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arXiv · 2501.03733

Positive self-commutators of positive operators

Roman Drnovšek·Marko Kandić
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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.

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Roman Drnovšek, Marko Kandić. 2025-01-07. Positive self-commutators of positive operators. https://arxiv.org/abs/2501.03733

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