Title: Re-parameterizing and reducing families of normal operators,
Abstract: We present a new proof of results of Kurdyka & Paunescu, and of Rainer, about
real-analytic multi-parameters generalizations of classical results by Rellich
and Kato about the reduction in families of univariate deformations of normal
operators over real or complex vector spaces of finite dimensions.
Given a real analytic family of normal operators over a finite dimensional
real or complex vector space, there exists a locally finite composition of
blowings-up with smooth centers re-parameterizing the given family such that at
each point of the source space of the re-parameterizing mapping, there exists a
neighbourhood of any given point over which exists a real analytic orthonormal
frame in which the pull back of the operator is in reduced form at every point
of the neighbourhood.
A free by-product of our proof is the local real analyticity of the
eigen-values, which in all prior works was a prerequisite step to get local
regular reducing bases.