Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T13:11:21.066Z Has data issue: false hasContentIssue false
  • English
  • Français

Generators of Nest Algebras

Published online by Cambridge University Press:  20 November 2018

W. E. Longstaff
W. E. Longstaff*
Affiliation:
University of Toronto, Toronto, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 3 , June 1974 , pp. 565 - 575
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Dixmier, J., Les algèbres d'opérateurs dans l'espace Hilbertien (Gauthier-Villars, Paris, 1957).Google Scholar
2. Erdos, J. A., On some non-s elf-adjoint algebras of operators, Ph.D. Thesis, Peterhouse College, Cambridge 1964.Google Scholar
3. Erdos, J. A., Unitary invariants for nests, Pacific J. Math. 23 (1967), 229–256.Google Scholar
4. Halmos, P. R., Measure theory (Van Nostrand, Princeton, 1955).Google Scholar
5. Kadison, R. V. and Singer, I. M., Triangular operator algebras, Amer. J. Math. 82 (1960), 227–259.Google Scholar
6. Kelley, J. L., General topology (Van Nostrand, Princeton, 1955).Google Scholar
7. von Neumann, J., Zür Algebra der Funktionaloperationen und Théorie der Normalen Operatoren, Math. Ann. 102 (1929), 370–427.Google Scholar
8. Radjavi, H. and Rosenthal, P., On invariant subspaces and reflexive algebras, Amer. J. Math. 91 (1969), 683–692.Google Scholar
9. Ringrose, J. R., On some algebras of operators, Proc. London Math. Soc. 15 (1965), 61–83.Google Scholar
10. Rosenthal, P., Completely reducible operators, Proc. Amer. Math. Soc. 19 (1968), 826–830.Google Scholar