A characterization of reflexive spaces of operators

Bračič, Janko, and Oliveira, Lina. "A characterization of reflexive spaces of operators." Czechoslovak Mathematical Journal 68.1 (2018): 257-266. <http://eudml.org/doc/294191>.

@article{Bračič2018,
abstract = {We show that for a linear space of operators $\{\mathcal \{M\}\}\subseteq \{\mathcal \{B\}\}(\mathcal \{H\}_1,\mathcal \{H\}_2)$ 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 $P\le \psi _1(P,Q)$ and $Q\le \psi _2(P,Q)$ for any pair $(P,Q)\in \{\rm Bil\}(\{\mathcal \{M\}\})$, and such that an operator $T\in \{\mathcal \{B\}\}(\mathcal \{H\}_1,\mathcal \{H\}_2)$ 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.},
author = {Bračič, Janko, Oliveira, Lina},
journal = {Czechoslovak Mathematical Journal},
keywords = {reflexive space of operators; order-preserving map},
language = {eng},
number = {1},
pages = {257-266},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A characterization of reflexive spaces of operators},
url = {http://eudml.org/doc/294191},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Bračič, Janko
AU - Oliveira, Lina
TI - A characterization of reflexive spaces of operators
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 1
SP - 257
EP - 266
AB - We show that for a linear space of operators ${\mathcal {M}}\subseteq {\mathcal {B}}(\mathcal {H}_1,\mathcal {H}_2)$ 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 $P\le \psi _1(P,Q)$ and $Q\le \psi _2(P,Q)$ for any pair $(P,Q)\in {\rm Bil}({\mathcal {M}})$, and such that an operator $T\in {\mathcal {B}}(\mathcal {H}_1,\mathcal {H}_2)$ 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.
LA - eng
KW - reflexive space of operators; order-preserving map
UR - http://eudml.org/doc/294191
ER -