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

  • Author(s): Farrell, Brendan
  • Strohmer, Thomas
  • et al.

Published Web Location

https://arxiv.org/pdf/math/0612033.pdf
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Abstract

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.

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