A Katznelson-Tzafriri theorem for measures

Abstract: This article generalises the well-known Katznelson-Tzafriri theorem for a $C_0$-semigroup $T$ on a Banach space $X$, by removing the assumption that a certain measure in the original result be absolutely continuous. In an important special case the rate of decay is quantified in terms of the growth of the resolvent of the generator of $T$. These results are closely related to ones obtained recently in the Hilbert space setting by Batty, Chill and Tomilov in [6]. The main new idea is to incorporate an assumption on the non-analytic growth bound $\zeta(T)$ which is equivalent to the assumption made in [6] if $X$ is a Hilbert space.