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Strong limits of normal operators

Published online by Cambridge University Press:  18 May 2009

John B. Conway  and
Donald W. Hadwin
John B. Conway
Affiliation:
Indiana University, Bloomington, Indiana 47405, U.S.A.
Donald W. Hadwin
Affiliation:
University of New Hampshire Durham, New Hampshire 03824, U.S.A.
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In [1, Theorem 3.3], E. Bishop proved that an operator S on a Hilbert space ℋ is subnormal if and only if there is a net of normal operators {Nα} that converges to S strongly (that is, ‖(Nα–S) f‖→ 0 for every f in ℋ). The proof that such a net exists if S is subnormal is not so difficult; in fact, a sequence of normal operators converging strongly to S can be found. Bishop's proof of the converse, however, is rather complicated and involves, among other things, some complicated arguments using operator-valued measures. The purpose of this note is to provide an easier proof of this part of the theorem. Our interest in finding such a proof was aroused by Paul Halmos.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 24 , Issue 1 , January 1983 , pp. 93 - 96
Copyright
Copyright © Glasgow Mathematical Journal Trust 1983

References

REFERENCES

1.Bishop, E., Spectral theory for operators on a Banach space, Trans. Amer. Math. Soc. 86 (1957), 414–445.CrossRefGoogle Scholar
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3.Hadwin, D., Completely positive maps and approximate equivalence, preprint.Google Scholar
4.Halmos, P. R., Normal dilations and extensions of operators, Summa Brasil. 2 (1950), 125–134.Google Scholar
5.Halmos, P. R., A Hilbert space problem book (Van Nostrand, 1967).Google Scholar