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I am looking to answer the question:

If $\mathcal{B}$ is a separable Banach space and $R: \mathcal{B}^*\to\mathcal{B}$ is a symmetric and positive operator, then $\phi: \mathcal{B}^*\to\mathbb{R}, \xi\mapsto \exp({-\frac{<\xi|R\xi>}{2}})$ is the characteristic function of a Gaussian measure $\mu$ on $\mathcal{B}$.

If this is false, then I am happy with answers saying it is true but you need these extra requirements on $\mathcal{B}$ or $R$. I am not interested in the Hilbert case nor the reflexive case.

In Bogachev's book "Gaussian Measures", it is stated that

However, not every nuclear, symmetric and nonnegative operator $S\in\mathcal{L}(X', X)$ is the covariance of a Gaussian measure on X.

This suggests that $R$ must be nuclear too, but reading chapter III of Vakhania's book "Probability distributions on Banach spaces" suggests to me that what I wrote is indeed sufficient due to the separability. This overflow post references the structure theorem, which states that all Gaussian measures on $\mathcal{B}$ form an Abstract Wiener space. Strook's books "Gaussian Measures in Finite and Infinite Dimension" and "Probability theory: an analytic view" didn't help me with the question.

Are there recent papers or modern books discussing this question in more detail?

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  • $\begingroup$ Shouldn't compactness of $R$ be a necessary condition (e.g. by Fernique and the dominated convergence theorem)? $\endgroup$
    – Rhys Steele
    Commented Oct 1 at 7:53
  • $\begingroup$ To find the right conditions, it may help to separate the question into two: (1) Is $\phi$ the characteristic function of a Gaussian cylindrical measure? (2) Is that cylindrical measure a (Radon) measure? $\endgroup$
    – user95282
    Commented Oct 1 at 12:48
  • $\begingroup$ Do you have a response to the answer below? $\endgroup$
    – Iosif Pinelis
    Commented Oct 2 at 15:43

1 Answer 1

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[...] suggests to me that what I wrote is indeed sufficient due to the separability.

This is not true. Indeed, by (say) Theorem 2.3.1 in Bogachev's book, the operator $R$ must be nuclear.

If this is false, then I am happy with answers saying it is true but you need these extra requirements on $\mathcal{B}$ or $R$.

As noted in Remark 3.11.24, with further references there, "the class of all nuclear, symmetric and nonnegative operators between $X'$ and $X$ coincides with the class of the covariance operators of Gaussian measures on $X$ precisely when $X$ is a space of type $2$."

As stated on p. 213 of Vakhania's book, "the [general] problem of d[e]scription of Gaussian measures [in terms of the characteristic functionals] is equivalent to the problem of description of the class of covariance operators of Gaussian measures. In the case of general Banach spaces the above important and interesting problem is still unsolved."

It appears to be unsolved even now.

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