property of the space, and that Hilbert space has cotype $2$ as well as $L^p$ spaces for $p\in [1,2]$ (but *not* for $p>2$).

5.  As observed in [@BG2 Example 4.2], (3) does not hold on an arbitrary Banach space $X$. Indeed, let $X:=\mathcal C_0([0,2\pi])=\{ f\in\mathcal C([0,2\pi]);\; f(0)=0\}$ and let $V:L^2(0,2\pi)\to X$ be the Volterra operator, $Vf(t)=\int_0^t f(s)\, ds$. There is a unique operator $T:X\to X$ such that $TV=VM_\phi$, where $M_\phi:L^2(0,2\pi)\to L^2(0,2\pi)$ is the multiplication operator associated with the function $\phi (t)=e^{it}$. The operator $T$ is given by the formula $$\label{Kal}
    Tf(t)=\phi(t)f(t)-\int_0^t \phi'(s)f(s)\, ds\, .$$ It is easy to check that $T$ has no eigenvalues. On the other hand, $T$ is strongly mixing with respect to the Wiener measure on $\mathcal C_0([0,2\pi])$.

As it turns out, ergodicity and weak mixing in the Gaussian sense are in fact equivalent (see e.g. [@G], or [@BG2 Theorem 4.1]). Hence, from Theorem \[WS\] we immediately get the following result. (A Gaussian measure $\mu$ is *nontrivial* if $\mu\neq\delta_0$).

\[characexistergod\] For a linear operator $T$ acting on a Banach space $X$ with cotype 2, the following are equivalent:

-   $T$ admits a nontrivial ergodic Gaussian measure;

-   there exists a closed, $T$-invariant subspace $Z\neq \{ 0\}$ such 