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)