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Zeitschrift für Analysis und ihre Anwendungen


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Volume 23, Issue 3, 2004, pp. 437–482
DOI: 10.4171/ZAA/1207

Published online: 2004-09-30

Wiener Algebras of Operators, and Applications to Pseudodifferential Operators

Vladimir S. Rabinovich[1] and Steffen Roch[2]

(1) Escuelo Superior de Mat y Fis del IPN, México, D.f., Mexico
(2) Technische Hochschule Darmstadt, Germany

\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.

Keywords: Wiener algebra, pseudodifferential operator, limit operator, Fredholmness

Rabinovich Vladimir S., Roch Steffen: Wiener Algebras of Operators, and Applications to Pseudodifferential Operators. Z. Anal. Anwend. 23 (2004), 437-482. doi: 10.4171/ZAA/1207