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Proceedings of the American Mathematical Society

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Invariant linear manifolds for CSL-algebras and nest algebras

Author: Alan Hopenwasser
Journal: Proc. Amer. Math. Soc. 129 (2001), 389-395
MSC (2000): Primary 47L35
Published electronically: August 29, 2000
MathSciNet review: 1707148
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Every invariant linear manifold for a CSL-algebra, $\operatorname{Alg} \mathcal{L}$, is a closed subspace if, and only if, each non-zero projection in $\mathcal{L}$ is generated by finitely many atoms associated with the projection lattice. When $\mathcal{L}$is a nest, this condition is equivalent to the condition that every non-zero projection in $\mathcal{L}$ has an immediate predecessor ( $\mathcal{L}^{\perp}$ is well ordered). The invariant linear manifolds of a nest algebra are totally ordered by inclusion if, and only if, every non-zero projection in the nest has an immediate predecessor.

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Additional Information

Alan Hopenwasser
Affiliation: Department of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487

Keywords: Nest algebra, CSL-algebra, invariant subspace, invariant linear manifold
Received by editor(s): June 15, 1998
Received by editor(s) in revised form: April 8, 1999
Published electronically: August 29, 2000
Additional Notes: The author would like to thank Ken Davidson for drawing his attention to the references regarding operator ranges.
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society