Linear factorization relations are derived for the matrix elements of quantum mechanical operators defined on some space H = H1⊕⋅2 which are diagonalizable on H1. The coefficients in these relationships do not depend on the operators per se but do depend on the representations in which the operators are diagonal. The formulation is very general with regard to the nature of the ’’input’’ information in the factorization. With each choice of input information there are associated consistency conditions. The consistency conditions, in turn, give rise to a flexibility in the form of the factorization relations. These relations are examined in detail for the operators of scattering theory which are local in the internal molecular coordinates. In particular, this includes S and T matrices in the energy sudden (ES) approximation. A similar development is given for the square of the magnitude of operator matrix elements appropriately averaged over ’’symmetry classes.’’ In the ES these relations apply to transition cross sections between symmetry classes. In particular, they apply to degeneracy averaged cross sections in situations where the symmetry classes correspond to energy levels.