Perturbation of complex polynomials and normal operators

Abstract

We study the regularity of the roots of complex monic polynomials P (t) of fixed degree depending smoothly on a real parameter t. We prove that each continuous parameterization of the roots of a generic C^∞ curve P (t) (which always exists) is locally absolutely continuous. Generic means that no two of the continuously chosen roots meet of infinite order of flatness. Simple examples show that one cannot expect a better regularity than absolute continuity. This result will follow from the proposition that for any t₀ there exists a positive integer N such that t ↦ P (t₀ ± (t – t₀)^N) admits smooth parameterizations of its roots near t₀. We show that Cⁿ curves P (t) (where n = deg P) admit differentiable roots if and only if the order of contact of the roots is ≥ 1. We give applications to the perturbation theory of normal matrices and unbounded normal operators with compact resolvents and common domain of definition: The eigenvalues and eigenvectors of a generic C^∞ curve of such operators can be arranged locally in an absolutely continuous way (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)