Reducibility of WCE operators on $L^2(\mathcal{F})$

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 $E^{\mathcal{A}}(\mathrm{\mid }u{\mathrm{\mid }}^2)$, 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 $\mathcal{F}$.