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 $$AB\ge BA\ge 0$$ A B ≥ B A ≥ 0 such that the commutator $$AB-BA$$ A B - B A 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 (Bračič et al., Positivity 14:431–439, 2010). We also obtain relevant results in the case of the Volterra and the Donoghue operator.
Positivity – Springer Journals
Published: Mar 28, 2016
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