THE FUGLEDE–PUTNAM THEOREM AND PUTNAM’S INEQUALITY FOR QUASI-CLASS (A, k) OPERATORS

An operator T ∈ B(H) is called quasi-class (A, k) if T ∗k(|T | − |T |)T k ≥ 0 for a positive integer k, which is a common generalization of class A. The famous Fuglede–Putnam’s theorem is as follows: the operator equation AX = XB implies A∗X = XB∗ when A and B are normal operators. In this paper, firstly we show that if X is a Hilbert-Schmidt operator, A is a quasi-class (A, k) operator and B∗ is an invertible class A operator such that AX = XB, then A∗X = XB∗. Secondly we consider the Putnam’s inequality for quasi-class (A, k) operators and we also show that quasisimilar quasi-class (A, k) operators have equal spectrum and essential spectrum. 1 College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, China. E-mail address: [email protected] 2 Department of Mathematics, Tongji University, Shanghai, 200092, China. E-mail address: [email protected] Date: Received: 28 January 2011; Revised: 27 April 2011; Accepted: 5 May 2011. ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary 47A63; Secondary 47B20.