An upper bound on the dimension of the reflexivity closure

Abstract:

Let ${\mathcal V},{\mathcal W}$ be linear spaces over an algebraically closed field, and let $\mathscr {S}$ be an $n$–dimensional subspace of linear operators that maps ${\mathcal V}$ into ${\mathcal W}$. We give a sharp upper bound for the dimension of the intersection of all images of nonzero operators from $\mathscr {S}$, namely $\dim ( \bigcap _{A\in \mathscr {S}\setminus \{0\}}\mathrm {Im} A ) \leq \dim {\mathcal V}-n+1$. As an application, we also give a sharp upper bound for the dimension of the reflexivity closure $\operatorname {Ref}\mathscr {S}$ of $\mathscr {S}$, namely $\dim ( \operatorname {Ref}\mathscr {S} ) \leq n(n+1)/2$.