Journal of Operator Theory
Volume 81, Issue 1, Winter 2019 pp. 175-194.
Effective perturbation theory for simple isolated eigenvalues of linear operators
Authors:
Benoit R. Kloeckner
Author institution: Universite Paris-Est, Laboratoire d'Analyse et de Matematiques Appliquees
(UMR 8050), UPEM, UPEC, CNRS, F-94010, Creteil, France
Summary: We propose a new approach to the spectral theory of perturbed linear operators
in the case of a simple isolated eigenvalue. We obtain two kinds 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-Esseen
inequalities with explicit constants, and on the other hand we apply the
radius bounds to transfer operators of intermittent maps, obtaining explicit
high-temperature regimes where a spectral gap occurs.
DOI: http://dx.doi.org/10.7900/jot.2017dec22.2179
Keywords: bounded operators, simple isolated eigenvalues, perturbations, spectral gap, effective inequalities
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