$k$-quasi $n$-power posinormal Weighted Composition and Cauchy Dual of Moore-Penrose inverse of Lambert Operators

In this paper we characterize $k$-quasi $n$-power posinormal composition operators and weighted composition operators on the Hilbert space $L^2(\mathrm{\Sigma })$. For Lambert conditional operators (of the form $T=M_{w}E{M}_u$), we establish necessary and sufficient conditions under which these Cauchy duals via the Moore-Penrose inverse become $k$-quasi $n$-power posinormal operators. Finally, we construct an explicit example of a $k$-quasi $n$-power posinormal weighted shift operator on a rooted directed tree.