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

Vishvesh Kumar; Shyam Swarup Mondal

paper

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

arXiv:1911.10554

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 compact 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 symbolic characterization for $r$ -nuclear, $0 < r \leq 1,$ pseudo-differential operators on $L^{p} (G / H)$ -space with applications to adjoint, product and trace formulae. The criteria here are given in terms of the concept of matrix-valued symbols defined on noncommutative analogue of phase space $G / H \times G / H .$ Finally, we present applications of aforementioned results in the context of heat kernels.