Bilattice

We show that for a linear space of operators #123;#92;mathcalM#125;#92;subseteq#123;#92;mathcalB#125;(H#95;1,H#95;2) the following assertions are equivalent. (i) #123;#92;mathcalM#125; is reflexive in the sense of Loginov--Shulman. (ii) There exists an order-preserving map 1 on a bilattice Bil(#123;#92;mathcalM#125;) of subspaces determined by #123;#92;mathcalM#125;, with P#92;leq#92;psi#95;1(P,Q) and Q#92;leq#92;psi#95;2(P,Q), for any pair (P,Q)#92;inBil(#123;#92;mathcalM#125;), and such that an operator T#92;in#123;#92;mathcalB#125;(H#95;1,H#95;2) lies in #123;#92;mathcalM#125; if and only if #92;psi#95;2(P,Q)T#92;psi#95;1(P,Q)#61;0 for all (P,Q)#92;inBil(#123;#92;mathcalM#125;).