More on positive commutators

Let A and B be positive operators on a Banach lattice E such that the commutator C=AB−BA is also positive. The paper continues the investigation of the spectral properties of C initiated in J. Bračič et al. (in press) [3]. If the sum A+B is a Riesz operator and the commutator C is a power compact operator, then C is a quasi-nilpotent operator having a triangularizing chain of closed ideals of E. If we assume that the operator A is compact and the commutator AC−CA is positive, the operator C is quasi-nilpotent as well. We also show that the commutator C is not invertible provided the resolvent set of C is connected.