  We study the renormalization of operators of the type ${\bar h_{v'}} \Gamma
G^{\mu\nu} h_v$ in the heavy-quark effective theory (HQET). We construct the
combinations of such operators that are renormalized multiplicatively, and
calculate their velocity-dependent anomalous dimensions at the one-loop order.
We then show that the virial theorem of the HQET is not renormalized, and that
in the limit of equal velocities the anomalous dimension of the chromo-electric
operator vanishes to all orders in perturbation theory. This implies an exact
relation between renormalization constants, which may help in a future
calculation of the two-loop anomalous dimension of the chromo-magnetic
operator.
