  We extend the estimates proved by Donnelly and Fefferman and by Lebeau and
Robbiano for sums of eigenfunctions of the Laplacian (on a compact manifold) to
estimates for sums of eigenfunctions of any positive and elliptic
pseudo-differential operator of positive order on a compact Lie group. Our
criteria are imposed in terms of the positivity of the corresponding
matrix-valued symbol of the operator. As an application of these inequalities
in the control theory, we obtain the null-controllability for diffusion models
for elliptic pseudo-differential operators on compact Lie groups.
