On the positive commutator in the radical

TL;DR

This paper proves that positive commutators between certain operators are in the radical of the generated Banach algebra, but also constructs counterexamples in higher dimensions, answering two open questions.

Contribution

It establishes conditions under which positive commutators are in the radical and provides counterexamples in Banach lattices, resolving two open problems.

Findings

Positive commutators between positive compact and positive operators are in the radical.

Counterexamples show commutators can lie outside the radical in higher dimensions.

Results extend to Volterra and Donoghue operators.

Abstract

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 $AB\geq BA\geq 0$ 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 [4]. We also obtain relevant results in the case of the Volterra and the Donoghue operator.