Positive operators as commutators of positive operators

It is known that a positive commutator $C=A B - B A$ between positive operators on a Banach lattice is quasinilpotent whenever at least one of $A$ and $B$ is compact. In this paper we study the question under which conditions a positive operator can be written as a commutator between positive operators. As a special case of our main result we obtain that positive compact operators on order continuous Banach lattices which admit order Pelczy\'nski decomposition are commutators between positive operators. Our main result is also applied in the setting of a separable infinite-dimensional Banach lattice $L^p(\mu)$ $(1<p<\infty)$.