that $$\overline{\rm span}\, \bigcup_{\lambda\in\TT\setminus D}\ker (T_{\vert Z}-\lambda)=Z$$ for every countable set $D\subset\TT$.

In this case, $T$ admits an ergodic Gaussian measure with support $Z$, for any such subspace $Z$.

If $T$ admits an ergodic Gaussian measure $\mu\neq\delta_0$, then $Z:={\rm supp}(\mu)$ is a non-zero $T$-invariant subspace, and $Z$ satisfies (b) by Theorem \[WS\] (3). The converse follows from Theorem \[WS\] (1).

For concrete applications, it is useful to formulate Theorem \[WS\] in terms of *$\TT$-eigenvector fields* for the operator $T$. A $\TT$-eigenvector field for $T$ is a map $E:\Lambda\to X$ defined on some set $\Lambda\subset\TT$, such that $$TE(\lambda)=\lambda E(\lambda)$$ for every $\lambda\in\Lambda$. (The terminology is not perfectly accurate: strictly speaking, $E(\lambda)$ is perhaps not a $\TT$-eigenvector because it is allowed to be $0$). Recall also that a closed set $\Lambda\subset\TT$ is *perfect* if it has no isolated points or, equivalently, if $V\cap\Lambda$ is uncountable for any open set $V\subset\TT$ such that $V\cap\Lambda\neq\emptyset$. Analogously, a closed set $\Lambda\subset \TT$ is said to be *$\mathcal U_0$-perfect* if $V\cap \Lambda$ is not a set of extended uniqueness for any open set $V$ such that $V\cap\Lambda\neq\emptyset$. (For example, any nontrivial closed arc is $\mathcal U_0$-perfect).

\[eigvectfield\] Let $X$ be a separable complex Fréchet space, and 