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 ${\lim }_{n\to \infty }\Vert T^n(I-T)\Vert =0$ if $T$ is a power-bounded operator on a Banach space and $σ(T) \cap \T \subseteq \{1\}$. Many variations and consequences of the original theorem have been proved subsequently, and we provide an account of this branch of operator theory.

References in corpus (2)

• Semi-uniform stability of operator semigroups and energy decay of damped waves
• A Katznelson-Tzafriri theorem for measures

Cited by in corpus (1)

• Robustness of Polynomial Stability with Respect to Sampling