Commutators of quasinilpotents and invariant subspaces

A. Katavolos; C. Stamatopoulos

Studia Mathematica (1998)

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

Abstract

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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.

How to cite

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Katavolos, 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 -

References

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