a one-sided limit is enough). Especially important for us will be the corresponding families of “small" sets of the circle; that is, the sets which are annihilated by every positive measure in the family under consideration (continuous measures, or Rajchman measures). Obviously, a Borel set $D\subset\TT$ is small for continuous measures if and only if it is countable. The small sets for Rajchman measures are the so-called *sets of extended uniqueness* or *sets of uniqueness for Fourier-Stieltjes series*, which have been extensively studied since the beginning of the 20th century (see [@KL]). The family of all sets of extended uniqueness is usually denoted by $\mathcal U_0$.

Our main results can now be summarized as follows.

\[WS\] Let $X$ be a complex separable Fréchet space, and let $T\in\mathfrak L(X)$.

1.  Assume that the $\TT$-eigenvectors are *perfectly spanning*, in the following sense: for any countable set $D\subset \TT$, the linear span of $\bigcup_{\lambda\in\TT\setminus D}\ker (T-\lambda)$ is dense in $X$. Then $T$ is weakly mixing in the Gaussian sense.

2.  Assume that the $\TT$-eigenvectors are *$\mathcal U_0$-perfectly spanning*, in the following sense: for any Borel set of extended uniqueness $D\subset \TT$, the linear span of $\bigcup_{\lambda\in\TT\setminus D}\ker (T-\lambda)$ is dense in $X$. Then 