We consider a positive operator A on a Hilbert lattice such that its self-commutator \(C = A^* A - A A^*\) is positive. If A is also idempotent, then it is an orthogonal projection, and so \(C = 0\) . Similarly, if A is power compact, then \(C = 0\) as well. We prove that every positive compact central operator on a separable infinite-dimensional Hilbert lattice \(\mathcal H\) is a self-commutator of a positive operator. We also show that every positive central operator on \(\mathcal H\) is a sum of two positive self-commutators of positive operators.
