On the orthogonality of the white noise, given on a ring of operators, relatively to multiplicative shifts

The article deals with a generalized Gaussian measure on the ring of all Hilbert–Shmidt operators $G_H$ on some separable Hilbert space $H$ with the characteristic functional
$$ \varphi(z)=\exp\biggl\{-\frac{1}{2}\langle z,z\rangle\biggr\},\ \text{where}\ \forall u,v\in G_H\colon\langle u,v\rangle=\operatorname{Sp}uv^*. $$
Conditions are studied for $\mu\sim\mu u^{-1}$ where $u\in X_H$, the set of all linear operators on $H$, and $\mu u^{-1}(F)=\mu(u^{-1}F)=\mu(v\colon uv\in F)$ for those Borel sets $F$ on $G_H$ for which this equality makes sense.