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Zbl 1219.47036
Gao, Fugen; Fang, Xiaochun
(Gao, Fu-gen; Fang, Xiao-chun)
The Fuglede-Putnam theorem and Putnam's inequality for quasi-class (A, k) operators. (English)
[J] Ann. Funct. Anal. AFA 2, No. 1, 105-113, electronic only (2011). ISSN 2008-8752/e

Summary: An operator $T\in B(H)$ is called quasi-class $(A,k)$ if $T^{\ast k}(|T^2| - |T^2|)T^k \geq 0$ for a positive integer $k$, which is a common generalization of class A. The famous Fuglede-Putnam theorem is as follows: the operator equation $AX = XB$ implies $A^\ast X = XB^\ast$ 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 $B^\ast$ is an invertible class A operator such that $AX = XB$, then $A^\ast X = XB^\ast$. Secondly, we consider 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.

MSC 2000:: *47B20 Subnormal operators, etc.
47A63 Operator inequalities, etc.

Keywords: Fuglede-Putnam's theorem; quasi-class $(A; k)$ operators; Putnam's inequality; quasisimilar

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