I'm currently studying Gaussian measures on Hilbert spaces and would like to find conditions under which convergence to a Gaussian measure with respect to a coarser topology induced by a covariance operator implies weak convergence with respect to the original topology. This question is an asymptotic version of this related question, where it was shown that if two measures agree on such a coarser topology, then they also agree on the original topology.
Full details are as follows:
Let $H$ be a real separable Hilbert space and $S$ a bijective, non-negative, self-adjoint trace-class operator on $H$ (for example the covariance operator of some Gaussian measure on $H$). Denote the inner product by $\left\langle \cdot,\cdot \right\rangle$ and define a new inner product $\left\langle \cdot,\cdot \right\rangle_1$ on $H$ by $$ \left\langle x,y \right\rangle_1 := \left\langle \sqrt{S} x, \sqrt{S} y \right\rangle, \qquad x,y \in H. $$ This inner product is weaker than $\left\langle \cdot,\cdot \right\rangle$, as for the induced norms we have that $\left\lVert \cdot \right\rVert_1 \leq \left\lVert T \right\rVert_{op} \left\lVert \cdot \right\rVert$. Let $H_1$ be the completion of $H$ with respect to $\left\lVert \cdot \right\rVert_1$. Then $H_0 \subseteq H_1$ densely and as a Borel set.
Let $\mathcal{B}(H)$ and $\mathcal{B}(H_1)$ denote the Borel sets of $H$ and $H_1$, with respect to the norms $\left\lVert \cdot \right\rVert$ and $\left\lVert \cdot \right\rVert_1$,respectively. Then $\mathcal{B}(H_1) \cap H \subseteq \mathcal{B}(H)$, where $\mathcal{B}(H_1) \cap H = \{ A \cap H \colon A \in \mathcal{B}(H_1) \}$.
Finally, let $\nu_n$ be a sequence of probability measures on $(H,\mathcal{B}(H))$ with trace class covariance operators and $\mu$ be a Gaussian measure on $(H,\mathcal{B}(H))$. Then $\mu$ is completely determined by its values on $\mathcal{B_1}\cap H$ (see the above linked question).
My question is: If $\nu_n$ converges weakly on $(H,\mathcal{B}(H_1)\cap H)$ to $\mu$ ("weakly" in the probabilistic sense, i.e. with respect to real-valued, bounded continuous functions), does this convergence also hold on $(H,\mathcal{B}(H))$, maybe under some additional conditions on the covariance operator defining the weaker inner product?
