Symbolic calculus and $M$-ellipticity of pseudo-differential operators on $\mathbb{Z}^n$

TL;DR

This paper develops a symbolic calculus for a class of pseudo-differential operators on the lattice $ Z^n$, introducing $M$-ellipticity, and analyzing their properties including parametrices, extensions, and Fredholmness.

Contribution

It introduces the concept of $M$-ellipticity for pseudo-differential operators on $ Z^n$ and constructs their parametrices, extending the theory of elliptic operators to the lattice setting.

Findings

Defined $M$-ellipticity for lattice pseudo-differential symbols

Constructed parametrices for $M$-elliptic operators

Analyzed the Fredholm properties and computed indices

Abstract

In this paper, we introduce and study a class of pseudo-differential operators on the lattice $\mathbb{Z}^n$. More preciously, we consider a weighted symbol class $M_{\rho, \Lambda}^m(\mathbb{ Z}^n\times \mathbb{T}^n), m\in \mathbb{R}$ associated to a suitable weight function $\Lambda$ on $\mathbb{ Z }^n$. We study elements of the symbolic calculus for pseudo-differential operators associated with $M_{\rho, \Lambda}^m(\mathbb{ Z}^n\times \mathbb{T}^n)$ by deriving formulae for the composition, adjoint, transpose. We define the notion of $M$-ellipticity for symbols belonging to $M_{\rho, \Lambda}^m(\mathbb{ Z}^n\times \mathbb{T}^n)$ and construct the parametrix of $M$-elliptic pseudo-differential operators. Further, we investigate the minimal and maximal extensions for $M$-elliptic pseudo-differential operators and show that they coincide on $\ell^2(\mathbb{Z}^n)$ subject to the $M$-ellipticity of symbols. We also determine the domains of the minimal and maximal operators. Finally, We discuss Fredholmness and compute the index of $M$-elliptic pseudo-differential operators on $\mathbb{Z}^n$.