Existence conditions for eigenvalue problems generated by compact multiparameter operators

Abstract

SynopsisLet 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 conerelated to solubility of the multiparameter eigenvalue problemwith 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.