We define the p-th moment operator of a Hilbert-valued random element and study its discriminative properties for p > 2 under Gaussian mixtures of homoscedastic random variables. We show that the optimal discriminant functional is associated with a spectral subspace of these operators, initially acting as a perturbation, while residing in a joint eigenspace of structurally identical kernels that support independent realizations within a shared functional space. In the empirical case, we investigate how increasing the sample size perturbs the moment operator, leading it to converge toward the optimal discriminant solution in the tail (lower-order eigenelements).