Weighted periodic and discrete pseudo-differential Operators

Abstract

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 $L^{2}\left( {\mathbb {T}}\right)$.