  We show that for a linear space of operators ${\mathcal M}\subseteq {\mathcal
B}(H_1,H_2)$ 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
  $Bil({\mathcal M})$ of subspaces determined by ${\mathcal M}$, with $P\leq
\psi_1(P,Q)$ and $Q\leq \psi_2(P,Q)$, for any pair $(P,Q)\in Bil({\mathcal
M})$, and
  such that an operator $T\in {\mathcal B}(H_1,H_2)$ lies in ${\mathcal M}$ if
and only if $\psi_2(P,Q)T\psi_1(P,Q)=0$ for all
  $(P,Q)\in Bil( {\mathcal M})$. This extends to reflexive spaces the
Erdos--Power type characterization of weakly closed bimodules over a nest
algebra.
