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 $OPS^0_{0,0}$. 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 $l^2(\sZ^{2N}, \, L^2(\sR^N))$. 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 $OPS^0_{0,0}$ in terms of their limit operators. Applications to Schr\"odinger operators with continuous potential and other partial differential operators are given.