Abstract
An operator $T \in B(\mathcal{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's 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 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.
Citation
Xiaochun Fang. Fugen Gao. "The Fuglede-Putnam theorem and Putnam's inequality for quasi-class $(A, k)$operators." Ann. Funct. Anal. 2 (1) 105 - 113, 2011. https://doi.org/10.15352/afa/1399900266
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