Three-space property for asymptotically uniformly smooth renormings

TL;DR

This paper establishes that the three-space property holds for asymptotically uniformly smooth renormings in Banach spaces, using Szlenk index techniques to extend smoothness properties from subspaces and quotients to the entire space.

Contribution

It proves the three-space property for asymptotically uniformly smooth renormings, providing new insights and applications in Banach space renorming theory.

Findings

The property holds for subspaces and quotients with asymptotic uniform smoothness.

Uses Szlenk index to analyze renorming properties.

Provides new applications to renorming theory.

Abstract

We prove that if $Y$ is a closed subspace of a Banach space $X$ such that $Y$ and $X/Y$ admit an equivalent asymptotically uniformly smooth norm, then $X$ also admits an equivalent asymptotically uniformly smooth norm. The proof is based on the use of the Szlenk index and yields a few other applications to renorming theory.