Traces of pseudo-differential operators on \(\mathbb{S }^{n-1}\) (Q1943002)

The authors consider pseudo-differential operators \(P\) on the unit sphere \(\mathbb{S}^{n-1}\) with measurable symbol \(\sigma\). On \(\mathbb{S}^1\), the operator \(P\) is represented by \[ Pf(\lambda)= \sum^{+\infty}_{n=-\infty} e^{in\lambda} \sigma(\lambda, n)\widehat f(n),\quad \lambda\in [-\pi,\pi], \] where \(\widehat f(n)\) are the Fourier coefficients of \(f\), and similarly on \(\mathbb{S}^{n-1}\), by using spherical harmonic polynomials. A characterization of Hilbert-Schmidt and trace class operators \(P\) is given in terms of the symbols \(\sigma\). A precise trace formula is obtained for trace class operators.