Power-bounded operators and related norm estimates
Functional Analysis
2013-06-04 v1 Operator Algebras
Abstract
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.
Cite
@article{arxiv.math/0211254,
title = {Power-bounded operators and related norm estimates},
author = {Nigel Kalton and Stephen Montgomery-Smith and Krzysztof Oleszkiewicz and Yuri Tomilov},
journal= {arXiv preprint arXiv:math/0211254},
year = {2013}
}
Comments
Also available at http://www.math.missouri.edu/~stephen/preprints/