A characterization of reflexive spaces of operators

Abstract

We show that for a linear space of operators M ⊆ B(H 1 , H 2 ) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ 1 , ψ 2 ) on a bilattice Bil(M) of subspaces determined by M, with P ≤ ψ 1 (P, Q) and Q ≤ ψ 2 (P, Q), for any pair (P, Q) ∈ Bil(M), and such that an operator T ∈ B(H 1 , H 2 ) lies in M if and only if ψ 2 (P, Q)T ψ 1 (P, Q) = 0 for all (P, Q) ∈ Bil(M). This extends to reflexive spaces the Erdos-Power type characterization of weakly closed bimodules over a nest algebra. 2010 Mathematics Subject Classification. Primary 47A15.

Bimodules of Banach space nest algebras

“…Those were characterized in terms of order homomorphisms on the corresponding nest, later to be called support functions (see [7]). Similar descriptions have been shown to hold in the case of reflexive operator algebras and, more generally, of reflexive spaces of operators (see [3], [11]). Davidson, Donsig and Hudson [7] went further to introduce the essential support function of a given norm closed bimodule, which allowed for finding the maximal and the minimal norm closed bimodule having the same essential support function.…”

Kernel maps and operator decomposition

“…Recall that a subset of P(H)× P(H) is said to be a bilattice if it is closed under the lattice operations (2.2) and contains the pairs (0, 0), (0, I), and (I, 0) (cf. [1,20]). The top and bottom elements of any bilattice are (I, 0), and (0, I), respectively.…”

“…Here, however, we are mainly interested in bilattices associated with a single operator in a way to be made precise below (see (2.3)). For more details on bilattices, the reader is referred to [1,8,13,20].…”

“…Adopting the notation of [1], we define the set BIL(T, L) by Observe that the set defined in (2.3) is a strongly closed bilattice which coincides with the intersection of the strongly closed bilattices L × L ⊥ and {(P, Q) ∈ P(H) × P(H) : QT P = 0} (cf. [1,20]). For each P ∈ L, define the projections ϕ T,L (P ), ψ T,L (P ) by…”

“…The idea of linking sets of operators with lattices and bilattices has been present in the literature, e.g., [6,8,10,11,13,20] and, more recently, [1]. However, this has been done mainly in connection with reflexivity which is not the approach here, since we are only interested in a single operator at a time.…”