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 $\tilde{A}$ be a linear operator, such that $\Vert (A-\tilde{A})A^{-\nu }\Vert <\infty$ $(0<\nu \le 1)$. It is assumed that $A$ has a simple eigenvalue. Under certain conditions $\tilde{A}$ also has a simple eigenvalue. We derive an estimate for $\Vert e(A)-e(\tilde{A})\Vert$, where $e(A)$ and $e(\tilde{A})$ are the normalized eigenvectors corresponding to these simple eigenvalues of $A$ and $\tilde{A}$, respectively. Besides, the perturbed operator $\tilde{A}$ 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.