POWER-BOUNDED OPERATORS AND RELATED NORM ESTIMATES
N. J. Kalton, Stephen Montgomery-Smith, Krzysztof Oleszkiewicz, Yuri Tomilov · Journal of the London Mathematical Society · 2004
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.