A characterization of reflexive spaces of operators

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 $Ψ=(ψ_1,ψ_2)$ on a bilattice $Bil({\mathcal M})$ of subspaces determined by ${\mathcal M}$, with $P\leq ψ_1(P,Q)$ and $Q\leq ψ_2(P,Q)$, 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 $ψ_2(P,Q)Tψ_1(P,Q)=0$ 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.

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