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Keywords:
reflexive space of operators; order-preserving map
reflexive space of operators; order-preserving map
Summary:
We show that for a linear space of operators ${\mathcal M}\subseteq {\mathcal B}(\scr {H}_1,\scr {H}_2)$ 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 ${\rm Bil}({\mathcal M})$ of subspaces determined by ${\mathcal M}$ with $P\leq \psi _1(P,Q)$ and $Q\leq \psi _2(P,Q)$ for any pair $(P,Q)\in {\rm Bil}({\mathcal M})$, and such that an operator $T\in {\mathcal B}(\scr {H}_1,\scr {H}_2)$ lies in ${\mathcal M}$ if and only if $\psi _2(P,Q)T\psi _1(P,Q)=0$ for all $(P,Q)\in {\rm Bil}( {\mathcal M})$. This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.
References:
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