Some classes of operators related to $(\lowercase{m},\lowercase{n})$-paranormal and $(\lowercase{m},\lowercase{n})^*$-paranormal operators

In this paper, we study new classes of operators $k$-quasi $(\lowercase{m},\lowercase{n})$-paranormal operator, $k$-quasi $(\lowercase{m},\lowercase{n})^*$-paranormal operator, $k$-qu\-asi $(\lowercase{m},\lowercase{n})$-class~ $\mathcal{Q}$ operator and $k$-quasi $(\lowercase{m},\lowercase{n})$-class~ $\mathcal{Q^{*}}$ operator which are the generalization of $(\lowercase{m},\lowercase{n})$-paranormal and $(\lowercase{m},\lowercase{n})^*$-paranormal operators. We give matrix characterizations for $k$-quasi $(\lowercase{m},\lowercase{n})$-paranormal and $k$-quasi $(\lowercase{m},\lowercase{n})^*$-paranormal operators. Also we study some properties of $k$-quasi $(\lowercase{m},\lowercase{n})$-class~ $\mathcal{Q}$ operator and $k$-quasi $(\lowercase{m},\lowercase{n})$-class~ $\mathcal{Q}^*$ operators. Moreover, these classes of composition operators on $L^2$ spaces are characterized.