Invertible preservers of certain decomposable tensors

Let U be an n-dimensional vector space over a field of characteristic 0. For each positive integer k < min{m,n}, let Jk be the set of all decomposable elements x1 … xm in the mth tensor product U such that dim x1, …, xm K. We characterize those nonsingular linear mappings T on U such that T(Jk) Jk.