Power-bounded operators and related norm estimates
  We consider whether L = limsup_{n to infty} n ||T^{n+1}-T^n|| < infty implies that the operator T is power bounded. We show that this is so if L<1/e, but it does not necessarily hold if L=1/e. As part of our methods, we improve a result of Esterle, showing that if sigma(T) = {1} and T != I, then liminf_{n to infty} n ||T^{n+1}-T^n|| >= 1/e. The constant 1/e is sharp. Finally we describe a way to create many generalizations of Esterle's result, and also give many conditions on an operator which imply that its norm is equal to its spectral radius. 
Nigel Kalton, Stephen Montgomery-Smith, Krzysztof Oleszkiewicz and   Yuri Tomilov