**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.
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