JournalszaaVol. 23, No. 3pp. 437–482

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
Download PDF

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.

Cite this article

Vladimir S. Rabinovich, Steffen Roch, Wiener Algebras of Operators, and Applications to Pseudodifferential Operators. Z. Anal. Anwend. 23 (2004), no. 3, pp. 437–482

DOI 10.4171/ZAA/1207