Superreflexivity and J-convexity of Banach spaces

TL;DR

This paper explores the relationship between superreflexivity and J-convexity in Banach spaces, providing a quantitative formulation that aids in understanding operator factorizations.

Contribution

It offers a new quantitative formulation of the equivalence between superreflexivity and J-convexity, enabling improved operator factorization analysis.

Findings

Quantitative equivalence between superreflexivity and J-convexity

Method to find factorization of S_n through X from larger N

Enhanced understanding of operator factorizations in Banach spaces

Abstract

A Banach space X is superreflexive if each Banach space Y that is finitely representable in X is reflexive. Superreflexivity is known to be equivalent to J-convexity and to the non-existence of uniformly bounded factorizations of the summation operators S_n through X. We give a quantitative formulation of this equivalence. This can in particular be used to find a factorization of S_n through X, given a factorization of S_N through [L_2,X], where N is `large' compared to n.