JournalszaaVol. 23 , No. 3DOI 10.4171/zaa/1207

Wiener Algebras of Operators, and Applications to Pseudodifferential Operators

  • Vladimir S. Rabinovich

    Escuelo Superior de Mat y Fis del IPN, México, D.f., Mexico
  • Steffen Roch

    Technische Hochschule Darmstadt, Germany
Wiener Algebras of Operators, and Applications to Pseudodifferential Operators cover

Abstract

\newcommand{\sR}{\mathbb R} \newcommand{\sZ}{\mathbb Z} We introduce a Wiener algebra of operators on L2(\sRN)L^2(\sR^N) which contains, for example, all pseudodifferential operators in the H\"ormander class OPS0,00OPS^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 l2(\sZ2N,L2(\sRN))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 L2(\sRN)L^2(\sR^N) of pseudodifferential operators in OPS0,00OPS^0_{0,0} in terms of their limit operators. Applications to Schr\"odinger operators with continuous potential and other partial differential operators are given.