Smooth operators for the action of SO(3) on L-2(S-2)

We define the ''signature'' of a bounded operator A on L-2(S-2) and prove that A is smooth for the action of SO(3) on L-2(S-2) if and only if its signature is smooth and any finite application of certain differential operators to it yields the signature of a bounded operator. Moreover, we show that the ''formal Fourier multipliers with bounded and smoothly variable coefficients'' are well defined bounded operators which are SO(3)-smooth.