Representation of measurable positive definite generalized Toeplitz Kernels inR

We define the signature of a bounded operatorA onL ² S ² and prove thatA is smooth for the action ofSO(3) onL 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 areSO(3)-smooth.