2025/01/07 by Roman Drnovšek, Drnovšek, Roman, Marko Kandić +1 · 1 citation
Mathematics · Computer Science · #Holomorphic and Operator Theory #Advanced Banach Space Theory #Matrix Theory and Algorithms
paper · doi:10.48550/arxiv.2501.03733
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.