  We characterize product Anosov diffeomorphisms in terms of the two-sided
limit shadowing property. It is proved that an Anosov diffeomorphism is a
product Anosov diffeomorphism if and only if any lift to the universal covering
has the unique two-sided limit shadowing property. Then we introduce two maps
in a suitable Banach space such that fixed points of these maps are related
with shadowing orbits on the universal covering.
