Geodesic
Strong limits of normal operators
Glasgow mathematical journal, Tome 24 (1983) no. 1, pp. 93-96

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Conway, John B.; Hadwin, Donald W. Strong limits of normal operators. Glasgow mathematical journal, Tome 24 (1983) no. 1, pp. 93-96. doi: 10.1017/S0017089500005115
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[1] 1.Bishop, E., Spectral theory for operators on a Banach space, Trans. Amer. Math. Soc. 86 (1957), 414–445. Google Scholar | DOI

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[4] 4.Halmos, P. R., Normal dilations and extensions of operators, Summa Brasil. 2 (1950), 125–134. Google Scholar

[5] 5.Halmos, P. R., A Hilbert space problem book (Van Nostrand, 1967). Google Scholar

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