Linear transformations of tensor products preserving a fixed rank

In this paper T is a linear transformation from a tensor product X Finto U V, where X, Y, U,V are vector spaces over an infinite field F. The main result gives a characterization of surjective transformations T for which there is a positive integer k(k < dim U,k< dim V) such that whenever zeXY has rank k then also TzeUV has rank k. It is shown that T = AB or T = So (C D) where A, 5, C, D are appropriate linear isomorphisms and S is the canonical isomorphism of V U onto UV.