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Báo cáo toán học: "Perturbation theory for definitizable operators in Krein spaces "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Lý thuyết nhiễu loạn cho các nhà khai thác definitizable trong không gian Krein | J. OPERATOR THEORY 9 1983 297 - 317 g Copyright by inCrest 1983 PERTURBATION THEORY FOR DEF1NITIZABLE OPERATORS IN KREIN SPACES H. LANGER and B. NAJMAN Definitizable selfadjoint operators in Krein spaces have a spectral function with possibly certain critical points. Besides the real spectrum they can also have a finite number of nonreal eigenvalues see 1 3 17 . It is the aim of this note to generalize some classical results on continuous perturbations of selfadjoint operators in Hilbert space in particular the Theorem of F. Rellich about the strong convergence of the spectral functions and the Theorem of H. F. Trotter T. Kato about the strong convergence of the corresponding unitary groups see 5 12 17 to the case of definitizable selfadjoint operators in Krein spaces. 2 contains some estimates for the resolvents of definitizable operators which are the main tools for the proofs in 3. The generalizations of F. Rellich s Theorem are given in 3. We first consider a sequence of definitizable selfadjoint operators An in a Krein space with definitiz-ing polynomials of uniformly bounded degrees which converges strongly in resolvent sense to a selfadjoint operator A. The Krein space structure that is the indefinite scalar product is also allowed to depend on n. In Theorem 3.1 we give conditions which ensure that for certain complex sets A the spectral projections E A of A converge strongly to the spectral projection E A of A. Subsequently Theorem 3.4 these results are specialized for the case of Pontrjagin spaces of fixed index X each selfadjoint operator in such a space has a definitizing polynomial of degree 2x . In some situations the spectral projections which correspond to nonreal eigenvalues converge even in norm while the operators themselves are only supposed to converge strongly see Remark 4 after Theorem 3.4 . In 4 we prove the above mentioned generalization of the Trotter Kato theorem under the additional stipulation that the definitizing polynomials are of .