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arXiv · 1705.09834

Power type $ξ$-Asymptotically uniformly smooth and $ξ$-asymptotically uniformly flat norms

R. M. Causey
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Abstract

For each ordinal $ξ$ and each $1<p<\infty$, we offer a natural, ismorphic characterization of those spaces and operators which admit an equivalent $ξ$-$p$-asymptotically uniformly smooth norm. We also introduce the notion of $ξ$-asymptotically uniformly flat norms and provide an isomorphic characterization of those spaces and operators which admit an equivalent $ξ$-asymptotically uniformly flat norm. Given a compact, Hausdorff space $K$, we prove an optimal renormong theorem regarding the $ξ$-asymptotic smoothness of $C(K)$ in terms of the Cantor-Bendixson index of $K$. We also prove that for all ordinals, both the isomorphic properties and isometric properties we study pass from Banach spaces to their injective tensor products. We study the classes of $ξ$-$p$-asymptotically uniformly smooth, $ξ$-$p$-asymptotically uniformly smoothable, $ξ$-asymptotically uniformly flat, and $ξ$-asymptotically uniformly flattenable operators. We show that these classes are either a Banach ideal or a right Banach ideal when assigned an appropriate ideal norm.

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R. M. Causey. 2017-06-05. Power type $ξ$-Asymptotically uniformly smooth and $ξ$-asymptotically uniformly flat norms. https://arxiv.org/abs/1705.09834

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