# Power type $\xi$-Asymptotically uniformly smooth and $\xi$-asymptotically uniformly flat norms

## TL;DR

This paper characterizes spaces and operators with $\xi$-$p$-asymptotically uniformly smooth or flat norms, introduces new notions, and explores their properties, including behavior under tensor products and renorming related to the Cantor-Bendixson index.

## Contribution

It provides isomorphic characterizations of $\xi$-$p$-asymptotically uniformly smooth and flat norms, introduces new classes of operators, and analyzes their ideal properties and stability under tensor products.

## Key findings

- Characterization of spaces with $\xi$-$p$-asymptotically uniformly smooth norms.
- Introduction of $\xi$-asymptotically uniformly flat norms and their properties.
- Stability of these properties under tensor products and renorming related to the Cantor-Bendixson index.

## Abstract

For each ordinal $\xi$ and each $1<p<\infty$, we offer a natural, ismorphic characterization of those spaces and operators which admit an equivalent $\xi$-$p$-asymptotically uniformly smooth norm. We also introduce the notion of $\xi$-asymptotically uniformly flat norms and provide an isomorphic characterization of those spaces and operators which admit an equivalent $\xi$-asymptotically uniformly flat norm. Given a compact, Hausdorff space $K$, we prove an optimal renormong theorem regarding the $\xi$-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 $\xi$-$p$-asymptotically uniformly smooth, $\xi$-$p$-asymptotically uniformly smoothable, $\xi$-asymptotically uniformly flat, and $\xi$-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.