Some developments around the Katznelson-Tzafriri theorem

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.