On Fuglede’s theorem and operator topologies

Rogers, Donald D.

doi:10.1090/S0002-9939-1979-0529207-9

On Fuglede's theorem and operator topologies

Author: Donald D. Rogers
Journal: Proc. Amer. Math. Soc. 75 (1979), 32-36
MSC: Primary 47B15
DOI: https://doi.org/10.1090/S0002-9939-1979-0529207-9
MathSciNet review: 529207
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Abstract: For each normal operator N and neighborhood E of 0 in the strong or weak operator topology there is a neighborhood D of 0 in the same topology such that $\left\Vert B \right\Vert \leqslant 1$ and in D imply ${N^\ast}B - B{N^\ast}$ in E. An example is given to show that D depends on N as well as E.

[1] E. Bishop, Spectral theory for operators on a Banach space, Trans. Amer. Math. Soc. 86 (1957), 414-445. MR 0100789 (20:7217)
[2] B. Fuglede, A commutativity theorem for normal operators, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 35-40. MR 0032944 (11:371c)
[3] P. R. Halmos, A Hilbert space problem book, Van Nostrand, Princeton, N. J., 1967. MR 0208368 (34:8178)
[4] E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R. I., 1957. MR 0089373 (19:664d)
[5] R. L. Moore, An asymptotic Fuglede theorem, Proc. Amer. Math. Soc. 50 (1975), 138-142. MR 0370247 (51:6474)
[6] M. Rosenblum, On a theorem of Fuglede and Putnam, J. London Math. Soc. 33 (1958), 376-377. MR 0099598 (20:6037)
[7] A. L. Shields, Weighted shift operators and analytic function theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R. I., 1974, pp. 49-128. MR 0361899 (50:14341)