Non-uniform stability for bounded semi-groups on Banach spaces

Abstract.

Let S(t) be a bounded strongly continuous semi-group on a Banach space B and – A be its generator. We say that S(t) is semi-uniformly stable when S(t)(A + 1)⁻¹ tends to 0 in operator norm. This notion of asymptotic stability is stronger than pointwise stability, but strictly weaker than uniform stability, and generalizes the known logarithmic, polynomial and exponential stabilities.

In this note we show that if S is semi-uniformly stable then the spectrum of A does not intersect the imaginary axis. The converse is already known, but we give an estimate on the rate of decay of S(t)(A + 1)⁻¹, linking the decay to the behaviour of the resolvent of A on the imaginary axis. This generalizes results of Lebeau and Burq (in the case of logarithmic stability) and Liu-Rao and Bátkai-Engel-Prüss-Schnaubelt (in the case of polynomial stability).