On Stability of C0-Semigroups

We prove that if T(t) is a C0-semigroup on a Hilbert space E, then (a) 1 ∈ ρ(T(ω)) if and only if sup$\{||\int^t_0 \text{exp}\{(2\pi ik)/\omega\}T(s)x ds||: t \geq 0, k \in \mathbf{Z}\}< \infty$, for all x ∈ E, and (b) T(t) is exponentially stable if and only if sup$\{||\int^t_0\text{exp}\{i\lamb...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2001-10, Vol.129 (10), p.2871-2879
Main Author: Phong, Vu Quoc
Format: Article
Language:English
Subjects:
Banach space
Differential equations
Hilbert spaces
Linear transformations
Mathematical theorems
Semigroups
Online Access:Get full text
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Summary:We prove that if T(t) is a C0-semigroup on a Hilbert space E, then (a) 1 ∈ ρ(T(ω)) if and only if sup$\{||\int^t_0 \text{exp}\{(2\pi ik)/\omega\}T(s)x ds||: t \geq 0, k \in \mathbf{Z}\}< \infty$, for all x ∈ E, and (b) T(t) is exponentially stable if and only if sup$\{||\int^t_0\text{exp}\{i\lambda t\}T(s)x ds||: t \geq 0, \lambda \in \mathbf{R}\} < \infty$, for all x ∈ E. Analogous, but weaker, statements also hold for semigroups on Banach spaces.
ISSN:0002-9939
1088-6826
DOI:10.1090/s0002-9939-01-05614-3