If $X\_t$ is a mean zero, square integrable process with covariance kernel $k(s,t),$ Mercer's theorem states that there exists an orthogonal basis $\{\phi\_i\}$ in $L^2$ and eigenvalues satisfying $$\int k(s,t)\phi\_n(s)ds=\lambda\_n \phi\_n(t)$$ and $$k(s,t)=\sum\_{i=1}^{\infty} \lambda\_n\phi\_n(s)\phi\_n(t).$$ Is there a generalized version of this theorem for vector valued random functions? If $X\_t$ is a vector valued process, does there exists a vector valued orthogonal basis $\{\phi\_i\}$ such that the same conclusions hold? Any help or references is appreciated. 



A generalized Mercer's Theorem?
The abstract of the following paper <https://arxiv.org/abs/1110.4017> states:


We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space $X$ into $\mathbb{C}^n$. Given a finite measure $\mu$ on $X,$ we represent the reproducing kernel $K$ as convergent series in terms of the eigenfunctions of a suitable compact operator depending on $K$ and $\mu$. Our result holds under the mild assumption that $K$ is measurable and the associated Hilbert space is separable. Furthermore, we show that $X$ has a natural second countable topology with respect to which the eigenfunctions are continuous and the series representing $K$ uniformly converges to $K$ on any compact subsets of $X\times X,$ provided that the support of $\mu$ is $X.$


It may be worth studying this paper and the references therein. Theorem 3.2 seems to be the main relevant result.


