Perturbation of complex polynomials and normal operators

✍ Scribed by Armin Rainer

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 203 KB
• Volume: 282
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200910837

✦ Synopsis

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 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)