  We analyze the low-energy behavior of scattering amplitudes involving
gravitons at loop level in four dimensions. The single-graviton soft limit is
controlled by soft operators which have been argued to separate into a
factorized piece and a non-factorizing infrared divergent contribution. In this
note we show that the soft operators responsible for the factorized
contributions are strongly constrained by gauge and Poincare invariance under
the assumption of a local structure. We show that the leading and subleading
orders in the soft-momentum expansion can not receive radiative corrections.
The first radiative correction occurs for the sub-subleading soft graviton
operator and is one-loop exact. It depends on only two undetermined
coefficients which should reflect the field content of the theory under
consideration.
