Factorization in independent sums of Haar system Hardy spaces

TL;DR

This paper introduces a new class of Hardy spaces built from independent copies of dyadic step function spaces and proves a factorization property of the identity operator through operators with large diagonals.

Contribution

It generalizes the Bourgain-Rosenthal-Schechtman space construction to Haar system Hardy spaces and establishes a factorization theorem for the identity operator.

Findings

The identity operator factors through any operator with large diagonal.

The identity factors either through the operator or its complement.

The approach combines finite-dimensional and infinite-dimensional techniques.

Abstract

We introduce a generalization of the Bourgain-Rosenthal-Schechtman $R_{\omega}^p$ space: Let $Y$ be a Haar system Hardy space, i.e., a separable rearrangement-invariant function space on the unit interval or an associated Hardy space defined via the square function (such as dyadic $H^1$). Then we define $Y_{\omega}$ as the closed linear span in $Y$ of independent distributional copies of the spaces $Y_n$ of dyadic step functions at scale $2^{-n}$. Combining finite-dimensional and infinite-dimensional techniques, we prove that the identity operator $I$ on $Y_{\omega}$ factors through every bounded linear operator $T$ on $Y_{\omega}$ which has large diagonal, and in general, the identity factors either through $T$ or through $I - T$.