Let $X$ be a random variable with values in a separable Hilbert space $\mathcal{H}$ with inner product $\langle \cdot, \cdot \rangle$. Assume that $X$ has a Gaussian distribution in the sense that
$$ \langle X, h\rangle $$
is Gaussian for every $h \in \mathcal{H}$. We define the expectation of $X$ as the element $\mu \in \mathcal{H}$ such that
$$ E(\langle X, h \rangle) = \langle \mu, h \rangle \quad \forall h \in \mathcal{H} $$
and the covariance operator as the Hilbert-Schmidt operator, $C$, from $\mathcal{H}$ to $\mathcal{H}$ given by
$$ \langle Cf ,g \rangle = E ( \langle X, f \rangle \langle X, g \rangle) - \langle \mu, f \rangle \langle \mu, g \rangle $$
What is the distribution of $\lVert X \rVert^2 = \langle X, X \rangle$? Does it even have a distribution? Can we express it using the mean and covariance?