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?