Dual variational approaches to multiparameter eigenvalue problems

Abstract

We study the abstract multiparameter problem T m x m = ∑ k n=1 λ n V mn x m , 0 ≠ x m ϵ H m , m = 1,…, k, for selfadjoint operators T m and V mn on separable Hilbert spaces H m . We develop dual variational approaches related to the polar duality between certain convex sets. The “primal” approach has been the basis of several investigations, particularly in the so-called “right definite” case. Here we use the “dual” approach to derive existence and comparison results for the “left definite” case.