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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
DOI: https://doi.org/10.1090/S0002-9939-00-05596-9
Published electronically: August 29, 2000
MathSciNet review: 1707148
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Abstract:

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.


References [Enhancements On Off] (What's this?)

  • 1. K. R. Davidson, Invariant operator ranges for reflexive algebras, J. Operator Theory 7 (1982), 101-107. MR 83e:47004
  • 2. -, Nest algebras, Pitman Research Notes in Mathematics Series, vol. 191, Longman Scientific and Technical, 1988. MR 90f:47062
  • 3. C. Foias, Invariant para-closed subspaces, Indiana Univ. Math. J. 20 (1971), 897-900. MR 53:3734
  • 4. -, Invariant para-closed subspaces, Indiana Univ. Math. J. 21 (1972), 887-906. MR 45:2516
  • 5. A. Hopenwasser, The equation ${T}x=y$ in a reflexive operator algebra, Indiana Univ. Math. J. 29 (1980), 121-126. MR 81c:47014
  • 6. R. V. Kadison, Irreducible operator algebras, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 273-276. MR 19:47e
  • 7. E. C. Lance, Some properties of nest algebras, Proc. London Math. Soc. (3) 19 (1969), 45-68. MR 39:3325
  • 8. W. Longstaff, Strongly reflexive lattices, J. London Math. Soc. (2) 11 (1975), 491-498. MR 52:15036
  • 9. S.-C. Ong, Invariant operator ranges of nest algebras, J. Operator Theory 3 (1980), 195-201. MR 81f:47008
  • 10. J. Ringrose, On some algebras of operators, Proc. London Math. Soc. (3) 15 (1965), 61-83. MR 30:1405

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Additional Information

Alan Hopenwasser
Affiliation: Department of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487
Email: ahopenwa@euler.math.ua.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05596-9
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

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