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

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 ${\mathbb{Z}}^n$.

24 page