The essential adjointness of pseudo-differential operators on ${\mathbb{Z}}^n$

Abstract

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 $(A,A^{†})$, where $A^{†}$ is the formal adjoint of $A$.

We simplified the proof of theorem 2.7. We added section 6. We added the reference [29]. We corrected a few typos