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On stability of $C_0$-semigroups

Author: Vu Quoc Phong
Journal: Proc. Amer. Math. Soc. 129 (2001), 2871-2879
MSC (2000): Primary 47D06
Published electronically: May 10, 2001
MathSciNet review: 1707013
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We prove that if $T(t)$ is a $C_0$-semigroup on a Hilbert space $E$, then (a) $1\in\rho(T(\omega))$ if and only if $\sup\{\Vert\int^t_0\exp\{(2\pi ik)/\omega\}T(s)x\,ds\Vert\colon t\geq 0, k\in\mathbf{Z}\}<\infty$, for all $x\in E$, and (b) $T(t)$ is exponentially stable if and only if $\sup\{\Vert\int^t_0\exp\{i\lambda t\}T(s)x\,ds\Vert\colon t\geq 0, \lambda\in\mathbf{R}\}<\infty$, for all $x\in E$. Analogous, but weaker, statements also hold for semigroups on Banach spaces.

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Additional Information

Vu Quoc Phong
Affiliation: Department of Mathematics, Ohio University, Athens, Ohio 45701

Received by editor(s): February 20, 1998
Received by editor(s) in revised form: May 26, 1999
Published electronically: May 10, 2001
Communicated by: David R. Larson
Article copyright: © Copyright 2001 American Mathematical Society