  In this work we establish sharp kernel conditions ensuring that the
corresponding integral operators belong to Schatten-von Neumann classes. The
conditions are given in terms of the spectral properties of operators acting on
the kernel. As applications we establish several criteria in terms of different
types of differential operators and their spectral asymptotics in different
settings: compact manifolds, operators on lattices, domains in ${\mathbb R}^n$
of finite measure, and conditions for operators on ${\mathbb R}^n$ given in
terms of anharmonic oscillators. We also give examples in the settings of
compact sub-Riemannian manifolds, contact manifolds, strictly pseudo-convex CR
manifolds, and (sub-)Laplacians on compact Lie groups.
