Invariant linear manifolds for CSL-algebras and nest algebras
HTML articles powered by AMS MathViewer
- by Alan Hopenwasser PDF
- Proc. Amer. Math. Soc. 129 (2001), 389-395 Request permission
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
- Kenneth R. Davidson, Invariant operator ranges for reflexive algebras, J. Operator Theory 7 (1982), no. 1, 101–107. MR 650195
- Kenneth R. Davidson, Nest algebras, Pitman Research Notes in Mathematics Series, vol. 191, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. Triangular forms for operator algebras on Hilbert space. MR 972978
- Ciprian Foiaş, Invariant para-closed subspaces, Indiana Univ. Math. J. 20 (1971), no. 10, 897–900. MR 399893, DOI 10.1512/iumj.1971.20.20074
- Ciprian Foiaş, Invariant para-closed subspaces, Indiana Univ. Math. J. 21 (1971/72), 887–906. MR 293439, DOI 10.1512/iumj.1972.21.21072
- Alan Hopenwasser, The equation $Tx=y$ in a reflexive operator algebra, Indiana Univ. Math. J. 29 (1980), no. 1, 121–126. MR 554821, DOI 10.1512/iumj.1980.29.29009
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- E. C. Lance, Some properties of nest algebras, Proc. London Math. Soc. (3) 19 (1969), 45–68. MR 241990, DOI 10.1112/plms/s3-19.1.45
- W. E. Longstaff, Strongly reflexive lattices, J. London Math. Soc. (2) 11 (1975), no. 4, 491–498. MR 394233, DOI 10.1112/jlms/s2-11.4.491
- Sing Cheong Ong, Invariant operator ranges of nest algebras, J. Operator Theory 3 (1980), no. 2, 195–201. MR 578939
- J. R. Ringrose, On some algebras of operators, Proc. London Math. Soc. (3) 15 (1965), 61–83. MR 171174, DOI 10.1112/plms/s3-15.1.61
Additional Information
- Alan Hopenwasser
- Affiliation: Department of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487
- Email: ahopenwa@euler.math.ua.edu
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 389-395
- MSC (2000): Primary 47L35
- DOI: https://doi.org/10.1090/S0002-9939-00-05596-9
- MathSciNet review: 1707148