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A characterization of reflexive spaces of operators

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Abstract

We show that for a linear space of operators M ⊆ B(H1, H2) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ1, ψ2) on a bilattice Bil(M) of subspaces determined by M with P ≤ ψ1(P,Q) and Q ≤ ψ2(P,Q) for any pair (P,Q) ∈ Bil(M), and such that an operator T ∈ B(H1, H2) lies in M if and only if ψ2(P,Q)Tψ1(P,Q) = 0 for all (P,Q) ∈ Bil(M). This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.

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Authors and Affiliations

  1. Naravoslovnotehniška Fakulteta, University of Ljubljana, Aškerčeva cesta 12, SI-1000, Ljubljana, Slovenia

    Janko Bračič

  2. 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

    Lina Oliveira

Authors
  1. Janko Bračič
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  2. Lina Oliveira
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Corresponding authors

Correspondence to Janko Bračič or Lina Oliveira.

Additional information

The first author was supported by the Slovenian Research Agency through the research program P2-0268. The second author was partially funded by FCT/Portugal through UID/MAT/04459/2013 and EXCL/MAT-GEO/0222/2012.

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Bračič, J., Oliveira, L. A characterization of reflexive spaces of operators. Czech Math J 68, 257–266 (2018). https://doi.org/10.21136/CMJ.2017.0456-16

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  • DOI: https://doi.org/10.21136/CMJ.2017.0456-16

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