  In this note we present criteria on both symbols and integral kernels
ensuring that the corresponding operators on compact manifolds belong to
Schatten classes. A specific test for nuclearity is established as well as the
corresponding trace formulae. In the special case of compact Lie groups, kernel
criteria in terms of (locally and globally) hypoelliptic operators are also
given. A notion of an invariant operator and its full symbol associated to an
elliptic operator are introduced. Some applications to the study of
$r$-nuclearity on $L^{p}$ spaces are also obtained.
