Abstract oscillation theorems for multiparameter eigenvalue problems

We prove abstract analogous of Klein's oscillation theorem by demonstrating the existence (and in some cases uniqueness) of eigenpairs with a given index for the multiparameter problem $\text{T}{}_{\mathrm{m}}\text{x}{}_{\mathrm{m}}\text{=}\text{∑}\text{n = 1}\text{k}\text{λ}{}_{\mathrm{n}}\text{V}{}_{\mathrm{mn}}\text{x}{}_{\mathrm{m}}$, 0 ≠ x_m?H_m, m = 1 … k. (1) Here T_m and V_mn are self-adjoint operators on Hilbert spaces H_m. The index is based on the number of negative eigenvalues of T_m ? ∑_{n = 1}^kλ_nV_mn and on the sign of the determinant δ₀ with (m, n)th entry (x_m, V_mnx_m). We assume that certain cofactors of δ₀ are positive, and we complement previous work of Sleeman on Sturm-Liouville systems, and of Binding and Browne on (1) in the case where δ₀ is positive.