A Katznelson-Tzafriri theorem for measures

TL;DR

This paper extends the Katznelson-Tzafriri theorem for $C_0$-semigroups on Banach spaces by removing the absolute continuity condition on measures, and relates decay rates to the resolvent growth of the generator.

Contribution

It generalizes the Katznelson-Tzafriri theorem to Banach spaces without requiring measure absolute continuity, introducing a new approach involving the non-analytic growth bound.

Findings

Removed the absolute continuity assumption on measures in the theorem

Quantified decay rates in terms of resolvent growth

Connected results to recent Hilbert space studies

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.