Communications of the
Korean Mathematical Society CKMS

To study the operator inequalities, the notions of class $A$ operators and quasi-class $A$ operators are developed up to recently. In this paper, quasi-$A(n,k)$ class operator for $n\geq 2$ and $k\geq 0$ is introduced as a new notion, which generalizes the quasi-class $A$ operator. We obtain some structural properties of these operators. Also we characterize quasi-$A(n,k)$ classes for $n$ and $k$ via backward extension of weighted shift operators. Finally, we give a simple example of quasi-$A(n,k)$ operators with two variables.

Keywords: class $A$, quasi-class $A$, weighted shift operator