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{e}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.
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