Journal of Spectral Theory


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Volume 7, Issue 1, 2017, pp. 191–199
DOI: 10.4171/JST/159

Published online: 2017-03-22

Rotations of eigenvectors under unbounded perturbations

Michael Gil'[1]

(1) Ben Gurion University, Beer Sheva, Israel

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.

Keywords: Linear operator, eigenvector, perturbation, differential operators

Gil' Michael: Rotations of eigenvectors under unbounded perturbations. J. Spectr. Theory 7 (2017), 191-199. doi: 10.4171/JST/159