A standard perturbation result states that perturbed eigenvalues and eigenprojections admit a perturbation series provided that the operator norm of the perturbation is smaller than a constant times the corresponding eigenvalue isolation distance. In this paper, we show that the same holds under a weighted condition, where the perturbation is symmetrically normalized by the square root of the reduced resolvent. This weighted condition originates from random perturbations where it leads to significant improvements. As an application, we derive higher-order perturbation expansions for the empirical covariance operator. For i.i.d. sub-Gaussian observations in a Hilbert space, we establish upper and lower error bounds for empirical eigenvalues and eigenprojections, which are also applicable in high-dimensional regimes.