This paper considers an operator A, acting on a Krein space, which is J- positive, J-selfadjoint, and similar to a selfadjoint operator (with respect to some positive definite inner product). The notation and definitions of part I [ibid. 2, 63-77 (1979; Zbl 0478.47020)] are used in this paper and are assumed here.
Various norms (depending on A) are introduced on the linear set \(D(A)\cap D(A^{-1})\), and various perturbations of A, characterized in terms of these norms, are considered.
In Theorem 1 it is shown that a [.,.]-symmetric perturbation which is small (norm less than 1) with respect to the appropriate norms produces an operator which is again J-positive and J-selfadjoint. Theorem 1 also includes criteria for preservation of regularity of critical points. If the perturbation is [.,.]-non-negative, then it need only be bounded with respect to the norms, and the perturbed operator will then also be similar to a selfadjoint operator. (This result is included in Theorem 2.)
In Theorem 3 the perturbation is assumed to be [.,.]-symmetric and compact with respect to the appropriate norms. The perturbed operator is then definitizable and J-selfadjoint, and a simple criterion is given for the regularity of the critical point zero of the perturbed operator.