Perturbation theory for normal operators

TL;DR

This paper extends and refines the theory of how eigenvalues and eigenvectors of normal operators depend smoothly or analytically on parameters, covering a broad class of regularity and infinite-dimensional settings.

Contribution

It generalizes existing results from self-adjoint to normal operators, providing a complete description of parameter dependence and demonstrating optimality of these results.

Findings

Complete description of eigenvalue and eigenvector dependence on parameters.

Extension of results from self-adjoint to normal operators.

Improved and optimal regularity results for eigen-structures.

Abstract

Let $E \ni x\mapsto A(x)$ be a $\mathscr{C}$-mapping with values unbounded normal operators with common domain of definition and compact resolvent. Here $\mathscr{C}$ stands for $C^\infty$, $C^\omega$ (real analytic), $C^{[M]}$ (Denjoy--Carleman of Beurling or Roumieu type), $C^{0,1}$ (locally Lipschitz), or $C^{k,\alpha}$. The parameter domain $E$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We completely describe the $\mathscr{C}$-dependence on $x$ of the eigenvalues and the eigenvectors of $A(x)$. Thereby we extend previously known results for self-adjoint operators to normal operators, partly improve them, and show that they are best possible. For normal matrices $A(x)$ we obtain partly stronger results.