ELLIPTICITY AND FREDHOLMNESS OF PSEUDO-DIFFERENTIAL OPERATORS ON l2(Zn)

The minimal operator and the maximal operator of an elliptic pseudo-differential operator with symbols on Z(n) x 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). The index of an elliptic pseudo-differential operator on Z(n) is also computed.

Automated summary

This article proves the coincidence of the minimal operator and the maximal operator of an elliptic pseudo-differential operator with symbols on Z(n)×T-n, and provides the domain in terms of a Sobolev space. It also demonstrates the equivalence of ellipticity and Fredholmness for pseudo-differential operators on Z(n) and computes the index of an elliptic pseudo-differential operator on Z(n).