Schatten-von Neumann properties for H\"ormander classes on compact Lie groups

Abstract

Let $G$ be a compact Lie group of dimension $n.$ In this work we characterise the membership of classical pseudo-differential operators on $G$ in the trace class ideal $S_{1}(L^2(G)),$ as well as in the setting of the Schatten ideals $S_{r}(L^2(G)),$ for all $r>0.$ In particular, we deduce Schatten characterisations of elliptic pseudo-differential operators of $(\rho,\delta)$-type for the large range $0\leq \delta<\rho\leq 1.$ Additional necessary and sufficient conditions are given in terms of the matrix-valued symbols of the operators, which are global functions on the phase space $G\times \widehat{G},$ with the momentum variables belonging to the unitary dual $\widehat{G}$ of $G$. In terms of the parameters $(\rho,\delta),$ on the torus $\mathbb{T}^n,$ we demonstrate the sharpness of our results showing the existence of atypical operators in the exotic class $\Psi^{-\varkappa}_{0,0}(\mathbb{T}^n),$ $\varkappa>0,$ belonging to all the Schatten ideals. Additional order criteria are given in the setting of classical pseudo-differential operators. We present also some open problems in this setting.