Schatten classes and traces on compact groups

TL;DR

This paper develops symbolic criteria for classifying invariant operators on compact groups within Schatten classes, providing trace formulas and examples involving sub-Laplacians on Lie groups, advancing understanding of operator properties in noncommutative harmonic analysis.

Contribution

It introduces new symbol-based criteria for Schatten class membership of invariant operators on compact groups, including trace formulas and applications to sub-Laplacians and non-elliptic operators.

Findings

Criteria for Schatten classes via operator symbols on $G\times\hat{G}$.

Trace formula established for Schatten class $S_1$ operators.

Applications to sub-Laplacians and non-elliptic operators on compact Lie groups.

Abstract

In this paper we present symbolic criteria for invariant operators on compact topological groups $G$ characterising the Schatten-von Neumann classes $S_{r}(L^{2}(G))$ for all $0<r\leq\infty$. Since it is known that for pseudo-differential operators criteria in terms of kernels may be less effective (Carleman's example), our criteria are given in terms of the operators' symbols defined on the noncommutative analogue of the phase space $G\times\hat{G}$, where $G$ is a compact topological (or Lie) group and $\hat{G}$ is its unitary dual. We also show results concerning general non-invariant operators as well as Schatten properties on Sobolev spaces. A trace formula is derived for operators in the Schatten class $S_{1}(L^{2}(G))$. Examples are given for Bessel potentials associated to sub-Laplacians (sums of squares) on compact Lie groups, as well as for powers of the sub-Laplacian and for other non-elliptic operators on SU(2)$\simeq\mathbb S^3$ and on SO(3).