Rates of decay in the classical Katznelson-Tzafriri theorem

TL;DR

This paper studies how quickly the decay occurs in the Katznelson-Tzafriri theorem for power-bounded operators when the spectrum intersects the unit circle only at 1, providing bounds based on resolvent growth.

Contribution

It establishes upper and lower bounds on decay rates related to resolvent growth, with optimal bounds in Banach spaces and partial results in Hilbert spaces.

Findings

Bounds on decay rates in terms of resolvent growth

Optimal bounds for Banach spaces

Partial bounds for Hilbert spaces

Abstract

Given a power-bounded operator $T$, the theorem of Katznelson and Tzafriri states that $\|T^n(I-T)\|\to0$ 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 $\theta\to0$. 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.