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Re-parameterizing and reducing families of normal operators

  • Vincent GrandjeanEmail author
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Abstract

We present a new proof of (a slight generalization of) recent results of Kurdyka and Paunescu, and of Rainer, which are multi-parameter versions of classical theorems of 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 normal operator L : FF over a connected real analytic real or complex vector bundle F of finite rank equipped with a fibered metric structure (Euclidean or Hermitian), there exists a locally finite composition of blowings-up of the basis manifold N, with smooth centers, σ : ÑN, such that at each point \(\widetilde y\) of the source manifold Ñ it is possible to find a neighborhood of \(\widetilde y\) over which there exists a real analytic orthonormal/unitary frame in which the pulled-back operator Lσ: σ*Fσ*F is in reduced form. We are working only with the eigenbouquet bundle of the operator and a free by-product of our proof is the local real analyticity of the eigenvalues, which in all prior works was a prerequisite step to get local regular reducing bases.

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© Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal do Ceará (UFC) Campus do PiciFortalezaBrasil

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