# Power type asymptotically uniformly smooth and asymptotically uniformly flat norms

## TL;DR

This paper characterizes $p$-asymptotic uniform smoothability and flattenability of operators and Banach spaces, showing these properties are preserved under tensor products and establishing Banach ideal structures for these classes.

## Contribution

It provides new characterizations of asymptotic uniform smoothness and flatness, and demonstrates their stability under tensor products and ideal norm structures.

## Key findings

- Injective tensor product of asymptotically uniformly smooth spaces remains asymptotically uniformly smooth.
- The classes of $p$-asymptotically uniformly smoothable and flattenable operators form Banach ideals.
- Many asymptotic uniform smoothness properties pass to tensor products.

## Abstract

We provide a short characterization of $p$-asymptotic uniform smoothability and asymptotic uniform flatenability of operators and of Banach spaces. We use these characterizations to show that many asymptotic uniform smoothness properties pass to injective tensor products of operators and of Banach spaces. In particular, we prove that the injective tensor product of two asymptotically uniformly smooth Banach spaces is asymptotically uniformly smooth. We prove that for $1<p<\infty$, the class of $p$-asymptotically uniformly smoothable operators can be endowed with an ideal norm making this class a Banach ideal. We also prove that the class of asymptotically uniformly flattenable operators can be endowed with an ideal norm making this class a Banach ideal.