  In the setting of the lattice $\mathbb{Z}^n$ we consider a
pseudo-differential operator $A$ whose symbol belongs to a class defined on
$\mathbb{Z}^n\times \mathbb{T}^n$, where $\mathbb{T}^n$ is the $n$-torus. We
realize $A$ as an operator acting between the discrete Sobolev spaces
$H^{s_j}(\mathbb{Z}^n)$, $s_j\in\mathbb{R}$, $j=1,2$, with the discrete
Schwartz space serving as the domain of $A$. We provide a sufficient condition
for the essential adjointness of the pair $(A,\,A^{\dagger})$, where
$A^{\dagger}$ is the formal adjoint of $A$.
