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

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. | J. OPERATOR THEORY 2 1979 3 37 Copyright by INCREST 1979 SOME RESULTS ON NORM-IDEAL PERTURBATIONS OF HILBERT SPACE OPERATORS DAN VOICULESCU The starting point for the present paper was a problem attributed in 4 to p. R. Halmos concerning Hilbert-Schmidt perturbations of normal operators for which we provide an affirmative answer every normal operator on a separable Hilbert space is a Hilbert-Schmidt perturbation of a diagonal normal operator. In fact we prove more namely that n-tuples of commuting hermitian operators for n 2 are .-perturbations of diagonal n-tuples of commuting hermitian operators. Thus for n-tuples of commuting hermitian operators with 2 the normideal cển is not the right analogue of the trace-class for n 1 where by a corollary of the Kato-Rosenblum theorem trace-class perturbations conserve up to unitary equivalence the absolutely continuous part. We exhibit in the present paper for each 02 a norm-ideal so that p 7when p n and ển and which seems to be the right replacement of the trace-class forn 2. We prove that for n 2 a n-tuple of commuting hermitian operators can be diagonalized after a . -perturbation if and only if its spectral measure is singular with respect to Lebes-gue-measure. Moreover under the additional assumption that the multiplicity function of the absolutely continuous part is integrable we prove that up to unitary equivalence the ab solutely continuous part is invariant with respect to -perturbations. This improves a part of the results of J. Voigt 12 concerning p-perturbations for p n and n 3. The method used to obtain these results grew out from the remark that the proof of the author s non-commutative Weyl-von Neumann type theorem 14 see also 2 can be adapted for norm-ideal perturbations other than compact provided there are quasicentral approximate units for which the almost - commutation property is satisfied in the norm of the given ideal. This reduces the diagonalization problem modulo a given norm-ideal for a n-tuple of