Rotations of eigenvectors under unbounded perturbations (Q528816)

Summary: 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 \(\|(A-\widetilde{A})A^{-\nu}\| < \infty\) \((0< \nu \leq 1)\). 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.