Locally linearly dependent operators and reflexivity of operator spaces

AbstractWe obtain two results on the existence of large subspaces of operators of small rank in locally linearly dependent spaces of operators. As a consequence we obtain an upper bound for the rank of operators belonging to a minimal locally linearly dependent space of operators. It has been known that the only obstruction to the reflexivity of a finite-dimensional operator space comes from the operators with small ranks. Our results improve known bounds on the minimal rank that guarantees the reflexivity