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

TL;DR

This paper develops symbolic criteria for classifying pseudo-differential operators on compact homogeneous spaces of groups within Schatten and nuclear classes, with applications to heat kernels and operator properties.

Contribution

It introduces new symbolic characterizations of Schatten-von Neumann and nuclear pseudo-differential operators on homogeneous spaces, extending operator theory on noncommutative phase spaces.

Findings

Criteria for Schatten class membership of operators.

Characterization of r-nuclear operators on L^p spaces.

Applications to heat kernel analysis.

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 $G/H \times \widehat{G/H}.$ Finally, we present applications of aforementioned results in the context of heat kernels.