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

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.