Nonuniform right definiteness

The multiparameter eigenvalue problem W_m(λ) x_m = x_m, $\text{W}{}_{\mathrm{m}}\text{(λ) = T}{}_{\mathrm{m}}\text{+}\text{∑}\text{n = 1}\text{k}\text{λ}{}_{\mathrm{n}}\text{V}{}_{\mathrm{mn}}$, m = 1,…, k, where /gl /gE $\text{C}$^k, x_m is a nonzero element of the separable Hilbert space H_m, and T_m and V_mn are compact symmetric is studied. Various properties, including existence and uniqueness, of λ = λⁱ ? $\text{C}$^k for which the i_mth greatest eigenvalue of W_m(λⁱ) equals one are proved. “Right definiteness” is assumed, which means positivity of the determinant with (m, n)th entry (y_m, V_mny_m) for all nonzero y_m?H_m, m = 1 … k. This gives a “Klein oscillation theorem” for systems of an o.d.e. satisfying a definiteness condition that is usefully weaker than in previous such results. An expansion theorem in terms of the corresponding eigenvectors x_mⁱ is also given, thereby connecting the abstract oscillation theory with a result of Atkinson.