Communications of the
Korean Mathematical Society CKMS

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

Keywords: $k$-quasi $(\lowercase{m},\lowercase{n})$-paranormal operator, $k$-quasi $ (\lowercase{m},\lowercase{n})$-class~ $\mathcal{Q}$ operator, $k$-quasi $(\lowercase{m},\lowercase{n})$-class~ $\mathcal{Q^{*}}$ operator, composition operators

MSC numbers: 47B20, 47B38

Supported by: The third author is supported by seed money project grant UO.No. 11874/2021/Admn, University of Calicut.