Power-bounded Operators and Related Norm Estimates

It is considered whether L= lim supn→∞ n‖Tn+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 infn→∞ 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.