  We have formulated and proved the Bogolyubov inequality for operators at zero
temperature. So far this inequality has been known for matrices, and we were
able to extend it to certain class of operators. We have also applied this
inequality to the system of interacting rotors. We have shown that if: {\em i)}
the dimension of the lattice is 1 or 2, {\em ii)} the interaction decreases
sufficiently fast with a distance, and {\em iii)} there is an energy gap over
the ground state, then the spontaneous magnetization in the ground state is
zero, i.e. there is no LRO in the system. We present also heuristic arguments
(of perturbation-theoretic nature) suggesting that one- and two-dimensional
system of interacting rotors has the energy gap independent of the system size
if the interaction is sufficiently small.
