Primary 47D03
  • DOI: https://doi.org/10.1090/S0002-9947-1990-1022866-5
  • MathSciNet review: 1022866