Nonlinear weakly sequentially continuous embeddings between Banach spaces

TL;DR

This paper investigates nonlinear embeddings between Banach spaces that are weakly sequentially continuous, revealing conditions under which such embeddings preserve spreading models and providing examples where coarse or uniform embeddings lack weak sequential continuity.

Contribution

It establishes a link between weakly sequentially continuous embeddings and spreading models, and constructs examples of embeddings that are coarse or uniform but not weakly sequentially continuous.

Findings

Weakly sequentially continuous embeddings impose conditions on spreading models.

Existence of Banach space pairs where coarse or uniform embeddings are not weakly sequentially continuous.

Main result relates asymptotic uniform convexity to embeddings via spreading models.

Abstract

In these notes, we study nonlinear embeddings between Banach spaces which are also weakly sequentially continuous. In particular, our main result implies that if a Banach space $X$ coarsely (resp. uniformly) embeds into a Banach space $Y$ by a weakly sequentially continuous map, then every spreading model $(e_n)_n$ of a normalized weakly null sequence in $X$ satisfies \[\|e_1+\ldots+e_k\|_{\overline{\delta}_Y}\lesssim\|e_1+\ldots+e_k\|_S,\] where $\overline{\delta}_Y$ is the modulus of asymptotic uniform convexity of $Y$. Among other results, we obtain Banach spaces $X$ and $Y$ so that $X$ coarsely (resp. uniformly) embeds into $Y$, but so that $X$ cannot be mapped into $Y$ by a weakly sequentially continuous coarse (resp. uniform) embedding.