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

TL;DR

This paper investigates the conditions under which pseudo-differential operators on the lattice bZ^n exhibit essential adjointness, extending the theory to discrete Sobolev spaces and symbols on bZ^n bT^n.

Contribution

It establishes a sufficient condition for the essential adjointness of pseudo-differential operators and their formal adjoints on discrete Sobolev spaces over bZ^n.

Findings

Provides a criterion for essential adjointness of operators on bZ^n.

Extends pseudo-differential operator theory to lattice and discrete Sobolev spaces.

Analyzes symbols defined on bZ^n bT^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 $A$.