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 lim n → ∞ | | T n ( I - T ) | | = 0 if T is a power-bounded operator on a Banach space and σ ( T ) ∩ T ⊆ { 1 } . Many variations and consequences of the original theorem have been proved subsequently, and we provide an account of this branch of operator theory.