Once more on positive commutators
Tom 211 / 2012
Studia Mathematica 211 (2012), 241-245 MSC: Primary 47B65, 47B47; Secondary 46B42. DOI: 10.4064/sm211-3-5
Let $A$ and $B$ be bounded operators on a Banach lattice $E$ such that the commutator $C = A B - B A$ and the product $BA$ are positive operators. If the product $AB$ is a power-compact operator, then $C$ is a quasi-nilpotent operator having a triangularizing chain of closed ideals of $E$. This answers an open question posed by Bračič et al. [Positivity 14 (2010)], where the study of positive commutators of positive operators was initiated.