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

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