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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The range of localization operators and lifting theorems for modulation and Bargmann-Fock spaces


Authors: Karlheinz Gröchenig and Joachim Toft
Journal: Trans. Amer. Math. Soc. 365 (2013), 4475-4496
MSC (2010): Primary 46B03, 42B35, 47B35
Published electronically: January 4, 2013
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Abstract: We study the range of time-frequency localization operators acting on modulation spaces and prove a lifting theorem. As an application we also characterize the range of Gabor multipliers, and, in the realm of complex analysis, we characterize the range of certain Toeplitz operators on weighted Bargmann-Fock spaces. The main tools are the construction of canonical isomorphisms between modulation spaces of Hilbert-type and a refined version of the spectral invariance of pseudodifferential operators. On the technical level we prove a new class of inequalities for weighted gamma functions.


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Additional Information

Karlheinz Gröchenig
Affiliation: Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria
Email: karlheinz.groechenig@univie.ac.at

Joachim Toft
Affiliation: Department of Computer Science, Physics and Mathematics, Linnæus University, Växjö, Sweden
Email: joachim.toft@lnu.se

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05836-9
PII: S 0002-9947(2013)05836-9
Keywords: Localization operator, Toeplitz operator, Bargmann-Fock space, modulation space, Sjöstrand class, spectral invariance, Hermite function
Received by editor(s): October 3, 2011
Received by editor(s) in revised form: March 22, 2012
Published electronically: January 4, 2013
Additional Notes: The first author was supported in part by the project P2276-N13 of the Austrian Science Fund (FWF)
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.