Power‐Bounded Operators and Related Norm Estimates

It is considered whether L = lim sup n → ∞ n ‖T n + 1 − T n ‖< ∞ 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 ‖T n + 1 − T n ‖⩾ 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.