The problem consists in the study of spectral properties by investigating the membership in Schatten-von Neumann ideals and in the ideal of nuclear operators in the sense of Grothendieck. We consider the analysis from two points of view: symbols and kernels. For the analysis with kernels the main tool used has been the factorisation method. For the study of the trace in terms of kernels it is customary to introduce a type of Hardy-Littlewood maximal function in order to average the kernel along the diagonal.
The main idea in the application of the factorisation method can be summarised in the following way:
If we know spectral properties of an operator and we know how it acts on the integral kernel of an integral operator , we can draw conclusions about the spectral properties of .
As applications of the obtained sharp conditions 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 of finite or infinite measure, and conditions for operators on 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.
Delgado J., Ruzhansky M., Schatten-von Neumann classes of integral operators, arxiv