Positive commutators of positive square-zero operators

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 ${\mathrm{\ell }}^p$ ($1\le p<\mathrm{\infty }$). In the last setting we need to extend the notion of the nonnegative rank of a nonnegative matrix.