Wiener algebras of operators, and applications to pseudodifferential operators

We introduce a Wiener algebra of operators on L²(ℝ ^N) which contains, for example, all pseudodifferential operators in the Hörmander class OPS_0,0⁰. A discretization based on the action of the discrete Heisenberg group associates to each operator in this algebra a band-dominated operator in a Wiener algebra of operators on l²(ℤ^2N, L²(ℝ^N)). The (generalized) Fredholmness of these discretized operators can be expressed by the invertibility of their limit operators. This implies a criterion for the Fredholmness on L²(ℝ^N) of pseudodifferential operators in OPS_0,0⁰ in terms of their limit operators. Applications to Schrödinger operators with continuous potential and other partial differential operators are given.