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

Abstract

For each ordinal ξ and each 1<p<∞, we offer a natural, isomorphic 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 renorming 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 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.