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

TL;DR

This paper characterizes when Lambert operators and their Cauchy duals are k-quasi n-power posinormal on Hilbert spaces, providing conditions and explicit examples of such operators.

Contribution

It offers necessary and sufficient conditions for Lambert operators' Cauchy duals to be k-quasi n-power posinormal, including explicit construction of examples.

Findings

Characterization of k-quasi n-power posinormal composition operators

Conditions for Cauchy duals of Lambert operators to be k-quasi n-power posinormal

Explicit example of a k-quasi n-power posinormal weighted shift operator

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.