  We prove a necessary and sufficient condition for the Liouville and strong
Liouville properties of the infinitesimal generator of a L\'evy process and
subordinate L\'evy processes. Combining our criterion with the necessary and
sufficient condition obtained by Alibaud et al., we obtain a characterization
of (orthogonal subgroup of) the set of zeros of the characteristic exponent of
the L\'evy process.
