Czechoslovak Mathematical Journal, online first, 10 pp.

A characterization of reflexive spaces of operators

Janko Bračič, Lina Oliveira

Received August 27, 2016.   First published March 30, 2017.

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

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

Classification (MSC 2010): 47A15

DOI: 10.21136/CMJ.2017.0456-16

Full text available as PDF.


References:
  [1] D. G. Han: On $\scr A$-submodules for reflexive operator algebras. Proc. Am. Math. Soc. 104 (1988), 1067-1070. DOI 10.2307/2047592 | MR 0969048 | Zbl 0694.47031
  [2] J. A. Erdos: Reflexivity for subspace maps and linear spaces of operators. Proc. Lond. Math. Soc., III Ser. 52 (1986), 582-600. DOI 10.1112/plms/s3-52.3.582 | MR 0833651 | Zbl 0609.47053
  [3] J. A. Erdos, S. C. Power: Weakly closed ideals of nest algebras. J. Oper. Theory 7 (1982), 219-235. MR 0658610 | Zbl 0523.47027
  [4] D. Hadwin: A general view of reflexivity. Trans. Am. Math. Soc. 344 (1994), 325-360. DOI 10.1090/S0002-9947-1994-1239639-4 | MR 1239639 | Zbl 0802.46010
  [5] P. R. Halmos: Reflexive lattices of subspaces. J. Lond. Math. Soc., II. Ser. 4 (1971), 257-263. DOI 10.1112/jlms/s2-4.2.257 | MR 0288612 | Zbl 0231.47003
  [6] K. Kliś-Garlicka: Reflexivity of bilattices. Czech. Math. J. 63 (2013), 995-1000. DOI 10.1007/s10587-013-0067-4 | MR 3165510 | Zbl 1313.47024
  [7] K. Kliś-Garlicka: Hyperreflexivity of bilattices. Czech. Math. J. 66 (2016), 119-125. DOI 10.1007/s10587-016-0244-3 | MR 3483227 | Zbl 06587878
  [8] P. Li, F. Li: Jordan modules and Jordan ideals of reflexive algebras. Integral Equations Oper. Theory 74 (2012), 123-136. DOI 10.1007/s00020-012-1982-8 | MR 2969043 | Zbl 1286.47046
  [9] A. I. Loginov, V. S. Sul'man: Hereditary and intermediate reflexivity of $W\sp*$-algebras. Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 1260-1273. (In Russian.) MR 0405124 | Zbl 0327.46073
  [10] V. Shulman, L. Turowska: Operator synthesis I. Synthetic sets, bilattices and tensor algebras. J. Funct. Anal. 209 (2004), 293-331. DOI 10.1016/S0022-1236(03)00270-2 | MR 2044225 | Zbl 1071.47066


Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris.cz.
Subscribers of Springer need to access the articles on their site, which is http://link.springer.com/journal/10587.


[List of online first articles] [Contents of Czechoslovak Mathematical Journal]