Effective perturbation theory for linear operators

TL;DR

This paper introduces a new perturbation theory approach for linear operators with isolated eigenvalues, providing explicit bounds for perturbation size and eigendata variations using the Implicit Function Theorem, applicable in general Banach spaces.

Contribution

It develops a novel method based on differential inequalities for spectral perturbation analysis, offering explicit bounds without assumptions on the Banach space.

Findings

Derived explicit radius bounds for perturbations

Established regularity bounds for eigendata variations

Applied results to Markov chains and transfer 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{\'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.