Existence and uniqueness of index multiparametric eigenvalues (Q1083672)

The authors study indexed eigenvalues \(\lambda =(\lambda_ 0,...,\lambda_ k)\in {\mathbb{R}}^{k+1}\) for the multiparameter problem \[ (*)\quad W_ m(\lambda)u_ m=0,\quad \| u_ m\| \quad =1,\quad m=1,...,k, \] where \[ W_ m(\lambda)=\lambda_ 0T_ m+\sum^{k}_{n=1}\lambda_ nV_{mn},\quad T_ m=V_{m0}-I_ m \] and the \(V_{mn}\) are compact symmetric operators on complex Hilbert spaces \(H_ m\). Let \(v_{mn}\) be the quadratic form induced by \(V_{mn}\), and let V(u) be the \(k\times k\) matrix with (m,n)th element \(v_{mn}(u_ m)\), \(1\leq m,n\leq k\). The index of \(\lambda\), at least in the case \(\lambda_ 0>0\), depends on the sign of det V(u) for \(u_ m\) in (*), and on the dimensions of maximal subspaces on which the \(W_ m(\lambda)\) are positive definite. Conditions are imposed on the \(v_{mn}\) to ensure existence and uniqueness of eigenvalues of specified norm and index. In particular, the known results for the ''left'' and ''right'' definiteness conditions in the literature, whether ''strong'' or not, are extended in a unified fashion. Meanwhile some additional results have been obtained by H. Volkmer in his habilitation thesis.