We study the spectrum of a non-negative operator A in a Krein space
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?