Rellich's perturbation theorem on hermitian matrices of holomorphic functions

Abstract

The principal axis theorem has a remarkable generalization in the perturbation theory of linear operators. Rellich [S] proved that a symmetric operator A(E) which depends analytically on a real parameter E has an orthonormal basis of eigenvectors depending also analytically on E. In this note we give a short proof of Rellich’s theorem based on the fact that the ring H(Q) of complex functions which are holomorphic in a region 0 is an elementary divisor domain. Let J be a real interval and let A denote the functions which are holomorphic on J, A = U{H(SZ), JcQ}. For W= (wii(z))~AkXn define