Pseudo-differential operators on homogeneous spaces of compact and Hausdorff groups

Let G be a compact Hausdorff group and let H be a closed subgroup of G. We introduce pseudo-differential operators with symbols on the homogeneous space G/H. We present a necessary and sufficient condition on symbols for which these operators are in the class of Hilbert-Schmidt operators. We also give a characterization of and a trace formula for the trace class pseudo-differential operators on the homogeneous space G/H.