JournalsjstVol. 7, No. 1pp. 191–199

Rotations of eigenvectors under unbounded perturbations

  • Michael Gil'

    Ben Gurion University, Beer Sheva, Israel
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Let AA be an unbounded selfadjoint positive definite operator with a discrete spectrum in a separable Hilbert space, and A~\widetilde A be a linear operator, such that (AA~)Aν<\|(A-\widetilde A)A^{-\nu}\| < \infty (0<ν1)(0< \nu\le 1). It is assumed that AA has a simple eigenvalue. Under certain conditions A~\widetilde A also has a simple eigenvalue. We derive an estimate for e(A)e(A~)\|e(A)-e(\widetilde A)\|, where e(A)e(A) and e(A~)e(\widetilde A) are the normalized eigenvectors corresponding to these simple eigenvalues of AA and A~\widetilde A, respectively. Besides, the perturbed operator A~\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 AA is a normal operator.

Cite this article

Michael Gil', Rotations of eigenvectors under unbounded perturbations. J. Spectr. Theory 7 (2017), no. 1, pp. 191–199

DOI 10.4171/JST/159