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Spectral operators and weakly compact homomorphisms in a class of Banach Spaces

Published online by Cambridge University Press:  18 May 2009

W. Ricker
Affiliation:
School of Mathematics and Physics, Macquarie University, North Ryde, 2113, Australia
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The purpose of this note is to present certain aspects of the theory of spectral operators in Grothendieck spaces with the Dunford-Pettis property, briefly, GDP-spaces, thereby elaborating on the recent note [10].

For example, the sum and product of commuting spectral operators in such spaces are again spectral operators (cf. Proposition 2.1) and a continuous linear operator is spectral if and only if it has finite spectrum (cf. Proposition 2.2). Accordingly, if a spectral operator is of finite type, then its spectrum consists entirely of eigenvalues. Furthermore, it turns out that there are no unbounded spectral operators in such spaces (cf. Proposition 2.4). As a simple application of these results we are able to determine which multiplication operators in certain function spaces are spectral operators.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1986

References

REFERENCES

1.Berkson, E. and Dowson, H. R., On reflexive scalar-type operators, J. London Math. Soc. (2), 8 (1974), 652656.CrossRefGoogle Scholar
2.Diestel, J. and Uhl, J. J. Jr, Vector measures (Math Surveys No. 15, Amer. Math. Soc, Providence, R.I., 1977).CrossRefGoogle Scholar
3.Dunford, N. and Schwartz, J., Linear operators III: Spectral operators (Wiley-Interscience, 1971).Google Scholar
4.Fixman, U., Problems in spectral operators, Pacific J. Math., 9 (1959), 10291051.CrossRefGoogle Scholar
5.Foguel, S. R., The relations between a spectral operator and its scalar part, Pacific J. Math., 8 (1958), 5165.CrossRefGoogle Scholar
6.Gillespie, T. A., Logarithms of L p translations, Indiana University Math. J., 24 (1975), 10371045.CrossRefGoogle Scholar
7.Hille, E. and Phillips, R. S., Functional analysis and semigroups (Amer. Math. Soc. Colloq. Publ. Vol. 31, Amer. Math. Soc, Providence, R.I., 1957).Google Scholar
8.Kluvanek, I. and Kováříková, M., Product of spectral measures, Czechoslovak Math. J., 17 (92) (1967), 248255.CrossRefGoogle Scholar
9.Lotz, H. P., Tauberian theorems for operators on L and similar spaces, Functional analysis: Surveys and recent results III, Bierstedt, K. D. and Fuchssteiner, B. (eds.), North HollandGoogle Scholar
10.Ricker, W., Spectral operators of scalar-type in Grothendieck spaces with the Dunford–Pettis property, Bull. London Math. Soc. 17 (1985), 268270.CrossRefGoogle Scholar
11.Schaefer, H. H., Topological vector spaces (Springer-Verlag, 1971).CrossRefGoogle Scholar
12.Shuchat, A., Vector measures and scalar operators in locally convex spaces, Michigan Math. J., 24 (1977), 303310.CrossRefGoogle Scholar
13.Shuchat, A., Spectral measures and homomorphisms, Rev. Roumaine Math. Pures Appl., 23 (1978), 939945.Google Scholar