Quantitative spectral stability for operators with compact resolvent

This paper deals with quantitative spectral stability for operators with compact resolvent acting on $L^2(X,m)$, where $(X,m)$ is a measure space. Under fairly general assumptions, we provide a characterization of the dominant term of the asymptotic expansion of the eigenvalue variation in this abstract setting. Many of the results about quantitative spectral stability available in the literature can be recovered by our analysis. Furthermore, we illustrate our result with several applications, e.g. quantitative spectral stability for a Neumann limit of a Robin problem, conformal transformations of Riemannian metrics, Dirichlet forms under the removal of sets of small capacity, and for families of Fourier-multipliers.