Quantitative spectral stability for compact operators

This paper deals with quantitative spectral stability for compact operators 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 Robin to Neumann problem, conformal transformations of Riemann metrics, Dirichlet forms under the removal of sets of small capacity, and for families of pseudo-differentials operators.