The Essential Adjointness of Pseudo-Differential Operators on \(\mathbb {Z}^n\)

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