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.