  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 G\r{a}rding's and Sharp G\r{a}rding'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
$L^{2}\left(\mathbb{T}\right)$.
