  We provide complete characterizations, on Banach spaces with cotype 2, of
those linear operators which happen to be weakly mixing or strongly mixing
transformations with respect to some nondegenerate Gaussian measure. These
characterizations involve two families of small subsets of the circle: the
countable sets, and the so-called sets of uniqueness for Fourier-Stieltjes
series. The most interesting part, i.e. the sufficient conditions for weak and
strong mixing, is valid on an arbitrary (complex, separable) Fr\'echet space.
