Journal of Operator Theory

Volume 64, Issue 1, Summer 2010 pp. 189-205.

Inverse-Closedness of a Banach Algebra of Integral Operators on the Heisenberg Group

Authors: Brendan Farrell (1) and Thomas Strohmer (2)
Author institution: (1) Heinrich-Hertz-Lehrstuhl fuer Informationstheorie und Theoretische Informationstechnik, Technische Universitaet Berlin, Einsteinufer 25, 10587 Berlin, Germany
(2) Departmentof Mathematics, University of California,Davis, CA 95616, U.S.A

Summary: Let $\mathbb{H}$ be the general, reduced Heisenberg group. Our main result establishes the inverse-closedness of a class of integral operators acting on $L^{p}(\mathbb{H})$, given by the off-diagonal decay of the kernel. As a consequence of this result, we show that if $\alpha_{1}I+S_{f}$, where $S_{f}$ is the operator given by convolution with $f$, $f\in L^{1}_{v}(\mathbb{H})$, is invertible in $\B(L^{p}(\mathbb{H}))$, then $(\alpha_{1}I+S_{f})^{-1}=\alpha_{2}I+S_{g}$, and $g\in L^{1}_{v}(\mathbb{H})$. We prove analogous results for twisted convolution operators and apply the latter results to a class of Weyl pseudodifferential operators. We briefly discuss relevance to mobile communications.

Keywords: Heisenberg group, integral operator, twisted convolution, pseudodifferential operator.

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