It is proved that the set Q of quasinilpotent elements in a Banach algebra is an ideal, i.e. equal to the Jacobson radical, if (and only if) the condition [Q,Q] ⊆ Q (or a similar condition concerning anticommutators) holds. In fact, if the inner derivation defined by a quasinilpotent element p maps Q into itself then p ∈ Rad A. Higher commutator conditions of quasinilpotents are also studied. It is shown that if a Banach algebra satisfies such a condition, then every quasinilpotent element has some fixed power in the Jacobson radical. These results are applied to topologically transitive representations. As a consequence, it is proved that a closed algebra of polynomially compact operators satisfying a higher commutator condition must have an invariant nest of closed subspaces, with "gaps" of bounded dimension. In particular, if [Q,Q] ⊆ Q, then the algebra must be triangularizable. An example is given showing that this may fail for more general algebras.
@article{bwmeta1.element.bwnjournal-article-smv128i2p159bwm,
author = {A. Katavolos and C. Stamatopoulos},
title = {Commutators of quasinilpotents and invariant subspaces},
journal = {Studia Mathematica},
volume = {129},
year = {1998},
pages = {159-169},
zbl = {0904.46037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv128i2p159bwm}
}
Katavolos, A.; Stamatopoulos, C. Commutators of quasinilpotents and invariant subspaces. Studia Mathematica, Tome 129 (1998) pp. 159-169. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv128i2p159bwm/
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