Dimension dependence of factorization problems: Hardy spaces and $SL_n^\infty$

TL;DR

This paper investigates how the complexity of factorization problems in Hardy spaces and $SL_n^ty$ depends on dimension, showing that the identity operator factors through certain operators with large diagonals, with linear dimension dependence.

Contribution

It establishes a quantitative factorization result in finite-dimensional Hardy spaces and $SL_n^ty$, revealing linear dimension dependence for the first time.

Findings

Identity operator factors through operators with large diagonals

Linear dependence of N on n for factorization

Quantitative bounds in finite-dimensional Hardy spaces

Abstract

Given $1 \leq p < \infty$, let $W_n$ denote the finite-dimensional dyadic Hardy space $H_n^p$, its dual or $SL_n^\infty$. We prove the following quantitative result: The identity operator on $W_n$ factors through any operator $T : W_N\to W_N$ which has large diagonal with respect to the Haar system, where $N$ depends \emph{linearly} on $n$.