Positive self-commutators of positive operators

Abstract

We consider a positive operator $A$ on a Hilbert lattice such that its self-commutator $C=A^{\ast }A-A{A}^{\ast }$ 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.

19 pages