in DaPrato/Zabczyk's book "Second Order Partial Differential Equations in Hilbert Spaces", there is a useful proposition (Prop. 1.2.8) about a particular calculation of a Gaussian integral in Hilbert spaces: Take a symmetric operator $M$ in a Hilbert space, a centred Gaussian measure $N_Q$ on $H$ with covariance operator $Q$ and assume $ \langle Q^\frac{1}{2} MQ^\frac{1}{2}u,u\rangle < \langle u , u\rangle$ for all $u\neq 0$. Then for $b\in H$,

$$\int_H\exp\left\{\frac{1}{2}\langle M y,y\rangle + \langle b,y\rangle\right \}N_Q(d y) = \frac{\exp\left\{\frac{1}{2}|(1-Q^\frac{1}{2}MQ^\frac{1}{2})^{-\frac{1}{2}}Q^\frac{1}{2}b|^2\right\}}{\sqrt{\det (1-Q^\frac{1}{2}MQ^\frac{1}{2})}}.$$

My problem is that I need to do a calculation of the form

$$\int_H \exp\left\{\frac{1}{2}\langle My,y\rangle + \langle b_1 + ib_2, y\rangle\right\}N_Q(dy).$$

This is almost the setting where I could use the formula above, but it cannot hold in the literal sense, with replacing $b = b_1 + b_2 i$, as the left hand side of the formula can (and will be) properly complex, whereas the right hand side is always real.

Does anyone know of a generalization of this formula? I did the calculation in 1d and there I get a *square* on the right hand side (which can be complex) instead of the *squared absolute value* (which will be real).