TAILIEUCHUNG - Báo cáo toán học: "Economical compact perturbations. I: Erasing normal eigenvalues "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Kinh tế nhỏ gọn nhiễu loạn. I: Tẩy xoá giá trị riêng bình thường. | Copyright by INCREST 1983 . o PERATOR THEORY 10 1983 289-306 ECONOMICAL COMPACT PERTURBATIONS. I ERASING NORMAL EIGENVALUES DOMINGO A. HERRERO 1. INTRODUCTION Several recent works on compact perturbations of Hilbert space operators deal with problems of the following type Given an operator T acting on a complex separable infinite dimensional Hilbert space and a certain property P there exists a compact operator K such that T K satisfies P . In certain cases . 1 11 14 it is shown that given 0 K can be chosen so that AT . In other cases . 1 13 the authors obtain a mere existence result with either no control at all or only a very rough estimate on the value of  . In the first part of this article it will be obtained the most economical value of X for a particular perturbation problem considered in 13 We shall need some notation. In what follows ĨẾựC will denote the algebra of all bounded linear operators acting on and will denote the ideal of all compact operators. Given T in .J ơ T and ffe T will denote the spectrum and the essential spectrum of T . the spectrum of the canonical image f T 4- Jf of T in the quotient Calkin algebra sđụ respectively . An isolated point Ằ 6 ơ T is a normal eigenvalue of T if the Riesz spectral dempotent associated to the clopen subset 2 of ơ T via functional calculus is a finite rank operator. This is equivalent to saying that Ằ is an isolated point of 7 T ne T . In 13 Lemma 6 J. G. Stampfli proved that given T in f J there exists AT in 3f .3i such that ơữ T K 0 where ơo R denotes the set of all normal eigenvalues of the operator R furthermore K can be chosen so that IIA II max dist Ấ ơe T Ằ e ơoíT isee Proposition below . But this estimate of the norm of the compact perturbation necessary to erase the normal eigenvalues of T is very crude in all cases. 290 DOMINGO A. HERRERO The result of 4 Proposition see also 7 Section suggests that one-half of the above value plus e always suffices and this is indeed the case.

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