Symbolic calculus and M-ellipticity of pseudo-differential operators on ℤn

Abstract

In this paper, we introduce and study a class of pseudo-differential operators on the lattice [Formula: see text]. More preciously, we consider a weighted symbol class [Formula: see text] associated to a suitable weight function [Formula: see text] on [Formula: see text]. We study elements of the symbolic calculus for pseudo-differential operators associated with [Formula: see text] by deriving formulae for the composition, adjoint and transpose. We define the notion of [Formula: see text]-ellipticity for symbols belonging to [Formula: see text] and construct the parametrix of [Formula: see text]-elliptic pseudo-differential operators. Further, we investigate the minimal and maximal extensions for [Formula: see text]-elliptic pseudo-differential operators and show that they coincide on [Formula: see text] subject to the [Formula: see text]-ellipticity of symbols. We also determine the domains of the minimal and maximal operators. Finally, we discuss Fredholmness and compute the index of [Formula: see text]-elliptic pseudo-differential operators on [Formula: see text].