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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Stability of individual elements under one-parameter semigroups


Authors: Charles J. K. Batty and Quôc Phóng Vù
Journal: Trans. Amer. Math. Soc. 322 (1990), 805-818
MSC: Primary 47D03
MathSciNet review: 1022866
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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 $ \vert\vert T(t)x\vert\vert \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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DOI: http://dx.doi.org/10.1090/S0002-9947-1990-1022866-5
PII: S 0002-9947(1990)1022866-5
Keywords: $ {C_0}$-semigroup, stability, residual spectrum, sun-reflexive, stabilization, almost periodic
Article copyright: © Copyright 1990 American Mathematical Society