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Generators of Nest Algebras

  • W. E. Longstaff (a1)

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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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References

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1. Dixmier, J., Les algèbres d'opérateurs dans l'espace Hilbertien (Gauthier-Villars, Paris, 1957).
2. Erdos, J. A., On some non-s elf-adjoint algebras of operators, Ph.D. Thesis, Peterhouse College, Cambridge 1964.
3. Erdos, J. A., Unitary invariants for nests, Pacific J. Math. 23 (1967), 229256.
4. Halmos, P. R., Measure theory (Van Nostrand, Princeton, 1955).
5. Kadison, R. V. and Singer, I. M., Triangular operator algebras, Amer. J. Math. 82 (1960), 227259.
6. Kelley, J. L., General topology (Van Nostrand, Princeton, 1955).
7. von Neumann, J., Zür Algebra der Funktionaloperationen und Théorie der Normalen Operatoren, Math. Ann. 102 (1929), 370427.
8. Radjavi, H. and Rosenthal, P., On invariant subspaces and reflexive algebras, Amer. J. Math. 91 (1969), 683692.
9. Ringrose, J. R., On some algebras of operators, Proc. London Math. Soc. 15 (1965), 6183.
10. Rosenthal, P., Completely reducible operators, Proc. Amer. Math. Soc. 19 (1968), 826830.
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Generators of Nest Algebras

  • W. E. Longstaff (a1)

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