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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 H be the general, reduced Heisenberg group. Our main result establishes the inverse-closedness of a class of integral operators acting on Lp(H), given by the off-diagonal decay of the kernel. As a consequence of this result, we show that if α1I+Sf, where Sf is the operator given by convolution with f, fL1v(H), is invertible in \B(Lp(H)), then (α1I+Sf)1=α2I+Sg, and gL1v(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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