A version of Lomonosov's theorem for collections of positive operators

It is known that for every Banach space $X$ and every proper $WOT$-closed subalgebra $\mathcal A$ of $L(X)$, if $\mathcal A$ contains a compact operator, then it is not transitive; that is, there exist non-zero $x\in X$ and $f\in X^*$ such that $\langle f,Tx\rangle=0$ for all $T\in\mathcal A$. In the case of algebras of adjoint operators on a dual Banach space, V. Lomonosov extended this result as follows: without having a compact operator in the algebra, one has $\bigl\lvert\langle f,Tx\rangle\bigr\rvert\le\lVert T_*\rVert_e$ for all $T\in\mathcal A$. In this paper, we prove a similar extension of a result of R. Drnovšek. Specifically, we prove that if $\mathcal C$ is a collection of positive adjoint operators on a Banach lattice $X$ satisfying certain conditions, then there exist non-zero $x\in X_+$ and $f\in X^*_+$ such that $\langle f,Tx\rangle\le\lVert T_*\rVert_e$ for all $T\in\mathcal C$.