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On the completeness of eigenvectors of right definite multiparameter problems

Published online by Cambridge University Press:  14 November 2011

Hans Volkmer
Hans Volkmer
Affiliation:
FB 6-Mathematik, Universität Essen GHS, Universitätsstr. 3, D-4300 Essen 1, BRD
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Synopsis

We prove that there exists a complete system of eigenvectors of the eigenvalue problem

for self-adjoint operators Tr and Vrs on separable Hilbert spaces Hr. It is assumed that

(i) the operators Tr have discrete spectrum;

(ii) the operators Vrs are bounded and commute for each r;

(iii) the operators Vrs have the definite sign factor property.

This theorem generalizes and improves a result of Cordes for two-parameter problems. The proof of the theorem depends on an approximation of the given eigenvalue problem by simpler problems, a technique which is related to Atkinson's proof of his expansion theorem.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 96 , Issue 1-2 , 1984 , pp. 69 - 78
Copyright
Copyright © Royal Society of Edinburgh 1984

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References

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