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The boundary of the numerical range

Published online by Cambridge University Press:  18 May 2009

G. de Barra
G. de Barra
Affiliation:
Department of Mathematics, Royal Holloway College, Eghanm Surrey, U.K.
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In [1] it was shown that for a compact normal operator on a Hilbert space the numerical range was the convex hull of the point spectrum. Here it is shown that the same holds for a semi-normal operator whose point spectrum satisfies a density condition (Theorem 1). In Theorem 2 a similar condition is shown to imply that the numerical range of a semi-normal operator is closed. Some examples are given to indicate that the condition in Theorem 1 cannot be relaxed too much.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 22 , Issue 1 , January 1981 , pp. 69 - 72
Copyright
Copyright © Glasgow Mathematical Journal Trust 1981

References

REFERENCES

1.de Barra, G., Giles, J. R. and Sims, B., On the numerical range of operators, J. London Math. Soc. (2) 5 (1972), 704–706.CrossRefGoogle Scholar
2.Embry, M. R., The numerical range of an operator, Pacific J. Math. 32 (1970), 647–650.CrossRefGoogle Scholar
3.Embry, M. R., Orthogonality and the numerical range, J. Math. Soc. Japan 27 (3) (1975), 405–411.CrossRefGoogle Scholar
4.MacCluer, C. R., On extreme points of the numerical range of normal operators, Proc. Amer. Math. Soc. 16 (1965), 1183–1184.CrossRefGoogle Scholar
5.Meng, Ching-Hwa, A condition that a normal operator have a closed numerical range, Proc. Amer. Math. Soc. 8 (1957), 85–88.CrossRefGoogle Scholar
6.Putnam, C. R., On the spectra of semi-normal operators, Trans. Amer. Math. Soc. 119 (1965), 509–523.CrossRefGoogle Scholar
7.Stampfli, J. G., Hyponormal operators, Pacific J. Math. 12 (1962), 1453–1458.CrossRefGoogle Scholar
8.Stampfli, J. G., Hyponormal operators and spectral density, Trans. Amer. Math. Soc. 117 (1965), 469–476.CrossRefGoogle Scholar
9.Stampfli, J. G., Extreme points of the numerical range of a hyponormal operator, Michigan Math. J. 13 (1966). 87–89.CrossRefGoogle Scholar