The Essential Adjointness of Pseudo-Differential Operators on Zn

Abstract

In the setting of the lattice Zn we consider a pseudo-differential operator A whose symbol belongs to a class defined on Zn×Tn, where Tn is the n-torus. We realize A as an operator acting between the discrete Sobolev spaces Hsj(Zn), sj∈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.