On power bounded operators

The main result asserts that, for any contraction T on an arbitrary Banach space X, ∥ Tⁿ − T^{n + 1} ∥ → 0 as n → ∞, if and only if the spectrum of T has no points on the unit circle except perhaps z = 1. This theorem is extended for ϑ(T)Tⁿ, where ϑ is a function of spectral synthesis on the unit circle. As an application, we generalize the so-called “zero-two” law of Ornstein and Sucheston and Zaharopol to positive contraction on a very large class of Banach lattices.