Schatten class and nuclear pseudo-differential operators on homogeneous spaces of compact groups

Abstract

Given a compact (Hausdorff) group G and a closed subgroup H of G, in this paper we present symbolic criteria for pseudo-differential operators on the compact homogeneous space G/H characterizing the Schatten–von Neumann classes \(S_r(L^2(G/H))\) for all \(0<r \le \infty .\) We go on to provide a symbolic characterization for r-nuclear, \(0< r \le 1,\) pseudo-differential operators on \(L^{p}(G/H)\) with applications to adjoint, product and trace formulae. The criteria here are given in terms of matrix-valued symbols defined on noncommutative analogue of phase space \(G/H \times \widehat{G/H}.\) Finally, we present an application of aforementioned results in the context of the heat kernels.