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

Abstract: In this paper we characterize (k)-quasi (n)-power posinormal composition operators and weighted composition operators on the Hilbert space (L^2(\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.