  In this paper we characterize the closed subspaces of $L^2(\mathcal{F})$ that
reduce the operators of the form $E^{\mathcal{A}}M_u$, in which $\mathcal{A}$
is a $\sigma$- subalgebra of $\mathcal{F}$. We show that $L^2(A)$ reduces
$E^{\mathcal{A}}M_u$ if and only if $E^{\mathcal{A}}(\chi_A)=\chi_A$ on the
support of $E^{\mathcal{A}}(|u|^2)$, where $A\in \mathcal{F}$. Also, some
necessary and sufficient conditions are provided for $L^2(\mathcal{B})$ to
reduces $E^{\mathcal{A}}M_u$, for the $\sigma$- subalgebra $\mathcal{B}$ of
$\mathcal{F}$.
