On reflexivity and hyperreflexivity for linear subspace of operators (Q1175588)

Let \(H_ 1\) and \(H_ 2\) be Hilbert spaces. For \({\mathcal P}_ 1\) and \({\mathcal P}_ 2\) denote the set of all closed subspaces of \(H_ 1\) and \(H_ 2\) respectively. For a subset \({\mathcal A}\) of \({\mathcal L}(H_ 1,H_ 2)\), the map of \({\mathcal A}\) which is denoted by Map \({\mathcal A}\) is defined as a map \(\varphi:{\mathcal P}_ 1\to{\mathcal P}_ 2\) given by \(\varphi(P)=[{\mathcal A}P]\), where \([{\mathcal A}P]\) denotes the closure of span\(\{Ax:\;A\in{\mathcal A},\;x\in P\}\). Given a map \(\varphi:{\mathcal P}_ 1\to{\mathcal P}_ 2\), we denote by \(op\varphi\) the set \(\{A\in{\mathcal L}(H_ 1,H_ 2):\;AP\subseteq\varphi(P),\;\forall P\in{\mathcal P}_ 1\}\). A subset \({\mathcal A}\) of \({\mathcal L}(H_ 1,H_ 2)\) is called to be reflexive if \({\mathcal A}=op\text{ Map }{\mathcal A}\). A weakly closed subspace \(M\) of \({\mathcal L}(H_ 1,H_ 2)\) is said to be a Von Neumann type subspace if \({\mathcal {MM}}^*{\mathcal M}\subseteq{\mathcal M}\). The author obtains some results on reflexivity for linear subspaces of \({\mathcal L}(H_ 1,H_ 2)\), and in particular he shows that any Von Neumann type subspace is reflexive.