Power-bounded operators and related norm estimates

Abstract

It is considered whether L = lim supn→∞ n∥T n+1 - Tn∥ < ∞ implies that the operator T is power-bounded. It is shown that this is so if L < 1/e, but it does not necessarily hold if L = 1/e. As part of the methods, a result of Esterle is improved, showing that if σ(T) = {1} and T ≠ I, then lim inf n-∞ n∥Tn+1 -Tn∥ ≥ 1/e. The constant 1/e is sharp. Finally, a way to create many generalizations of Esterle's result is described, and also many conditions are given on an operator which imply that its norm is equal to its spectral radius.