2015 | OriginalPaper | Chapter
Estimates on Non-uniform Stability for Bounded Semigroups
Author : Thomas Duyckaerts
Published in: Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics
Publisher: Springer International Publishing
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Let S(t) be a bounded strongly continuous semigroup on a Banach space, with generator −A. Assume that the spectrum of A has empty intersection with the imaginary axis. In [6], Charles J.K. Batty and the author have given an estimate of the decay of the operator norm of $$S(t)(1+A)^{-1}$$ , as t tends to infinity, in terms of asymptotic bounds of the resolvent of A on the imaginary axis. In this note, we give another proof of this result. The original proof relied on a trick appearing in an analytic proof of the prime number theorem by D. Newman, which we do not use here.