  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.
