A characterization of reflexive spaces of operators

Abstract

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 $P\le {\psi }_1(P,Q)$ and $Q\le {\psi }_2(P,Q)$, 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.