Non-negative operators in Krein spaces and rank one perturbations

The presented thesis addresses problems in perturbation theory of operators in Krein spaces and is settled in the area of functional analysis. We study the spectrum of a non-negative operator A in a Krein space (K, [·, ·]) under rank one perturbations in resolvent sense. The following two questions are answered: (i) How does the spectral multiplicity in a gap of the essential spectrum of A change under rank one perturbations? (ii) How does the Jordan structure at isolated eigenvalues of A change under rank one perturbations? More precisely, how does the number and the length of Jordan chains of A at a given eigenvalue change under a rank one perturbation? To show these results we use amongst others boundary triplets for symmetric operators in Krein spaces and associated Weyl functions, realisations of generalized Nevanlinna-, D0-, and D1-functions, and algebraic properties of Krein spaces.