Weakly closed Lie modules of nest algebras

A collection of subspaces of a Hilbert space is called a nest if it is totally ordered by inclusion. The set of all bounded linear operators leaving invariant each member of a given nest forms a weakly-closed algebra, called a nest algebra. Nest algebras were introduced by J. R. Ringrose in [9]. The present paper is concerned with generating nest algebras as weakly-closed algebras, and in particular with the following question which was first raised by H. Radjavi and P. Rosenthal in [8], viz: Is every nest algebra on a separable Hilbert space generated, as a weakly-closed algebra, by two operators? That the answer to this question is affirmative is proved by first reducing the problem using the main result of [8] and then by using a characterization of nests due to J. A. Erdos [2].

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Perturbation of a nest algebra module

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410006059x ◽

1983 ◽

Vol 93 (2) ◽

pp. 303-306 ◽

Cited By ~ 1

Author(s):

Sotirios Karanasios

Keyword(s):

General Situation ◽

Compact Perturbation ◽

Nest Algebra ◽

Nest Algebras ◽

Weakly Closed ◽

Carry Over ◽

Algebra Module ◽

Algebra Of Operators

Fall, Arveson and Muhly(4) characterized the compact perturbation of nest algebras. In fact they proved that the compact perturbation of a nest algebra corresponding to a nest of projections is the algebra of operators which are quasitriangular relative to this nest. Erdos and Power(3) investigated weakly closed ideals and modules of nest algebras and these exhibit properties that are very close to the properties of the nest algebras themselves. They also showed that in certain cases, as in the case when the homomorphism which determines the nest algebra module is continuous, the results of Fall, Arveson and Muhly carry over to the more general situation. In this paper we provide a characterization of the compact perturbation of any nest algebra module.

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