Root vectors for geometrically simple multiparameter eigenvalues (Q1880350)

An \(n\)-parameter system (\(n\geq 2\)) of the form \(\lambda\) \[ W_i(\lambda)=\sum_{j=0}^n A_{ij}\lambda_j,\quad i\in N, \] is considered. Here \(A_{ij}\) are bounded linear operators acting on a separable Hilbert space \(H_i\) over the complex numbers, \(A_{i0}\) are closed densely defined operators with domain in \(H_i\), \(\lambda =(\lambda_0,\dots,\lambda_n)\) are parameters, \(i\in N\), and \(N = \{1,\dots,n\}\). A class of multiparameter eigenvalue problems \[ W_i(\lambda)x_i=0,\quad x_i\neq0,\quad\lambda\in{\mathbb C}\setminus\{0\} \] is studied. Bases for lower order root subspaces, at geometrically simple eigenvalues of Fredholm type of arbitrary finite index, are computed in terms of the underlying multiparameter system.