Commutators of quasinilpotents and invariant subspaces

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.