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 \((A,\,A^{\dagger })\), where \(A^{\dagger }\) is the formal adjoint of A.