Perturbation theory for normal operators

Let $E\ni x\mapsto A(x)$ be a $\mathcal{C}$-mapping with values unbounded normal operators with common domain of definition and compact resolvent. Here $\mathcal{C}$ stands for $C^{\mathrm{\infty }}$, $C^{\omega }$ (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 $\mathcal{C}$-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.