Perturbation of complex polynomials and normal operators

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 t 0 there exists a positive integer N such that t ↦ P ( t 0 ± ( t – t 0 ) N ) admits smooth parameterizations of its roots near t 0 . We show that C n 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)