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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
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
Information
Glasgow Mathematical Journal , Volume 28 , Issue 2 , July 1986 , pp. 215 - 222
Copyright
Copyright © Glasgow Mathematical Journal Trust 1986

References

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