Let $\{X_t\}_{t \in \mathbb{N}}$ be a centered Gaussian process, let $X$ be the space of process trajectories (equipped with the pushforward measure) and let $T\colon X \to X$ be the shift operation defined by $$T(x)(t) = x(t + 1)$$
Finally, let $F$ be the linear operator on $L^2(X)$ defined by $F(f) = f \circ T$. If a measurable set $A \subseteq X$ is a $T$-invariant in the sense that $T^{-1}(A) \Delta A$ has measure zero, then the indicator function of $A$ is an element of $L^2(X)$ verifying $F(\mathbb{1}_A) = \mathbb{1}_A$ almost certainly. We see from this that the process is ergodic exactly when the fixed points of $F$ are only the constant functions.
Now, instead of $L^2(X)$, it is often easier to think about the so-called reproducing kernel Hilbert space of the process. This space is naturally isomorphic to the closed span of the variables $X_t$ in $L^2(X)$, which in turn is stable under $F$. When the process itself is ergodic, $F$ only has the zero function as a fixed point on the RKHS. My question is: is a converse also true? That is, can ergodicity of the process be characterized by the behavior of $F$ on the RKHS of the process?