Wiener algebras of operators, and applications to pseudodifferential operators (Q879691)

Summary: We introduce a Wiener algebra of operators on \(L^2(\mathbb{R}^N)\) which contains, for example, all pseudodifferential operators in the Hörmander class \(OPS^0_{0,0}\). 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 \(l^2(\mathbb{Z}^{2N}, \, L^2(\mathbb{R}^N))\). 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 \(L^2(\mathbb{R}^N)\) of pseudodifferential operators in \(OPS^0_{0,0}\) in terms of their limit operators. Applications to Schrödinger operators with continuous potential and other partial differential operators are given.