Linear rank one preservers between spaces of matrices with zero trace

Let X,Y be a pair of vector spaces over a field F associated with a bilinear form such that (x,y)=0 for all y in Y, implies that x=0. Let (X⊗Y)0 be the subspace of X⊗Y spanned by all decomposable elements x⊗y with (x,y)=0. Let U,V be any two vector spaces over F. In this note, we study linear mappings from (X⊗Y)0 to U⊗V that send nonzero decomposable elements to nonzero decomposable elements and some of its consequences.