An operator $T$ is said to be $k$-quasi class $\mathcal{A}^{\ast}_{n}$ operator if \break $T^{*k}\left( \vert T^{n+1} \vert ^{\frac{2}{n+1}}-\vert T^{*} \vert^{2}\right) T^{k}\geq 0,$ for some positive integers $n$ and $k$. In this paper, we prove that the $k$-quasi class $\mathcal{A}^{\ast}_{n}$ operators have Bishop$^{,}$s property $(\beta)$. Then, we give a necessary and sufficient condition for $T\otimes S$ to be a $k$-quasi class $\mathcal{A}^{\ast}_{n}$ operator, whenever $T$ and $S$ are both non-zero operators. Moreover, it is shown that the Riesz idempotent for a non-zero isolated point $\lambda_{0}$ of a $k$-quasi class $\mathcal{A}^{\ast}_{n}$ operator $T$ say $\mathcal{R}_i$, is self-adjoint and $ran(\mathcal{R}_i)=ker(T-\lambda_{0})=ker(T-\lambda_{0})^{*}$. Finally, as an application in the last section, a necessary and sufficient condition is given in such a way that the weighted conditional type operators on $L^{2}(\Sigma)$, defined by $T_{w,u}(f):= w E(uf)$, belong to $k$-quasi- $\mathcal{A}^{\ast}_{n}$ class.
