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

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