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Let $X$ be a topological linear space over $\mathbb R$ which is complete and Hausdorff with a dual space that separates points. Let $k : X^* \to X$ be an arbitrary covariance operator. i.e., any continuous linear map which is symmetric and nonnegative-definite: $\psi[k\varphi] = \varphi[k\psi]$ and $\varphi[k\varphi] \ge 0$.

Suppose that the set $kX^* \subseteq X$ is separable.

Does there exist a probability measure $\mathbb P$ on $X$ which has $k$ as its covariance operator?

Theorem 4.4.6 of Vakhania (1981) answers this question in the affirmative when the space $X$ is separable and reflexive, but I do not know the answer in the non-reflexive case.

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    $\begingroup$ Note that there may not exist a Gaussian measure $\mathbb P$-- such operators are called "Gaussian covariance operators", not to be confused with so-called Gaussian kernels of the form $c(i,i') = \operatorname{exp}( -\|i-i'\|^2 / 2\sigma^2 )$. $\endgroup$
    – Tom LaGatta
    Commented Feb 1, 2014 at 0:57

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