COMPACT AND FINITE RANK PERTURBATIONS OF SELFADJOINT OPERATORS IN KREIN SPACES WITH APPLICATIONS TO

A selfadjoint operator A in a Krein space (K, (ò, ò)) is called de?nitizable if the resolvent set ?(A) is nonempty and there exists a polynomial p such that (p(A)x, x) ? 0 for all x ? dom (p(A)). It was shown in (L1) and (L5) that a de?nitizable operator A has a spectral function EA which is de?ned for all real intervals the boundary points of which do not belong to some ?nite subset of the real axis. With the help of the spectral function the real points of the spectrum ?(A) of A can be classi?ed in points of positive and negative type and critical points: A point ? ? ?(A) ? R is said to be of positive type (negative type) if ? is contained in some open interval ? such that EA(?) is de?ned and (EA(?)K, (ò, ò)) (resp. (EA (?)K, ?(ò, ò))) is a Hilbert space. Spectral points of A which are not of de?nite type, that is, not of positive or negative type, are called critical points. The set of critical points of A is ?nite; every critical point of A is a zero of any polynomial p with the ôde?nitizingö property mentioned above. Spectral points of positive and negative type can also be characterized with the help of approximative eigensequences (see (LcMM), (LMM), (J6)), which allows, in a convenient way, to carry over the sign type classi?cation of spectral points to non-de?nitizable selfadjoint operators and relations in Krein spaces.