TAILIEUCHUNG - Báo cáo toán học: "Preservation of essential matrix ranges by compact perturbations "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Bảo quản của phạm vi ma trận cần thiết bởi nhiễu loạn nhỏ gọn. | Copyright by ỈNUREST 1982 J. OPERATOR THEORY 8 1982 299-317 PRESERVATION OF ESSENTIAL MATRIX RANGES BY COMPACT PERTURBATIONS VERN I. PAULSEN 1. INTRODUCTION In 12 it was shown that if T is a bounded operator on a separable Hilbert space then for any positive integer II there exists a compact operator K such that the first n matrix ranges of T K are the same as the first n essential matrix ranges of T. They then asked if for every T there exists a compact operator K such that all the matrix ranges of T K are the same as all the essential matrix ranges of T. In this paper we obtain several partial results concerning this question. We prove that to answer the above question it is sufficient to assume that T is block-diagonal Corollary . For a block-diagonal operator we give a complete characterization of whether of not such a compact perturbation exists in terms of a distance formula Theorem . These results allow US to give a somewhat different proof of the result of 12 referred to above Theorem . For a general operator we attempt to show that the existence of such a compact operator is completely determined by the essential matrix ranges of the operator. We find a formula involving the essential matrix ranges which implies the existence of the desired compact Theorem . In the converse direction we only succeed in showing that if the essential matrix ranges of some operator fail to satisfy this formula then there exists an operator for which no such compact perturbation can be found Theorem . Finally we show that the original question can be re-formulated strictly in terms of sets of finite matrices Corollary . 2. BLOCK DIAGONALIZATIONS Let . Í denote a separable complex Hilbert space the bounded linear operators on the ideal of compact operators and Q f the quotient Calkin algebra. For T in .iff. we shall use T to denote its image in Q JỌ We shall use Mn to denote the c -algebra of n X n matrices. 3C0 VERN I. PAULSEN If H and Ỗ9 are C .

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