tailieunhanh - Báo cáo toán học: "Some results on norm-ideal perturbations of Hilbert space operators. II "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một số kết quả trên nhiễu loạn tiêu chuẩn lý tưởng của các nhà khai thác không gian Hilbert. II. | J. OPERATOR THEORY 5 1981 77-100 Copyright by INCREST 1981 SOME RESULTS ON NORM-IDEAL PERTURBATIONS OF HILBERT SPACE OPERATORS. II DAN VOICULESCU The present paper is a continuation of the stud we began in 10 of normideal perturbations based on the invariant ci7 t . The results we obtain here when applied to A perturbations of commuting n-tuples of hermitian operators yield for n 2 the invariance up to unitary equivalence of the absolutely continuous part and for 77 3 an existence theorem for wave-operators. Let US mention that our result on wave-operators improves the result of J. Voigt 11 . Namely the ideal 7 is strictly bigger than the union of the ideals with p n and the wave-operators we consider are more general. Moreover we show that for 7 3 all the wave-operators are equal. Our approach to these questions is based on the following considerations. For a commuting 7-tuple of hermitian operators the singular part can be characterized as the greatest invariant subspace such that the restriction of the 77-tuple admits a quasicentral approximate unit relative to the ideal cê . This leads US to consider a generalization in this sense of the decomposition into singular and absolutely continuous part to the framework of representations of c -algebras and arbitrary normideals Under certain conditions the non-vanishing of k Ịi on the analogue of the absolutely continuous part implies that sequences of elements of Gí which commute asymptotically with the algebra considered must converge strongly to zero which is the source for our results on the existence of wave-operators. The present paper has three sections. In 1 we give the generalization of the decomposition into singular and absolutely continuous part and prove three theorems which represent our results on the invariance of the generalized absolutely continuous part and the existence of waveoperators. In 2 we apply the results of 1 to commuting 77-tuples of hermitian operators. Besides that we show how the .