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On power-bounded prespectral operators

Published online by Cambridge University Press:  20 January 2009

H. R. Dowson
H. R. Dowson
Affiliation:
University of Glasgow, Glasgow G12 8QW
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Foguel (8) and Fixman (7) independently proved that an invertible spectral operator, which is power-bounded, is of scalar type. Their proofs rely heavily on a result of Dunford on spectral operators whose resolvents satisfy a growth condition. (See Lemma 3.16 of (6, p. 609).) Observe that the resolvent of an invertible power-bounded operator T satisfies an inequality of the form

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 20 , Issue 2 , September 1976 , pp. 173 - 175
Copyright
Copyright © Edinburgh Mathematical Society 1976

References

REFERENCES

(1) Berkson, E., A characterization of scalar-type operators on reflexive Banach spaces, Pacific J. Math. 13 (1963), 365–373.CrossRefGoogle Scholar
(2) Berkson, E. and Dowson, H. R., Prespectral operators, Illinois J. Math. 13 (1969), 291–315.CrossRefGoogle Scholar
(3) Bonsall, F. F. and Duncan, J., Numerical Ranges II (London Math. Soc. Lecture Note Series No. 10, 1973).CrossRefGoogle Scholar
(4) Dowson, H. R., A commutativity theorem for prespectral operators, Illinois J. Math. 17 (1973), 525–532.CrossRefGoogle Scholar
(5) Dowson, H. R., Logarithms of prespectral operators, J. London Math. Soc. (2) 9 (1974), 57–64.CrossRefGoogle Scholar
(6) Dunford, N., Spectral theory II. Resolutions of the identity, Pacific J. Math. 2 (1952), 559–614.CrossRefGoogle Scholar
(7) Fixman, U., Problems in spectral operators, Pacific J. Math. 9 (1959), 1029–1051.CrossRefGoogle Scholar
(8) Foguel, S. R., The relations between a spectral operator and its scalar part, Pacific J. Math. 8 (1958), 51–65.CrossRefGoogle Scholar
(9) Lumer, G., Spectral operators, hermitian operators and bounded groups, Acta Sci. Math. (Szeged) 25 (1964), 75–85.Google Scholar