  We study a class of non-protected local composite operators which occur in
the R symmetry singlet channel of the OPE of two stress-tensor multiplets in
{\cal N}=4 SYM. At tree level these are quadrilinear scalar dimension four
operators, two single-traces and two double-traces. In the presence of
interaction, due to a non-trivial mixing under renormalization, they split into
linear combinations of conformally covariant operators. We resolve the mixing
by computing the one-loop two-point functions of all the operators in an {\cal
N}=1 setup, then diagonalizing the anomalous dimension matrix and identifying
the quasiprimary operators. We find one operator whose anomalous dimension is
negative and suppressed by a factor of 1/N^2 with respect to the anomalous
dimensions of the Konishi-like operators. We reveal the mechanism responsible
for this suppression and argue that it works at every order in perturbation
theory. In the context of the AdS/CFT correspondence such an operator should be
dual to a multiparticle supergravity state whose energy is less than the sum of
the corresponding individual single-particle states.
