Weighted periodic and discrete Pseudo-Differential Operators

In this paper, we study elements of symbolic calculus for pseudo-differential operators associated with the weighted symbol class $M_{ρ, Λ}^m(\mathbb{ T}\times \mathbb{Z})$ (associated to a suitable weight function $Λ$ 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 $M_{ρ, Λ}^0(\mathbb{ T}\times \mathbb{Z})$ 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 $T_σ u=f$ in $L^{2}\left(\mathbb{T}\right)$.

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