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

In this paper, we introduce and study a class of pseudo-differential operators on the lattice ℤ n . More preciously, we consider a weighted symbol class M ρ , Λ m (ℤ n × n) , m ∈ ℝ associated to a suitable weight function Λ on ℤ n . We study elements of the symbolic calculus for pseudo-differential operators associated with M ρ , Λ m (ℤ n × n) by deriving formulae for the composition, adjoint and transpose. We define the notion of M -ellipticity for symbols belonging to M ρ , Λ m (ℤ n × 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 ℓ 2 (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 ℤ n . [ABSTRACT FROM AUTHOR]