Ellipticity and Fredholmness of pseudo-differential operators on ${\mathrm{\ell }}^2(\text{Zn})$

The minimal operator and the maximal operator of an elliptic pseudo-differential operator with symbols on ${\mathbb{Z}}^n\times {\mathbb{T}}^n$ are proved to coincide and the domain is given in terms of a Sobolev space. Ellipticity and Fredholmness are proved to be equivalent for pseudo-differential operators on ${\mathbb{Z}}^n$. The index of an elliptic pseudo-differential operator on ${\mathbb{Z}}^n$ is also computed.