Once More on Positive Commutators

TL;DR

This paper proves that for certain positive operators on a Banach lattice, if their product is power-compact, then their commutator is quasi-nilpotent and admits a triangularization, answering an open question in operator theory.

Contribution

It establishes a new result linking positivity, power-compactness, and quasi-nilpotency of commutators on Banach lattices, resolving a previously open problem.

Findings

Commutator C is quasi-nilpotent under given conditions.

C admits a triangularizing chain of closed ideals.

The theorem addresses an open question in the field.

Abstract

Let A and B be bounded operators on a Banach lattice E such that the commutator C=AB-BA and the product BA are positive operators. If the product AB is a power-compact operator, then C is a quasi-nilpotent operator having a triangularizing chain of closed ideals of E. This theorem answers an open question posed in a paper by Bra\v{c}i\v{c}, Drnov\v{s}ek, Farforovskaya, Rabkin and Zem\'{a}nek.