Reflexivity and hyperreflexivity of operator spaces. (Q1874586)

Let \(V\) be a complex linear space and \(H\) be a complex Hilbert space. Denote by \(L(V)\) and \(B(H)\) the space of all linear transformations on \(V\) and the space of all bounded linear operators on \(H\), respectively. The authors prove that if \({\mathcal S}\subset L(V)\) is an \(n\)-dimensional subspace such that every nonzero transformation in \(\mathcal S\) has rank greater than or equal to \(2n-1\), then \(\mathcal S\) is algebraically reflexive. This is an improvement of a result due to \textit{L. Ding} [Proc. Am. Math. Soc. 124, 3101--3108 (1996; Zbl 0861.47023)]. It is also proved that if \({\mathcal S}\subset B(H)\) is an \(n\)-dimensional subspace such that every nonzero operator in \(\mathcal S\) has rank greater than or equal to \(2n-1\), then \(\mathcal S\) is hyperreflexive.