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On commutative non-self-adjoint operator algebras

Published online by Cambridge University Press:  18 May 2009

R. H. Kelly
R. H. Kelly
Affiliation:
University of Glasgow, Glasgow G12 8QQ
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A proof is given here of a theorem of Sarason [9, Theorem 2], the proof being valid in an arbitrary (non-separable) complex Hilbert space. Sarason's proof uses a theorem and lemma of Wermer which may both fail when the separability hypothesis is omitted [3]. By using a special case of Sarason's theorem and another result of Sarason [10, Lemma 1] a simplified and shortened proof is given of a result of Scroggs [11, Corollary 1].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 15 , Issue 1 , March 1974 , pp. 54 - 59
Copyright
Copyright © Glasgow Mathematical Journal Trust 1974

References

REFERENCES

1.Bade, W. G., On Boolean algebras of projections and algebras of operators, Trans. Amer. Math. Soc. 80 (1955), 345–367.CrossRefGoogle Scholar
2.Banach, S., Operations linéaires (New York, 1955).Google Scholar
3.Dowson, H. R. and Moeti, G. L. R., Property (P) for normal operators, Proc. Roy. Irish Acad. Sect A, 73 (1973), 159–167.Google Scholar
4.Dunford, N. and Schwartz, J. T., Linear operators, Part 3 (New York, 1971).Google Scholar
5.Halmos, P. R., Introduction to Hilbert space and the theory of spectral multiplicity (New York, 1951).Google Scholar
6.Radjavi, H. and Rosenthal, P., On invariant subspaces and reflexive algebras, Amer. J. Math. 91 (1969), 683–692.CrossRefGoogle Scholar
7.Ringrose, J. R., Lecture notes on von Neumann algebras, University of Newcastle upon Tyne (1966–67).Google Scholar
8.Rudin, W. R., Real and complex analysis (New York, 1966).Google Scholar
9.Sarason, D., Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511–517.CrossRefGoogle Scholar
10.Sarason, D., Weak-star density of polynomials, J. Reine Angew. Math. 252 (1972), 1–15.Google Scholar
11.Scroggs, J. E., Invariant subspaces of a normal operator, Duke Math. J. 26 (1959), 95–111.CrossRefGoogle Scholar