The Fuglede-Putnam theorem and Putnam's inequality for‎ ‎quasi-class $(A‎, ‎k)$operators

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.