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Commutators of quasinilpotents and invariant subspaces

100%

Katavolos A., Stamatopoulos C.

Studia Mathematica

1998

tom 128

nr 2

159-169

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.

The decomposability of operators relative to two subspaces

81%

Katavolos A., Lambrou M. S., Longstaff W. E.

Studia Mathematica

1993

tom 105

nr 1

25-36

Let M and N be nonzero subspaces of a Hilbert space H satisfying M ∩ N = {0} and M ∨ N = H and let T ∈ ℬ(H). Consider the question: If T leaves each of M and N invariant, respectively, intertwines M and N, does T decompose as a sum of two operators with the same property and each of which, in addition, annihilates one of the subspaces? If the angle between M and N is positive the answer is affirmative. If the angle is zero, the answer is still affirmative for finite rank operators but there are even trace class operators for which it is negative. An application gives an alternative proof that no distance estimate holds for the algebra of operators leaving M and N invariant if the angle is zero, and an analogous result is obtained for the set of operators intertwining M and N.

Ograniczanie wyników

2 Studia Mathematica

2 Katavolos A.

1 Lambrou M. S.

1 Longstaff W. E.

1 Stamatopoulos C.

1 1998

1 1993

Wyniki wyszukiwania

Commutators of quasinilpotents and invariant subspaces

The decomposability of operators relative to two subspaces