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Strong Stability of Bounded Evolution Families and Semigroups

✍ Scribed by Charles J.K. Batty; Ralph Chill; Yuri Tomilov

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
212 KB
Volume
193
Category
Article
ISSN
0022-1236

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✦ Synopsis

We prove several characterizations of strong stability of uniformly bounded evolution families ðUðt; sÞÞ t5s50 of bounded operators on a Banach space X , i.e. we characterize the property lim t!1 jjUðt; sÞxjj ¼ 0 for all s50 and all x 2 X . These results are connected to the asymptotic stability of the well-posed linear nonautonomous Cauchy problem ' u uðtÞ ¼ AðtÞuðtÞ; t5s50; uðsÞ ¼ x;

x 2 X : ( In the autonomous case, i.e. when Uðt; sÞ ¼ Tðt À sÞ for some C 0 -semigroup ðTðtÞÞ t50 , we present, in addition, a range condition on the generator A of ðTðtÞÞ t50 which is sufficient for strong stability. This condition is more general than the condition in the ABLV-Theorem involving countability of the imaginary part of the spectrum of A.

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