On stability of C 0 C_0 -semigroups

We prove that if T(t)T(t) is a C0C_0-semigroup on a Hilbert space EE, then (a) 1∈ρ(T(ω))1\in \rho (T(\omega )) if and only if sup{‖∫0texp⁡{(2πik)/ω}T(s)xds‖: t≥0,k∈Z}>∞\sup \{\|\int ^t_0\exp \{(2\pi ik)/\omega \}T(s)x\,ds\|\colon \ t\geq 0, k\in \mathbf {Z}\}>\infty, for all x∈Ex\in E, and (b)...

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Published in:Proceedings of the American Mathematical Society 2001-05, Vol.129 (10), p.2871-2879
Main Author: Phong, Vu Quoc
Format: Article
Language:English
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Research article
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Summary:We prove that if T(t)T(t) is a C0C_0-semigroup on a Hilbert space EE, then (a) 1∈ρ(T(ω))1\in \rho (T(\omega )) if and only if sup{‖∫0texp⁡{(2πik)/ω}T(s)xds‖: t≥0,k∈Z}>∞\sup \{\|\int ^t_0\exp \{(2\pi ik)/\omega \}T(s)x\,ds\|\colon \ t\geq 0, k\in \mathbf {Z}\}>\infty, for all x∈Ex\in E, and (b) T(t)T(t) is exponentially stable if and only if sup{‖∫0texp⁡{iλt}T(s)xds‖: t≥0,λ∈R}>∞\sup \{\|\int ^t_0\exp \{i\lambda t\}T(s)x\,ds\|\colon \ t\geq 0, \lambda \in \mathbf {R}\}>\infty, for all x∈Ex\in 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