Some results on the reflexivity of operator subspaces (Q1272740)

Let \(L(H)\) denote the algebra of bounded linear operators on a Hilbert space \(H\) and \(L_*\) the trace ideal of \(L(H)\). Then \(L(H)\) is identified with \((L_*)^*\). For a subset \(S\) of \(L(H)\), the preannihilator of \(S\) is denoted by \(S_{\perp}\). For \(f,g\in L_*\), \(f\sim g\) means that \(f-g\in S_{\perp}\). Similarly if \((f_{ij})\) and \((g_{ij})\), \(1\leq i\leq m\), \(1\leq j\leq n\) are two operator matrices with \(f_{ij}\) and \(g_{ij}\) in \(L_*\) then \((f_{ij})\sim (g_{ij})\) provided \(f_{ij}\sim g_{ij}\) for all \(i\), \(j\). For a subspace \(S\) of \(L(H)\), let \[ \text{ref}_n(S)= \{T\in L(H): T^{(n)}(x)\in [S^{(n)}(x)],\;\forall x\in H^{(n)}\}. \] \(S\) is said to be \(n\)-reflexive if \(\text{ref}_n(S)= S\). The authors introduce the properties \(B^n_{p,q}\) and \(\widetilde B^n_{p,q}\) as follows: Let \(S\) be a linear subspace of \(L(H)\) and \(n\), \(p\), \(q\) be positive integers. \(S\) is said to have property \(B^n_{p,q}\) if for every set \(\{f_{ij}: 0\leq i<p, 0\leq j<q\}\) of finite rank operators, there exist sequences \(\{x_{ik}\}\) and \(\{y_{kj}\}\) in \(H\) such that \[ f_{ij}\sim \sum^n_{k=1} x_{ik}\otimes y_{kj},\quad 0\leq i<p,\quad 0\leq j<q. \] \(S\) has property \(\widetilde B^n_{p,q}\) if for every \(\varepsilon> 0\), there exists \(\delta>0\) such that, for any set \(\{f_{ij}, 0\leq i<p, 0\leq j<q\}\) and sequences \(\{x_{ik}'\}\) and \(\{y_{kj}'\}\) in \(H\) satisfying \[ \Biggl\|f_{ij}- \sum^n_1 x_{ik}'\otimes y_{kj}'\Biggr\|< \delta, \] there are sequences \(\{x_{ik}\}\) and \(\{y_{kj}\}\) in \(H\) such that \(\|x_{ik}'- x_{ik}\|< \varepsilon\), \(\|y_{kj}'- y_{kj}\|< \varepsilon\) and \(f_{ij}\sim \sum^n_1 x_{ik}\otimes y_{kj}\). The main results of this paper can be described as follows: Let \(S\) be a weakly closed subspace of \(L(H)\). Then (i) \(S\) is \((2n+1)\)-reflexive if \(S\) has the property \(B^n_{1,1}\); (ii) \(S\) is \(2n\)-reflexive if \(S\) has the property \(B^n_{1,2}\) and (iii) \(S\) is \(n\)-reflexive if \(S\) has the property \(\widetilde B^n_{2,3}\). These results generalize some of the results of \textit{H. Bercovici}, \textit{C. Foiaş} and \textit{C. Pearcy} [Mich. Math. J. 33, 119-126 (1986; Zbl 0594.47037)].