A short proof of the Y. Katznelson’s and L. Tzafriri’s theorem

Abstract

A short proof is given to the following theorem of Y. Katznelson and L. Tzafriri: Let T T be a power-bounded operator in a Banach space E E . Then lim n → ∞ | | T n + 1 − T n | | = 0 {\lim _{n \to \infty }}||{T^{n + 1}} - {T^n}|| = 0 if and only if σ ( T ) ∩ { z ∈ C : | z | = 1 } ⊂ { 1 } \sigma (T) \cap \{ z \in \mathbb {C}:|z| = 1\} \subset \{ 1\} .