Title: Effective perturbation theory for linear operators,
Abstract: We propose a new approach to the spectral theory of perturbed linear
operators , in the case of a simple isolated eigenvalue. We obtain two kind of
results: "radius bounds" which ensure perturbation theory applies for
perturbations up to an explicit size, and "regularity bounds" which control the
variations of eigendata to any order. Our method is based on the Implicit
Function Theorem and proceeds by establishing differential inequalities on two
natural quantities: the norm of the projection to the eigendirection, and the
norm of the reduced resolvent. We obtain completely explicit results without
any assumption on the underlying Banach space. In companion articles, on the
one hand we apply the regularity bounds to Markov chains, obtaining
non-asymptotic concentration and Berry-Ess{é}en inequalities with explicit
constants, and on the other hand we apply the radius bounds to transfer
operator of intermittent maps, obtaining explicit high-temperature regimes
where a spectral gap occurs.