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The Fuglede-Putnam theorem and Putnam’s inequality for quasi-class (A, k) operators. (English) Zbl 1219.47036
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.

47B20Subnormal operators, hyponormal operators, etc.
47A63Operator inequalities
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