Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators

Let $t\mapsto A(t)$ for $t\in T$ be a $C^M$-mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here $C^M$ stands for $C^{\text{om}}$ (real analytic), a quasianalytic or non-quasianalytic Denjoy-Carleman class, $C^{\mathrm{\infty }}$, or a H"older continuity class $C^{0,\text{al}}$. The parameter domain $T$ is either $\mathbb{R}$ or ${\mathbb{R}}^n$ or an infinite dimensional convenient vector space. We prove and review results on $C^M$-dependence on $t$ of the eigenvalues and eigenvectors of $A(t)$.