Wiener Algebras of Operators, and Applications to Pseudodifferential Operators

Abstract

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