A Characterization of Compact Weighted SG Pseudo-Differential Operators

In this paper, we first prove a version of the Gohberg’s lemma for pseudo-differetial operators with weighted SG M-symbol. Then by using Gohberg’s lemma, we give a necessary and sufficient condition for SG pseudo-differential operators with symbol in class \(M_{\rho ,\Lambda }^{0,0}\) to be compact from \(L^2(\mathbb {R}^n)\) into \(L^2(\mathbb {R}^n)\) .