Factorization of the identity through operators with large diagonal

TL;DR

This paper investigates whether the identity operator on certain Banach spaces factors through operators with large diagonal, revealing space-dependent behaviors and providing new insights into operator factorizations.

Contribution

It demonstrates that on Gowers' unconditional Banach space, the identity does not always factor through such operators, while on mixed-norm Hardy spaces, it always does, highlighting space-specific properties.

Findings

Counterexample on Gowers' space where factorization fails

Universal factorization property on mixed-norm Hardy spaces

Extension of previous results on $L^p$ spaces to other Banach spaces

Abstract

Given a Banach space~$X$ with an unconditional basis, we consider the following question: does the identity on~$X$ factor through every operator on~$X$ with large diagonal relative to the unconditional basis? We show that on Gowers' unconditional Banach space, there exists an operator for which the answer to the question is negative. By contrast, for any operator on the mixed-norm Hardy spaces $H^p(H^q)$, where $1 \leq p,q < \infty$, with the bi-parameter Haar system, this problem always has a positive solution. The spaces $L^p, 1 < p < \infty$, were treated first by Andrew~[{\em Studia Math.}~1979].