Comparison cones for multiparameter eigenvalue problems

We consider the multiparameter eigenvalue problem (T_r + ∑_{s = 1}^kλ_sV_rs) x_r = 0, x_r ≠ 0, 1 ? r ? k, where T_r and V_rs are self-adjoint linear operators on Hilbert spaces H_r, the V_rs being bounded. The problem may be posed in either ⊕_{r = 1}^kH_r or ⊕_{r = 1}^kH_r and we develop variational approaches for both settings. We explore the rôles played in both settings by $\text{C ={λ ∈ R}{}^{\mathrm{k}}\text{|}\text{∑}\text{k}\text{s=1}\text{λ}{}_{\mathrm{s}}\text{(V}{}^{\mathrm{rs}}\text{x}{}_{\mathrm{r}}\text{,x}{}_{\mathrm{r}}\text{? 0}$ for some nonozero and related cones in $\text{R}$^k. We also compare certain geometrical conditions on C with analytical definiteness conditions already in the literature.