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Transactions of the American Mathematical Society

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



Tauberian theorems and stability of one-parameter semigroups

Authors: W. Arendt and C. J. K. Batty
Journal: Trans. Amer. Math. Soc. 306 (1988), 837-852
MSC: Primary 47D05; Secondary 34G10
MathSciNet review: 933321
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Abstract: The main result is the following stability theorem: Let $ \mathcal{T} = {(T(t))_{t \geqslant 0}}$ be a bounded $ {C_0}$-semigroup on a reflexive space $ X$. Denote by $ A$ the generator of $ \mathcal{T}$ and by $ \sigma (A)$ the spectrum of $ A$. If $ \sigma (A) \cap i{\mathbf{R}}$ is countable and no eigenvalue of $ A$ lies on the imaginary axis, then $ {\lim _{t \to \infty }}T(t)x = 0$ for all $ x \in X$.

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Keywords: Tauberian theorems, $ {C_0}$-semigroup, stability, power bounded, Laplace transform, residual spectrum
Article copyright: © Copyright 1988 American Mathematical Society

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