$\xi$-asymptotically uniformly smooth, $\xi$-asymptotically uniformly convex, and $(\beta)$ operators

Ryan M. Causey; Stephen J. Dilworth

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arXiv · 1607.01362

$\xi$-asymptotically uniformly smooth, $\xi$-asymptotically uniformly convex, and $(\beta)$ operators

Ryan M. Causey·Stephen J. Dilworth

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Abstract

For each ordinal $\xi$, we define the notions of $\xi$-asymptotically uniformly smooth and $w^*$-$\xi$-asymptotically uniformly convex operators. When $\xi=0$, these extend the notions of asymptotically uniformly smooth and $w^*$-asymptotically uniformly convex Banach spaces. We give a complete description of renorming results for these properties in terms of the Szlenk index of the operator, as well as a complete description of the duality between these two properties. We also define the notion of an operator with property $(\beta)$ of Rolewicz which extends the notion of property $(\beta)$ for a Banach space. We characterize those operators the domain and range of which can be renormed so that the operator has property $(\beta)$ in terms of the Szlenk index of the operator and its adjoint.

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BibTeX RIS

Ryan M. Causey, Stephen J. Dilworth. 2016-07-05. $\xi$-asymptotically uniformly smooth, $\xi$-asymptotically uniformly convex, and $(\beta)$ operators. https://arxiv.org/abs/1607.01362

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