Nigel Kalton, Stephen Montgomery-Smith, Krzysztof Oleszkiewicz and Yuri Tomilov, Power-bounded operators and related norm estimates. Journal of London Math. Soc. 70, (2004), 463-478. 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 \(T \ne I\), then \(\liminf_{n\to\infty} n {\|T^{n+1}-T^n\|} \ge 1/e\). 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. (tex, dvi, ps, pdf.)


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