Power-bounded operators and related norm estimates

It is considered whether L= limsup(n-->infinity)nparallel toT(n+1) - Tnparallel to < infinity 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 sigma(T) = {1} and T not equal I; then lim inf(n--> infinity) nparallel toT(n+1) - T-n parallel to greater than or equal to 1/e. The constant 1/e is sharp. Finally, a way to create many general izat ions 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.