On the nonnegative rank of positive operators

Roman Drnovšek; Marko Kandić

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arXiv · 2605.21232

On the nonnegative rank of positive operators

Roman Drnovšek·Marko Kandić

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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.

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BibTeX RIS

Roman Drnovšek, Marko Kandić. 2026-05-20. On the nonnegative rank of positive operators. https://arxiv.org/abs/2605.21232

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