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 : F → F 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 : Ñ → N, 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 L ◦ σ: σ*F → σ*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.
This is a preview of subscription content, log in to check access.
A. Belotto, E. Bierstone, V. Grandjean and P. Milman, Resolution of singularities of the cotangent sheaf of a singular variety, Advances in Mathematics 307 (2017), 780–832.MathSciNetCrossRefzbMATHGoogle Scholar
E. Bierstone and P. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Inventiones Mathematicae 128 (1997), 207–302.MathSciNetCrossRefzbMATHGoogle Scholar
H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Annals of Mathematics 79 (1964), 109–203; 79 (1964), 205–326.MathSciNetCrossRefzbMATHGoogle Scholar
T. Kato, Perturbation Theory for Linear Operators, Grundlehren der Mathematischen Wissenschaften, Vol. 132, Springer-Verlag, Berlin–New York, 1976.Google Scholar