Commutators of quasinilpotents and invariant subspaces
Studia Mathematica, Tome 128 (1998) no. 2, pp. 159-169
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
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.
A. Katavolos. Commutators of quasinilpotents and invariant subspaces. Studia Mathematica, Tome 128 (1998) no. 2, pp. 159-169. doi: 10.4064/sm-128-2-159-169
@article{10_4064_sm_128_2_159_169,
author = {A. Katavolos},
title = {Commutators of quasinilpotents and invariant subspaces},
journal = {Studia Mathematica},
pages = {159--169},
year = {1998},
volume = {128},
number = {2},
doi = {10.4064/sm-128-2-159-169},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/sm-128-2-159-169/}
}
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