We consider Gaussian measures $μ, \tildeμ$ on a separable Hilbert space, with fractional-order covariance operators $A^{-2β}$ resp. $\tilde{A}^{-2\tildeβ}$, and derive necessary and sufficient conditions on $A, \tilde{A}$ and $β, \tildeβ > 0$ for I. equivalence of the measures $μ$ and $\tildeμ$, and II. uniform asymptotic optimality of linear predictions for $μ$ based on the misspecified measure $\tildeμ$. These results hold, e.g., for Gaussian processes on compact metric spaces. As an important special case, we consider the class of generalized Whittle-Matérn 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-Matérn fields.