Czechoslovak Mathematical Journal, Vol. 68, No. 1, pp. 257-266, 2018
A characterization of reflexive spaces of operators
Janko Bračič, Lina Oliveira
Received August 27, 2016. First published March 30, 2017.
Abstract: 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.
Keywords: reflexive space of operators; order-preserving map
Affiliations: Janko Bračič, Naravoslovnotehniška Fakulteta, University of Ljubljana, Aškerčeva cesta 12, SI-1000 Ljubljana, Slovenia, e-mail: janko.bracic@fmf.uni-lj.si; Lina Oliveira, Center for Mathematical Analysis, Geometry and Dynamical Systems, and Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal, e-mail: linaoliv@math.tecnico.ulisboa.pt
