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 Zn. More preciously, we consider a weighted symbol class M-rho Lambda(m) (Z(n) xT(n)), m is an element of R associated to a suitable weight function Lambda on Z(n). We study elements of the symbolic calculus for pseudo-differential operators associated with M-rho Lambda(m) (Z(n) xT(n)) by deriving formulae for the composition, adjoint and transpose. We define the notion of M-ellipticity for symbols belonging to M-rho Lambda(m) (Z(n) xT(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).