### A characterization of reflexive spaces of operators

We show that for a linear space of operators $\mathcal{M}\subseteq \mathcal{B}({\mathscr{H}}_{1},{\mathscr{H}}_{2})$ the following assertions are equivalent. (i) $\mathcal{M}$ is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map $\Psi =({\psi}_{1},{\psi}_{2})$ on a bilattice $\mathrm{Bil}\left(\mathcal{M}\right)$ of subspaces determined by $\mathcal{M}$ with $P\le {\psi}_{1}(P,Q)$ and $Q\le {\psi}_{2}(P,Q)$ for any pair $(P,Q)\in \mathrm{Bil}\left(\mathcal{M}\right)$, and such that an operator $T\in \mathcal{B}({\mathscr{H}}_{1},{\mathscr{H}}_{2})$ lies in $\mathcal{M}$ if and only if ${\psi}_{2}(P,Q)T{\psi}_{1}(P,Q)=0$ for all $(P,Q)\in \mathrm{Bil}\left(\mathcal{M}\right)$. This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.