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Báo cáo toán học: "On the spectral picture of an operator "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên hình ảnh quang phổ của một nhà điều hành. | J. OPERATOR THEORY 4 1980 119-132 Copyright by INCREST 1980 ON THE SPECTRAL PICTURE OF AN OPERATOR BERNARD CHEVREAU 1. INTRODUCTION LetXJ be a separable infinite dimensional complex Hilbert space and let ÍỂựẩ denote the algebra of all bounded linear operators onye. The concept of the spectral picture of an operator introduced by Pearcy in 8 has proved to be useful in various ways. For example one of the main theorems of the Brown-Douglas-Fillmore theory 4 can be stated in terms of spectral pictures thus Two essentially normal operators in ỵụe are compalent if and only if they have the same spectral picture. See 8 for definitions. Also the Romanian characterization of quasitriangular operators 1 can be formulated concisely in terms of spectral pictures An operator in yỵye is quasitriangular if and only if its spectral picture contains no negative number. The purpose of this paper is to make a contribution toward our understanding of the notion of spectral picture. In Section 2 we give a concise exposition of the unpublished result of John Conway to the effect that all spectral pictures are possible. In Section 3 we discuss the behavior of the concept of spectral picture with respect to the relation of quasisimilarity and give a definitive result in an important case. The remainder of the paper Sections 4 and 5 may be considered as an attempt to study the continuity properties of the spectral picture. To make this question precise a metric is introduced in Section 4 on the set of all spectral pictures. The main result of Section 5 Theorem 5.6 can be paraphrased by saying that the map sending an operator to its spectral picture is pathologically discontinuous at every operator in As a by-product of this negative result we show that the norm--closure of the set of all n-cyclic operators has empty interior. We conclude this section by some terminology. For our purposes it will be convenient to use a slightly more formal definition of spectral picture than the one .