TAILIEUCHUNG - Báo cáo toán học: "Compact perturbations of definitizable operators. II "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Nhiễu loạn nhỏ gọn của nhà khai thác | J. OPERATOR THEORY 8 1982 3-18 Copyright by INCREST 1982 COMPACT PERTURBATIONS OF DEFIN1TIZABLE OPERATORS. Il p. JONAS The aim of the present note is to continue the study of perturbations of defi-nitizable J-selfadjoint operators in a Krein space from 10 . In this paper we assume that the unperturbed operator A is a . -positive J-selfadjoint operator similar to a sclfadjoint operator with respect to some positive definite scalar product . We first consider in Theorem 1 perturbations which produce . -positive J-selfadjoint operators again. These perturbations are assumed to have in a certain sense a small size with respect to A see . Theorem 1 extends a result of K. Veselió 15 for A having a bounded inverse. On the one hand in our context Theorem 1 is a tool for the investigation of perturbations which in some sense are compact. On the other hand it is of interest on its own right. It includes conditions for the preservation of regularity of the critical points and hence criteria for the perturbed operator B to be similar to a selfadjoint operator. Theorem 2 which is a consequence of Theorem 1 specifies a class of -positive perturbations which preserve the similarity to a selfadjoint operator. In Theorem 3 we consider a class of perturbations which produce definitizable -selfadjoint operators. These perturbations are defined by some compactness properties with respect to the unperturbed operator A. Though for a general operator A not every finite-dimensional operator belongs to the perturbations considered here this class is of interest for another reason There is a simple criterion for the regularity of the critical point 0 of the perturbed operator. For the perturbations considered here the difference of the resolvents of A and B B the perturbed J-self-adjoint operator p Ạ 0 is compact in a common point of their resolvent sets. We remark that this property by a result of 10 is not sufficient for B to be defi-nitizable. Roughly speaking Theorem 3 is giving a

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