Geodesic
A characterization of reflexive spaces of operators
Czechoslovak Mathematical Journal, Tome 68 (2018) no. 1, pp. 257-266 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.21136/CMJ.2017.0456-16
Classification : 47A15
Keywords: reflexive space of operators; order-preserving map 
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     title = {A characterization of reflexive spaces of operators},
     journal = {Czechoslovak Mathematical Journal},
     pages = {257--266},
     year = {2018},
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Bračič, Janko; Oliveira, Lina. A characterization of reflexive spaces of operators. Czechoslovak Mathematical Journal, Tome 68 (2018) no. 1, pp. 257-266. doi: 10.21136/CMJ.2017.0456-16

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