In this paper, we study elements of symbolic calculus for pseudo-differential operators associated with the weighted symbol class \(M_{\rho , \Lambda }^m({\mathbb {T}}\times {\mathbb {Z}})\) (associated to a suitable weight function \(\Lambda \) on \({\mathbb {Z}}\) ) by deriving formulae for the asymptotic sums, composition, adjoint, transpose. We also construct the parametrix of M-elliptic pseudo-differential operators on \({\mathbb {T}}\) . Further, we prove a version of Gohberg’s lemma for pseudo-differetial operators with weighted symbol class \(M_{\rho , \Lambda }^0({\mathbb {T}}\times {\mathbb {Z}})\) and as an application, we provide a sufficient and necessary condition to ensure that the corresponding pseudo-differential operator is compact on \(L^2({\mathbb {T}})\) . Finally, we provide Gårding’s and Sharp Gårding’s inequality for M-elliptic operators on \({\mathbb {Z}}\) and \({\mathbb {T}}\) , respectively, and present an application in the context of strong solution of the pseudo-differential equation \(T_{\sigma } u=f\) in \(L^{2}\left( {\mathbb {T}}\right) \) .
