Equivalent Gaussian measures

Question

Let $\mu$ be a gaussian measure with eigenpair $\{e_k,2^{-k}\}$ and $\nu$ with eigenpair $\{ Te_k,2^{-k}\}$. Here, $T$ is the unitary operator given by $Tx = x - 2\left\langle x,v \right\rangle v$. Now, they are absolutely continuous with respect to each other if $I - G^{-1/2}G'G^{-1/2}$ is trace class where $G$ and $G'$ are covariance operators of $\mu$ and $\nu$ respectively (according to Kuo). We can see that it happens for any value of $v$. But, if $\mu$ and $\nu$ are absolutely continuous wrt each other, they should have the same Cameron-Martin space. That is clearly not true for all values of $v$. What is going wrong here? Also, Kuo does state other conditions but they are satisfied if $I - G^{-1/2}G'G^{-1/2}$ is trace class.

Answer 1

Did you check that your operator $I-G^{-1/2}G'G^{-1/2}$ is of the trace class? I did some calculations and for me it is not always of the trace-class, just for $n$ from the Cameron-Martin space of $\mu$.

p.s. I used to watch the lectures of Ecole Normale of Paris by the italian mathematician Guiseppe Da Prato and also his book ``Second Order Partial Differential Equations in Hilbert Spaces''. Requirment there, is the operator above should be of the Hilbert-Schmidt class (however the trace class is enough).

p.s. Also I never heard that criterion for equivalence of Gaussian Measures, is the equivalence of Cameron-Martin Spaces.