Effective perturbation theory for linear operators
Benoît Kloeckner
TL;DR
The paper addresses analytic perturbation theory for a simple isolated eigenvalue of a bounded operator on a Banach space, providing explicit, non-asymptotic radius bounds and higher-order regularity bounds for eigendata. It develops a direct Banach-space approach using the Implicit Function Theorem and a differential-inequality framework on the pair of quantities $tau_L$ and $gamma_L$, yielding uniform analyticity of eigenvalues, eigenvectors, and eigenforms with explicit derivative formulas and remainder estimates up to third order. The main contributions include a sharp radius bound $r_0 = 1/(6\tau_0\gamma_0)$ guaranteeing an Analytic Simple Isolated Eigenvalue (ASIE), explicit first- through third-order Taylor expansions with finite remainders, and quantitative spectral-gap stability results under perturbations with practical corollaries. These results apply to non-self-adjoint operators such as transfer operators and Markov averaging operators, enabling effective non-asymptotic conclusions in dynamical systems and probability, including Berry-Esseen-type concentration bounds. The methodology emphasizes a direct operator-theoretic route over contour-integral methods, offering transparent constants and scalability to higher-order derivatives through a simple differential-inequality comparison principle.
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.