  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 true under a
weighted condition, where the perturbation is symmetrically normalized by the
square-root of the reduced resolvent. This weighted condition originates in
random perturbations where it leads to significant improvements.
