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.

17 pages, to appear in Journal of Integral Equations and Operator Theory

References in corpus (2)

• A Katznelson-Tzafriri theorem for measures
• Some improvements of the Katznelson-Tzafriri theorem on Hilbert space

Cited by in corpus (7)

• Optimal rates of decay for operator semigroups on Hilbert spaces
• Quantified versions of Ingham's theorem
• Semi-uniform stability of operator semigroups and energy decay of damped waves
• Stability theory for semigroups using $(L^p,L^q)$ Fourier multipliers
• A Katznelson-Tzafriri theorem for measures
• Some developments around the Katznelson-Tzafriri theorem
• Asymptotic behaviour of coupled systems in discrete and continuous time