$X$ with full support with respect to which $T$ has (P). The problem of determining which operators are ergodic in the Gaussian sense was investigated by E. Flytzanis ([@F]), in a Hilbert space setting. The fundamental idea of [@F] is that one has to look at the *$\TT$-eigenvectors* of the operator, the eigenvectors associated with eigenvalues of modulus $1$: roughly speaking, ergodicity is equivalent to the existence of “sufficiently many $\TT$-eigenvectors and eigenvalues". This is of course to be compared with the now classical eigenvalue criterion for hypercyclicity found by G. Godefroy and J. Shapiro ([@GS]), which says in essence that an operator having enough eigenvalues inside and outside the unit circle must be hypercyclic.

The importance of the $\TT$-eigenvectors is easy to explain. Indeed, it is almost trivial that if $T\in\mathfrak L(X)$ is an operator whose $\TT$-eigenvectors span a dense subspace of $X$, then $T$ admits an invariant Gaussian measure with full support: choose a sequence of $\TT$-eigenvectors $(x_n)_{n\geq 0}$ (say $T(x_n)=\lambda_n x_n$) with dense linear span such that $\sum_0^\infty\Vert x_n\Vert<\infty$ for every continuous semi-norm $\Vert\,\cdot\,\Vert$ on $X$, and let $\mu$ be the distribution of the random variable $\xi=\sum_0^\infty g_n x_n$. That $\mu$ is $T$-invariant follows from the linearity 