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Existence conditions for eigenvalue problems generated by compact multiparameter operators
Published online by Cambridge University Press: 14 November 2011
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
References
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