  We show that the linear map defined by multiplication with a general
bi-homogeneous form between two bi-graduated pieces of the first cohomology of
a nonsingular quadric in the projective space is of maximal rank. This is the
first non trivial case of a more general open problem on natural linear maps
between vector spaces of tensors defined in terms of multiplications and
contractions. An interpretation in terms of bi-homogeneous linear differential
operators with polynomial coefficients is also given.
