On the positive commutator in the radical

In this paper we prove that a positive commutator between a positive compact operator A and a positive operator B is in the radical of the Banach algebra generated by A and B. Furthermore, on every at least three-dimensional Banach lattice we construct finite rank operators A and B satisfying \(AB\ge BA\ge 0\) such that the commutator \(AB-BA\) is not contained in the radical of the Banach algebra generated by A and B. These two results now completely answer to two open questions published in (Bra?i? et al., Positivity 14:431–439, 2010). We also obtain relevant results in the case of the Volterra and the Donoghue operator.