Rates of decay in the classical Katznelson-Tzafriri theorem

Abstract

Given a power-bounded operator $T$, the theorem of Katznelson and Tzafriri states that $\Vert T^n(I-T)\Vert \to 0$ as $n\to \infty$ if and only if the spectrum $\sigma (T)$ of $T$ intersects the unit circle $\mathbb{T}$ in at most the point 1. This paper investigates the rate at which decay takes place when $\sigma (T)\cap \mathbb{T}=\{1\}$. The results obtained lead in particular to both upper and lower bounds on this rate of decay in terms of the growth of the resolvent operator $R({\mathrm{e}}^{\mathrm{i}\theta },T)$ as $\theta \to 0$. In the special case of polynomial resolvent growth, these bounds are then shown to be optimal for general Banach spaces but not in the Hilbert space case.

24 pages, to appear in Journal d'Analyse Mathématique

References in corpus (2)

• A quantified Tauberian theorem for sequences
• Some improvements of the Katznelson-Tzafriri theorem on Hilbert space

Cited by in corpus (8)

• Ritt operators and convergence in the method of alternating projections
• Local decay of $C_0$-semigroups with a possible singularity of logarithmic type at zero
• Optimal rates of decay in the Katznelson-Tzafriri theorem for operators on Hilbert spaces
• Some developments around the Katznelson-Tzafriri theorem
• Asymptotics for periodic systems
• A quantified Tauberian theorem and local decay of $C_0$-semigroups
• Robustness of Polynomial Stability with Respect to Sampling
• The asymptotic behaviour of the Cesàro operator