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Existence conditions for eigenvalue problems generated by compact multiparameter operators

  • Paul Binding (a1), Patrick J. Browne (a1) and Lawrence Turyn (a2)

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.

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References

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1Binding, P. and Browne, P. J.. Spectral properties of two-parameter eigenvalue problems. Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), 157173.
2Binding, P., Browne, P. J. and Turyn, L.. Existence conditions for two-parameter eigenvalue problems. Proc. Roy. Soc. Edinburgh Sect. A 91 (1981), 1530.
3Carmichael, R. D.. Boundary value and expansion problems; formulation of various transcendental problems. Amer. J. Math. 43 (1921), 232270.
4Pell, A. J.. Linear equations with two parameters. Trans. Amer. Math. Soc. 23 (1922), 198211.
5Rockafellar, R. T.. Convex Analysis (Princeton: Princeton University Press, 1972).
6Tuyrn, L.. Sturm-Liouville problems with several parameters. J. Differential Equations 38 (1980), 239259.

Existence conditions for eigenvalue problems generated by compact multiparameter operators

  • Paul Binding (a1), Patrick J. Browne (a1) and Lawrence Turyn (a2)

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