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A characterisation and two examples of Riesz operators

Published online by Cambridge University Press:  18 May 2009

T. A. Gillespie  and
T. T. West
T. A. Gillespie
Affiliation:
Yale University and Edinburgh UniversityTrinity College, Dublin
T. T. West
Affiliation:
Yale University and Edinburgh UniversityTrinity College, Dublin
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A Riesz operator is a bounded linear operator on a Banach space which possesses a Riesz spectral theory. These operators have been studied in [5] and [6]. In §2 of this paper we characterise Riesz operators in terms of their resolvent operators. In [6] it was shown that every Riesz operator on a Hilbert space can be decomposed into the sum of compact and quasi-nilpotent parts. §3 contains an example to show that these parts cannot, in general, be chosen to commute. In §4 the eigenset of a Riesz operator is defined. It is a sequence of quadruples each of which consists of an eigenvalue, the corresponding spectral projection, index and nilpotent part. This sequence satisfies certain obvious conditions, and the question arises of the existence of a Riesz operator which has such a sequence as its eigenset. We give an example of an eigenset which has no corresponding Riesz operator.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 9 , Issue 2 , July 1968 , pp. 106 - 110
Copyright
Copyright © Glasgow Mathematical Journal Trust 1968

References

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