Positive self-commutators of positive operators

TL;DR

This paper investigates conditions under which positive operators on Hilbert lattices can be expressed as self-commutators, revealing that positive compact central operators are always such self-commutators.

Contribution

It proves that every positive compact central operator on a separable infinite-dimensional Hilbert lattice is a self-commutator of a positive operator, and that positive central operators are sums of two such self-commutators.

Findings

Positive compact central operators are self-commutators.

Positive central operators are sums of two positive self-commutators.

Idempotent positive operators are orthogonal projections.

Abstract

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.