Positive operators as commutators of positive operators
Volume 245 / 2019
Studia Mathematica 245 (2019), 185-200
MSC: Primary 47B07, 47B47; Secondary 47B65, 46B42.
DOI: 10.4064/sm170703-26-9
Published online: 20 July 2018
Abstract
It is known that a positive commutator $C=A B - B A$ between positive operators on a Banach lattice is quasinilpotent whenever at least one of $A$ and $B$ is compact. We study the question under which conditions a positive operator can be written as a commutator of positive operators. As a special case of our main result we find that positive compact operators on order continuous Banach lattices which admit order Pełczyński decomposition are commutators of positive operators. Our main result is also applied in the setting of a separable infinite-dimensional Banach lattice $L^p(\mu )$ $(1 \lt p \lt \infty )$.
