Perturbation of Complex Polynomials and Normal Operators

We study the regularity of the roots of complex monic polynomials P (t) of degree n depending smoothly on a real parameter t. If P (t) is C∞ and no two of the continuously chosen roots meet of infinite order of flatness, then there exists a locally absolutely continuous parameterization of the roots. Simple examples show that the conclusion is best possible. This result will follow from the proposition that for any t0 there exists a positive integer N such that t 7→ P (±(t−t0) ) admits smooth parameterizations of its roots near t0. Provided that P (t) is C, the roots may be parameterized differentiably if and only if whenever roots meet they meet of order at least 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 C∞ curve of such operators can be arranged locally in an absolutely continuous way, provided that no two of the eigenvalues meet of infinite order of flatness.