On the positive commutator in the radical

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 $$AB\ge BA\ge 0$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

On the positive commutator in the radical

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Publisher
Springer Journals
Copyright
Copyright © 2016 by Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0409-1
Publisher site
See Article on Publisher Site

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 $$AB\ge BA\ge 0$$ 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.

Journal

PositivitySpringer Journals

Published: Mar 28, 2016

References

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