  Let $\{X_k\}_{k \in \mathbb{Z}}$ be a stationary Gaussian process with values
in a separable Hilbert space $\mathcal{H}_1$, and let $G:\mathcal{H}_1 \to
\mathcal{H}_2$ be an operator acting on $X_k$. Under suitable conditions on the
operator $G$ and the temporal and cross-sectional correlations of $\{X_k\}_{k
\in \mathbb{Z}}$, we derive a central limit theorem (CLT) for the normalized
partial sums of $\{G[X_k]\}_{k \in \mathbb{Z}}$. To prove a CLT for the Hilbert
space-valued process $\{G[X_k]\}_{k \in \mathbb{Z}}$, we employ techniques from
the recently developed infinite dimensional Malliavin-Stein framework. In
addition, we provide quantitative and continuous time versions of the derived
CLT. In a series of examples, we recover and strengthen limit theorems for a
wide array of statistics relevant in functional data analysis, and present a
novel limit theorem in the framework of neural operators as an application of
our result.
