Some developments around the Katznelson-Tzafriri theorem

This paper is a survey article on developments arising from a theorem proved by Katznelson and Tzafriri in 1986 showing that lim𝑛→∞||𝑇𝑛(𝐼−𝑇)||=0limn→∞||Tn(I−T)||=0 if T is a power-bounded operator on a Banach space and σ(𝑇)∩𝕋⊆{1}σ(T)∩T⊆{1}. Many variations and consequences of the original theorem ha...