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

TL;DR

This paper proves the inverse-closedness of a class of integral operators with off-diagonal decay on the Heisenberg group, ensuring their inverses also have similar decay properties, with applications to pseudodifferential operators and communications.

Contribution

It establishes inverse-closedness for integral operators on the Heisenberg group and characterizes inverses of convolution-type operators within the same algebra.

Findings

Inverse-closedness of integral operators with off-diagonal decay on $ ext{Heisenberg group$.

Characterization of inverses of convolution operators as similar operators with kernels in the same function space.

Application of results to Weyl pseudodifferential operators and relevance to mobile communications.

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.