Some developments around the Katznelson-Tzafriri theorem

TL;DR

This survey reviews various developments and extensions of the Katznelson-Tzafriri theorem, which characterizes the asymptotic behavior of power-bounded operators on Banach spaces.

Contribution

It provides a comprehensive account of the numerous variations and consequences derived from the original theorem since 1986.

Findings

Multiple variations of the theorem have been established.

Extensions include broader classes of operators and spectral conditions.

The theorem's implications for operator theory are extensive.

Abstract

This paper is a survey article on developments arising from a theorem proved by Katznelson and Tzafriri in 1986 showing that $\lim_{n\to\infty} \|T^n(I-T)\| =0$ if $T$ is a power-bounded operator on a Banach space and $\sigma(T) \cap \T \subseteq \{1\}$. Many variations and consequences of the original theorem have been proved subsequently, and we provide an account of this branch of operator theory.