On the Nonnegative Rank of Positive Operators

In this paper, we introduce the concept of a nonnegative rank of a positive operator 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 , 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 [6, Theorem 4.1]. We also provide an example of a positive rank‐three operator on the Banach lattice with an infinite nonnegative rank.