# Commutators of quasinilpotents and invariant subspaces

A. Katavolos; C. Stamatopoulos

Studia Mathematica (1998)

- Volume: 128, Issue: 2, page 159-169
- ISSN: 0039-3223

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topKatavolos, A., and Stamatopoulos, C.. "Commutators of quasinilpotents and invariant subspaces." Studia Mathematica 128.2 (1998): 159-169. <http://eudml.org/doc/216481>.

@article{Katavolos1998,

abstract = {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.},

author = {Katavolos, A., Stamatopoulos, C.},

journal = {Studia Mathematica},

keywords = {higher commutator conditions of quasinilpotents; quasinilpotent elements in a Banach algebra; ideal; Jacobson radical; inner derivation; polynomially compact operators; triangularizable},

language = {eng},

number = {2},

pages = {159-169},

title = {Commutators of quasinilpotents and invariant subspaces},

url = {http://eudml.org/doc/216481},

volume = {128},

year = {1998},

}

TY - JOUR

AU - Katavolos, A.

AU - Stamatopoulos, C.

TI - Commutators of quasinilpotents and invariant subspaces

JO - Studia Mathematica

PY - 1998

VL - 128

IS - 2

SP - 159

EP - 169

AB - 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.

LA - eng

KW - higher commutator conditions of quasinilpotents; quasinilpotent elements in a Banach algebra; ideal; Jacobson radical; inner derivation; polynomially compact operators; triangularizable

UR - http://eudml.org/doc/216481

ER -

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