Kernel and symbol criteria for Schatten classes and r-nuclearity on compact manifolds

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 invariant operator and its full symbol associated with an elliptic operator are introduced. Some applications to the study of r-nuclearity on L-P spaces are also obtained. (C) 2014 The Authors. Published by Elsevier Masson SAS on behalf of Academie des sciences.