  We consider Gaussian measures $\mu, \tilde{\mu}$ on a separable Hilbert
space, with fractional-order covariance operators $A^{-2\beta}$ resp.
$\tilde{A}^{-2\tilde{\beta}}$, and derive necessary and sufficient conditions
on $A, \tilde{A}$ and $\beta, \tilde{\beta} > 0$ for I. equivalence of the
measures $\mu$ and $\tilde{\mu}$, and II. uniform asymptotic optimality of
linear predictions for $\mu$ based on the misspecified measure $\tilde{\mu}$.
These results hold, e.g., for Gaussian processes on compact metric spaces. As
an important special case, we consider the class of generalized
Whittle-Mat\'ern Gaussian random fields, where $A$ and $\tilde{A}$ are elliptic
second-order differential operators, formulated on a bounded Euclidean domain
$\mathcal{D}\subset\mathbb{R}^d$ and augmented with homogeneous Dirichlet
boundary conditions. Our outcomes explain why the predictive performances of
stationary and non-stationary models in spatial statistics often are
comparable, and provide a crucial first step in deriving consistency results
for parameter estimation of generalized Whittle-Mat\'ern fields.
