Hölder equality for conditional expectations with application to linear monotone operators

In a standard space $L_p=L_p(\Omega,\mathfrak{A},P)$, $1\le p<\infty$, for a given factor $f$ and a $\sigma$-algebra $\mathfrak{B}\subseteq\mathfrak{A} $, a certain criterion is derived for a conditional expectation $x(X)=E(Xf\,|\,\mathfrak{B})$ to represent a continuous linear operator over $X\in L_p$. As an application, the above representation (with the corresponding factor $f\ge 0$) is considered for a general linear monotone operator $x(X)$, $X\in K$, given on an arbitrary subcone $K\subseteq L_p^+ $ in $L_p^+ =\{X\in L_p:X\ge 0\}$.