On the class of $k$-quasi-$(n,m)$-power normal operators

We introduce a family of operators called the $k$-quasi-$(n,m)$-power normal operators. Such includes normal, $n$-normal and $(n,m)$-power An operator $T \in {\mathcal B}({\mathcal H})$ is said to be if it satisfies $$T^{*k}\bigg(T^nT^{*m}-T^{*m}T^n\bigg)T^k=0,$$ where $k,n$ $m$ are natural numbers. Firstly, some basic structural properties this established with help special kind matrix representation associated such Secondly, of\linebreak algebraically discussed. Thirdly, we consider study tensor products A necessary sufficient condition for $T\otimes S$ given, when \neq0$ $S\neq0$.