Discrete Fourier Analysis and spectral properties

We present some recent results on the study of Schatten-von Neumann properties for
 operators on compact manifolds. We will explain the  point of view of kernels and full symbols. In both cases

one relies on a suitable Discrete Fourier analysis depending on the domain.

We will also discuss about operators on $L^p$ spaces by using the notion of nuclear operator in the sense of

Grothendieck and deduce Grothendieck-Lidskii trace formulas in terms of the matrix-symbol. We present examples

 for fractional powers of differential operators.  (Joint work with Michael Ruzhansky)