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

TL;DR

This paper characterizes the subspaces of L^2 spaces that reduce certain weighted conditional expectation operators, providing necessary and sufficient conditions for reducibility based on the structure of subalgebras and indicator functions.

Contribution

It offers a complete characterization of reducing subspaces for WCE operators on L^2 spaces, extending understanding of their structure and conditions for reducibility.

Findings

L^2(A) reduces E^{A} M_u iff E^{A}( ext{indicator}_A)= ext{indicator}_A on the support of E^{A}(|u|^2)

Provides necessary and sufficient conditions for L^2(B) to reduce E^{A} M_u

Characterizes reducibility in terms of conditional expectations and indicator functions

Abstract

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 $\mathcal{F}$.