TAILIEUCHUNG - Báo cáo toán học: "The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. II "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Commutativity Fuglede lý modulo lớp Hilbert Schmidt và các chức năng tạo ra cho các nhà khai thác ma trận. II. | Copyright by INCREST 1981 J. OPERATOR THEORY 5 1981 3 16 THE FUGLEDE COMMUTATIVITY THEOREM MODULO THE HILBERT-SCHMIDT CLASS AND GENERATING FUNCTIONS FOR MATRIX OPERATORS II GARY WEISS Let Xd denote a separable complex Hilbert space and let denote the class of all bounded linear operators acting on XP. Let X F denote the class of compact operators in and let Cp denote the Schatten p-class 0 p oo with II lip 1 p co denoting the associated p-norm. Hence c2 is the Hilbert-Schmidt class and Cj is the trace class. In 5 we pointed out connections between a problem of I. D. Berg 1 namely Is every normal operator the sum of a diagonalizable operator and a Hilbert-Schmidt operator and several statements regarding normal operators Hilbert-Schmidt operators and trace class operators. Some of these statements were proven and some were left open questions. Here we settle the main question 5 statement 3 and obtain a generalization and we ask several new questions. Theorem 1. If N-Ị N2 are normal operators and X is a bounded operator then NjX XN2 c _ N X XN c2. In particular NxX XN2 e c2 implies NỊX XNỊ e c2. We give a proof of this theorem which blends two earlier proofs. The first proof used generating functions and a kind of distribution theory. The second proof was entirely operator theoretic. The first proof was the original proof and suggests certain methods and generalizations. The second proof was a more recent proof that the author constructed from the first proof at the urging of Dan Voiculescu. It was felt that an operator theoretic proof was important. In 5 Theorem 2c we proved that to prove Theorem 1 it suffices to assume Nỵ N2 Mv the operator of multiplication by p where p e L T Mv acts on L2 T and for every complex number c m z p z c 0. Proof. In 5 The Main Construction we defined the generating function for the matrix operator X x j to be the formal Fourier series F z w 4 GARY WEISS co Xij-z w7. In other words the entries of the matrix operator are precisely I -00

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