Minimal rank and reflexivity of operator spaces

Abstract:

Let ${\mathcal {S}}$ be an $n$-dimensional space of linear operators between the linear spaces $U$ and $V$ over an algebraically closed field $\mathbb {F}$. Improving results of Larson, Ding, and Li and Pan we show the following. Theorem. Let $S_1, \ldots , S_n$ be a basis of $\mathcal {S}$. Assume that every nonzero operator in $\mathcal {S}$ has rank larger than $n$. Then a linear operator $T : U \to V$ belongs to $\mathcal {S}$ if and only if for every $u\in U$, $Tu$ is a linear combination of $S_1 u , \ldots , S_n u$.