Generalized Fuglede-Putnam Theorem and $m$-quasi-class $A(k)$ operators

For a bounded linear operator $T$ acting on acomplex infinite dimensional Hilbert space $\h,$ we say that $T$is $m$-quasi-class $A(k)$ operator for $k>0$ and $m$ is apositive integer (abbreviation $T\in\QAkm$) if$T^{*m}\left((T^*|T|^{2k}T)^{\frac{1}{k+1}}-|T|^2\right)T^m\geq0.$ The famous {\it Fuglede-Putnam theorem} asserts that: the operator equation$AX=XB$ implies $A^*X=XB^*$ when $A$ and $B$ are normal operators.In this paper, we prove that if $T\in \QAkm$ and $S^*$ isan operator of class $A(k)$ for $k>0$. Then $TX=XS$, where $X\in\bh$ is an injective with dense range implies $XT^*=S^*X$.