In this paper we consider those
linear transformations from one tensor product of vector spaces to another which
carry nonzero decomposable tensors into nonzero decomposable tensors. We obtain a
general decomposition theorem for such transformations. If we suppose further that
the transformation maps the space into itself then we have a complete structure
theorem in the following two cases: (1) the transformation is onto, and (2) the field is
algebraically closed and the tensor space is a product of finite dimensional
vector spaces. The main results are contained in Theorems 3.5 and 3.8 which
state that the transformation T : U1⊗⋯⊗ Un→ U1⊗⋯⊗ Un has the form
T(x1⊗⋯⊗ xn) = T1(xπ(1)) ⊗⋯⊗ Tn(x2ζ(n)) where Ti: Uπ(i)→ Ui are
nonsingular and π is a permutation. Case (2) generalizes a theorem of Marcus and
Moyls.
