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Normal-equivalent operators and operators with dual of scalar-type

Published online by Cambridge University Press:  20 January 2009

M. B. Ghaemi
M. B. Ghaemi
Affiliation:
Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, UK (mbg@maths.gla.ac.uk)
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Abstract

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If T ∈ L(X) is such that T′ is a scalar-type prespectral operator, then Re T′ and Im T′ are both dual operators. It is shown that that the possession of a functional calculus for the continuous functions on the spectrum of T is equivalent to T′ being scalar-type prespectral of class X, thus answering a question of Berkson and Gillespie.

Keywords

hermitiannormalscalar typeprespectraloperatorfunctional calculus
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 2 , June 2000 , pp. 261 - 268
Copyright
Copyright © Edinburgh Mathematical Society 2000

References

1. Behrends, E., Normal operators and multipliers on complex Danach spaces and a symmetry property of L1-predual spaces, Israel J. Math. 47 (1984), 23–28.CrossRefGoogle Scholar
2. Berkson, E. and Dowson, H. R., Prespectral operators, Illinois J. Math. 13 (1969), 291–315.CrossRefGoogle Scholar
3. Berkson, E. and Gillespie, T. A., Absolutely continuous function of two variables and well-bounded operators, J. Lond. Math. Soc. (2) 30 (1984), 305–321.Google Scholar
4. Bonsall, F. F. and Duncan, J., Complete normed algebras (Springer, 1973).CrossRefGoogle Scholar
5. Doust, I. and Delaubenfels, R., Functional calculus, integral representation, and Banach space geometry, Quaestiones Math. 17(2) (1994), 161–171.CrossRefGoogle Scholar
6. Dowson, H. R., Spectral theory of linear operators (Academic Press, 1978).Google Scholar
7. Dowson, H. R., Gillespie, T. A. and Spain, P. G., A commutativity theorem for hermitian operators, Math. Ann. 220 (1976), 215–217.CrossRefGoogle Scholar
8. Dunford, N., Spectral operators. Pacific J. Math. 4 (1954), 321–354.CrossRefGoogle Scholar
9. Kluvánek, I., Characterization of scalar-type spectral operators. Arch. Math. (Brno) 2 (1966), 153–156.Google Scholar
10. Pełczynski, A., Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228.CrossRefGoogle Scholar
11. Spain, P. G., On scalar-type spectral operators, Proc. Camb. Phil. Soc. 69 (1971), 409–410.CrossRefGoogle Scholar