Positive commutators of positive square-zero operators

TL;DR

This paper investigates which nonnegative matrices can be expressed as commutators of positive square-zero matrices, extending the analysis to infinite-dimensional operators on Banach lattices and introducing a new concept of nonnegative rank.

Contribution

It provides a characterization of such commutators in finite and infinite dimensions, including the extension of nonnegative rank to infinite-dimensional operators.

Findings

Characterization of nonnegative matrices as commutators of positive square-zero matrices.

Extension of results to operators on Banach lattices like $L^p[0,1]$ and $\,ell^p$.

Introduction of a generalized notion of nonnegative rank for infinite-dimensional operators.

Abstract

In this paper we first consider the question which nonnegative matrices are commutators of nonnegative square-zero matrices. Then, we treat infinite-dimensional analogues of these results for operators on the Banach lattices $L^p[0,1]$ and $\ell^p$ ($ 1 \leq p < \infty$). In the last setting we need to extend the notion of the nonnegative rank of a nonnegative matrix.