  We study perturbative and instanton corrections to the Operator Product
Expansion of the lowest weight Chiral Primary Operators of N=4 SYM_4. We
confirm the recently observed non-renormalization of various operators (notably
of the double-trace operator with dimension 4 in the 20 irrep of SU(4)), that
appear to be unprotected by unitarity restrictions. We demonstrate the
splitting of the free-field theory stress tensor and R-symmetry current in
supermultiplets acquiring different anomalous dimensions in perturbation theory
and argue that certain double-trace operators also undergo a perturbative
splitting into operators dual to string and two-particle gravity states
respectively. The instanton contributions affect only those double-trace
operators that acquire finite anomalous dimensions at strong coupling. For the
leading operators of this kind, we show that the ratio of their anomalous
dimensions at strong coupling to the anomalous dimensions due to instantons is
the same number.
