Strategically reproducible bases and the factorization property

TL;DR

This paper introduces strategically reproducible bases in Banach spaces, demonstrating their significance in operator factorization and providing examples in classical spaces like Lebesgue, Hardy, and L^1 spaces.

Contribution

It defines strategically reproducible bases, shows their role in operator factorization, and provides multiple classical space examples where these bases occur.

Findings

Haar system is strategically reproducible in several classical spaces

Operators with large diagonal are factors of the identity in these bases

Strategic reproducibility is preserved under unconditional sums

Abstract

We introduce the concept of strategically reproducible bases in Banach spaces and show that operators which have large diagonal with respect to strategically reproducible bases are factors of the identity. We give several examples of classical Banach spaces in which the Haar system is strategically reproducible: multi-parameter Lebesgue spaces, mixed-norm Hardy spaces and most significantly the space $L^1$. Moreover, we show the strategical reproducibility is inherited by unconditional sums.