Lambert conditional operators on \(L^2(\Sigma )\) (Q2302320)

Given a complete \(\sigma\)-finite measure space \((X,\Sigma,\mu)\), and a sub-\(\sigma\)-finite algebra \(\mathcal{A}\), the Lambert conditional operator \(T_{(w,u)}\) is defined on \(L^2(\Sigma)\) by \(T_{(w,u)}(f)=wE(uf)\), where \(E\) is the conditional expectation with respect to \(\mathcal{A}\). The authors consider bounded Lambert conditional operators \(T_{(w,u)}\) on \(L^2(\Sigma)\), They prove that, for these operators, the class of normal, quasinormal, \(p\)-hyponormal, centered, binormal and \(n\)-normal coincide, that all \(n\)th iterates of the \(\lambda\)-Aluthge transform of \(T_{(w,u)}\) are normal, and that the latter coincides with the Duggal transformation of \(T_{(w,u)}\). Other topics are treated by the authors, such as the reverse order law for the Moore-Penrose inverse and the characterization of closed range, positive, partial isometry, and orthogonal projection.