Wiener Algebras of Operators, and Applications to Pseudodifferential Operators

Abstract

\newcommand{\sR}{\mathbb R} \newcommand{\sZ}{\mathbb Z} We introduce a Wiener algebra of operators on \( L^2(\sR^N) \) which contains, for example, all pseudodifferential operators in the H\"ormander 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 \( l^2(\sZ^{2N}, \, L^2(\sR^N)) \). 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 \( L^2(\sR^N) \) of pseudodifferential operators in $OP{S}_{0,0}^0$ in terms of their limit operators. Applications to Schr\"odinger operators with continuous potential and other partial differential operators are given.