Schatten classes and traces on compact groups

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 $0 \lt r \leq \infty$. 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 $SO(3)$.

Keywords

compact Lie groups, topological groups, pseudodifferential operators, eigenvalues, trace formula, Schatten classes

2010 Mathematics Subject Classification

Primary 35S05. Secondary 22E30, 43A75, 47B06.

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The first author was supported by Marie Curie IIF 301599 and by the Leverhulme Grant RPG-2014-02. The second author was supported by EPSRC Leadership Fellowship EP/G007233/1 and by EPSRC Grant EP/K039407/1. No new data was collected or generated during the course of the research.

Received 15 March 2013

Accepted 5 February 2016

Published 9 November 2017