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Existence conditions for eigenvalue problems generated by compact multiparameter operators

Published online by Cambridge University Press:  14 November 2011

Paul Binding ,
Patrick J. Browne  and
Lawrence Turyn
Paul Binding
Affiliation:
Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, CanadaT2N 1N4
Patrick J. Browne
Affiliation:
Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, CanadaT2N 1N4
Lawrence Turyn
Affiliation:
Department of Mathematics, Wright State University, Dayton, Ohio 45435, U.S.A.
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Synopsis

Let T, V1,…, Vk denote compact symmetric linear operators on a separable Hilbert space H, and write W(λ) = T + λ1V1 + … + λkVk, λ = (λ1, …, λk) ϵ ℝk. We study conditions on the cone

related to solubility of the multiparameter eigenvalue problem

with W(λ)−I nonpositive definite. The main result is as follows.

Theorem. If 0 ∉ V, then (*) is soluble for any T. If 0 ∈ V, then there exists T such that (*) is insoluble.

We also deduce analogous results for problems involving self-adjoint operators with compact resolvent.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 96 , Issue 3-4 , 1984 , pp. 261 - 274
Copyright
Copyright © Royal Society of Edinburgh 1984

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References

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