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Let A and B be completely decomposable nonnegative matrices such that the commutator AB− BA is also a nonnegative matrix. We prove that the set {A,B} is completely decomposable, i.e., there exists a permutation matrix P such that PAP−1 and PBP−1 are upper triangular matrices. We show similar results for collections of completely decomposable nonnegative matrices. We also find conditions on commutators under which a given operator on a Riesz space is necessarily scalar. Mathematics subject classification (2010): 15A48, 20M20.

Positive commutators and collections of operators