On stability of 𝐶₀-semigroups

Phong, Vu

doi:10.1090/S0002-9939-01-05614-3

On stability of $C_0$-semigroups
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Proc. Amer. Math. Soc. 129 (2001), 2871-2879 Request permission

Abstract:

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 $\sup \{\|\int ^t_0\exp \{(2\pi ik)/\omega \}T(s)x ds\|\colon \ t\geq 0, k\in \mathbf {Z}\}<\infty$, for all $x\in E$, and (b) $T(t)$ is exponentially stable if and only if $\sup \{\|\int ^t_0\exp \{i\lambda t\}T(s)x ds\|\colon \ t\geq 0, \lambda \in \mathbf {R}\}<\infty$, for all $x\in E$. Analogous, but weaker, statements also hold for semigroups on Banach spaces.

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