On the nonnegative rank of positive operators

TL;DR

This paper extends the concept of nonnegative rank to positive operators between ordered vector spaces, establishing conditions under which nonnegative rank equals rank and providing a counterexample at rank three.

Contribution

It introduces a new definition of nonnegative rank for positive operators and proves equivalence with rank for ranks up to two, along with a counterexample at rank three.

Findings

Nonnegative rank and rank agree for operators with rank at most two.

Provided an example of a rank-three positive operator with infinite nonnegative rank.

Extended the concept of nonnegative rank to infinite-dimensional settings.

Abstract

In this paper we introduce the concept of a nonnegative rank of a positive operator $T\colon X\to Y$ between ordered vector spaces. In the case of nonnegative matrices, our definition agrees with the standard definition of a nonnegative rank. Under some natural and mild assumptions on the cone $Y_+$, we prove that the nonnegative rank and the rank agree whenever the rank is at most two. This can be considered as the infinite-dimensional version of \cite[Theorem 4.1]{CR93}. We also provide an example of a positive rank-three operator on the Banach lattice $C[0,1]$ with an infinite nonnegative rank.exceed $\lceil 6\min\{m,n\}/7\rceil$.