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On Individual Stability of C0-Semigroups

J. M. A. M. van Neerven
Proceedings of the American Mathematical Society
Vol. 130, No. 8 (Aug., 2002), pp. 2325-2333
Stable URL: http://www.jstor.org/stable/2699469
Page Count: 9
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
On Individual Stability of C0-Semigroups
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Abstract

Let {T(t)}t≥ 0 be a C0-semigroup with generator A on a Banach space X. Let x0 ∈ X be a fixed element. We prove the following individual stability results. (i) Suppose X is an ordered Banach space with weakly normal closed cone C and assume there exists t0 ≥ 0 such that T(t)x0 ∈ C for all t ≥ t0. If the local resolvent $\lambda \mapsto (\lambda - A)^{-1}x_0$ admits a bounded analytic extension to the right half-plane $\{Re \lambda > 0\}$, then for all μ ∈ ϱ(A) and x* ∈ X* we have $\underset{t \rightarrow \infty}{\text{lim}} \langle T(t)(\mu - A)^{-1}x_0, x*\rangle = 0.$ (ii) Suppose E is a rearrangement invariant Banach function space over [0, ∞) with order continuous norm. If x*0 ∈ X* is an element such that $t \mapsto \langle T(t)x_0,x*_0\rangle$ defines an element of E, then for all μ ∈ ϱ(A) and β ≥ 1 we have $\underset{t\rightarrow \infty} \tex{lim}\langle T(t)(\mu - A)^{-\beta} x_0, x*_0\rangle = 0.$

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