Da Prato/Zabczyk "Second Order Partial Differential Equations in Hilbert Spaces" states the following lemma (this is a reformulation of proposition 1.3.11 in their book):
Let $\mu = \mathcal N(0,C)$ be a Gaussian measure on a separable Hilbert space $\mathcal H$.
If $\Gamma \colon \mathcal H \to \mathcal H$ is self-adjoint and such that $C^{1/2}\Gamma C^{1/2}$ is trace class on $\mathcal H$ and $\langle x, C^{1/2}\Gamma C^{1/2}x \rangle < \|x\|^2$ for all $x\in\mathcal H$. Then for $\nu =\mathcal N(0, (C^{-1}-\Gamma)^{-1})$ we have $$ \frac{\mathrm d \mu}{\mathrm d\nu}(u) = \frac{\exp\left(-\frac 12 \langle \Gamma u , u \rangle\right)}{\sqrt{\det(I - C^{1/2}\Gamma C^{1/2})}}. $$
I am interested in a generalization of this result to the Banach space case, i.e. when $\mathcal H$ is replaced by a separable Banach space $X$ and $\mu$ being a measure on this space. Does anyone know a reference for that or whether this can be done at all?
