Rotations of eigenvectors under unbounded perturbations

Abstract

Let A be an unbounded selfadjoint positive definite operator with a discrete spectrum in a separable Hilbert space, and \widetilde A be a linear operator, such that \|(A-\widetilde A)A^{-\nu}\| < \infty (0< \nu\le 1) . It is assumed that A has a simple eigenvalue. Under certain conditions \widetilde A also has a simple eigenvalue. We derive an estimate for \|e(A)-e(\widetilde A)\| , where e(A) and e(\widetilde A) are the normalized eigenvectors corresponding to these simple eigenvalues of A and \widetilde A , respectively. Besides, the perturbed operator \widetilde A can be non-selfadjoint. To illustrate that estimate we consider a non-selfadjoint differential operator. Our results can be applied in the case when A is a normal operator.