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Jan Nesemann studies relatively bounded perturbations of selfadjoint operators in Krein spaces with real spectrum. The main results provide conditions which guarantee the spectrum of the perturbed operator to remain real. Similar results are established for relatively formbounded perturbations and for pseudoFriedrichs extensions. The author pays particular attention to the case when the unperturbed selfadjoint operator has infinitely many spectral gaps, either between eigenvalues or, more generally, between separated parts of the spectrum.
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