A Note on Perturbations of Selfadjoint Operators in Krein Spaces

Abstract

Let H be a separable Krein space, A a definitizable selfadjoint operator in H and B a selfadjoint operator in H with nonempty resolvent set. Let the difference of the resolvents of A and B belong to some Schatten-von Neumann ideal S_p, 1 ≤p < ∞, of compact operators in H ([3]). If, in addition, A is fundamentally reducible ([1]), then B possesses a spectral function with singularities and the set S of the spectral singularities of B has no more than a finite number of accumulation points. More precisely, the set S’ of the accumulation points of S ∪ (σ(B) \ R) is contained in the union of the set of critical points and the nonreal spectrum of A. For any interval [a,b] with a,b ∉ S and [a,b] n S’ = Ø, the restriction of B to the spectral subspace corresponding to [a,b] is a definitizable operator. In [5] such an operator B was called definitizable over R̄\ S’ (see the definition given below). For bounded operators this result was proved by H. Langer in [7] (even for the Macaev ideal S _ω instead of S _p), for its generalization to unbounded operators see [5].