On the completeness of eigenvectors of right definite multiparameter problems

Abstract

SynopsisWe prove that there exists a complete system of eigenvectors of the eigenvalue problemfor self-adjoint operators Tr and Vrs on separable Hilbert spaces Hr. It is assumed that(i) the operators Tr have discrete spectrum;(ii) the operators Vrs are bounded and commute for each r;(iii) the operators Vrs have the definite sign factor property.This theorem generalizes and improves a result of Cordes for two-parameter problems. The proof of the theorem depends on an approximation of the given eigenvalue problem by simpler problems, a technique which is related to Atkinson's proof of his expansion theorem.