Re-parameterizing and reducing families of normal operators

TL;DR

This paper introduces a new proof technique for families of normal operators, showing they can be locally re-parameterized to achieve a simplified form with real analytic eigenvalues, extending classical results.

Contribution

It provides a novel proof of re-parameterization results for families of normal operators, ensuring local real analytic eigenvalues and reducing the operators to a simplified form.

Findings

Existence of local re-parameterizations via blowings-up with smooth centers.

Operators can be reduced to a form with a real analytic orthonormal frame.

Eigenvalues are shown to be locally real analytic as a by-product.

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.