On admissible translations of weighted Gaussian measures

A family of Gaussian measures $\{\mu_{\Sigma\alpha_iA_i}(dx)\}$ in a Hilbert space $H$ with characteristic functional $\exp\Bigr\{-\frac12\sum_{i=1}^m\alpha_i(A_iz,z)\Bigr\}$ and their weighted sums (1) is considered. The set of admissible translations of (1) (as well as each $\mu_{\Sigma\alpha_iA_i}$) is shown to coincide with $\Bigl(\sum_{i=1}^mA_i\Bigr)^{1/2}H$. For $m=1$ an expression for the corresponding density is found.