<p>In this paper we characterize <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k\)</EquationSource> </InlineEquation>-quasi <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation>-power posinormal composition operators and weighted composition operators on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2(\mu )\)</EquationSource> </InlineEquation> space. Also, we study <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(k\)</EquationSource> </InlineEquation>-quasi <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation>-power posinormal operators in the view point of Cauchy dual of Lambert conditional operators using the Moore-Penrose inverse. Moreover we give example for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(k\)</EquationSource> </InlineEquation>-quasi <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation>-power posinormal weighted shift operator on directed tree.  </p>
In this paper we characterize \(k\)-quasi \(n\)-power posinormal composition operators and weighted composition operators on \(L^2(\mu )\) space. Also, we study \(k\)-quasi \(n\)-power posinormal operators in the view point of Cauchy dual of Lambert conditional operators using the Moore-Penrose inverse. Moreover we give example for \(k\)-quasi \(n\)-power posinormal weighted shift operator on directed tree.