2^{-n}\sigma_n$ is in $\mathcal B$ and ${\rm supp}(\sigma)=\Lambda$.

The results
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To state our results we need two more definitions.

\[defBperfspan\] Let $T$ be an operator acting on a complex separable Fréchet space $X$, and let $\mathcal B\subset\mathcal M(\TT)$. We say that the $\TT$-eigenvectors of $T$ are *$\mathcal B$-perfectly spanning* if, for any Borel $\mathcal B$-small set $D\subset \TT$, the linear span of $\bigcup_{\lambda\in\TT\setminus D}\ker (T-\lambda)$ is dense in $X$. When $\mathcal B$ has the form $\mathcal F_+^{-1}(\mathbf S)$, the terminology *$\mathbf S$-perfectly spanning* is used.

Thus, perfect spanning is $\mathcal B$-perfect spanning for the family of continuous measures, and $\mathcal U_0$-perfect spanning is $\mathcal B$-perfect spanning for the family of Rajchman measures.

At some places, we will encounter sets which are *analytic* but perhaps non Borel. Recall that a set $D$ in some Polish space $Z$ is [analytic]{} if one can find a Borel relation $B\subset Z\times E$ (where $E$ is Polish) such that $z\in D\Leftrightarrow\exists u\;:\; B(z,u)$. If the spanning property of the above definition holds for every analytic $\mathcal B$-small set $D$, we say that the $\TT$-eigenvectors of $T$ are [$\mathcal B$-perfectly spanning]{} *for analytic sets*.

We shall say that a family $\mathbf S\subset\ell^\infty(\ZZ_+)$ is *$c_0$-like* if 