Ellipticity and Fredholmness of pseudo-differential operators on ℓ²(ℤⁿ)

Abstract

The minimal operator and the maximal operator of an elliptic pseudo-differential operator with symbols on Z n × T n \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 Z n \mathbb {Z}^n . The index of an elliptic pseudo-differential operator on Z n \mathbb {Z}^n is also computed.