  In this note, we want to investigate the question, given a projective
algebraic scheme X/k and coherent sheaves F, E on X, when do global
differential operators of order N greater than zero, between E and F exist. We
investigate in particular the case of projective n-space and prove that for
large N always global differential operators between E and F exist. We also
show that the dimension of global operators of order N grows like a polynomial
in N of degree 2n. Finally, we define algebraic elliptic operators in the
classical sense and show that for "most" algebraic smooth complete varieties,
they do not exist on locally free sheaves.
