Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


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
MathSciNet review: 3055702
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. H. Ando and Y. Morimoto.
    Wick calculus and the Cauthy problem for some dispersive equations.
    Osaka J. Math., 39(1):123-147, 2002. MR 1883917 (2003b:35219)
  • 2. V. Bargmann.
    On a Hilbert space of analytic functions and an associated integral transform.
    Comm. Pure Appl. Math., 14:187-214, 1961. MR 0157250 (28:486)
  • 3. F. A. Berezin.
    Wick and anti-Wick symbols of operators.
    Mat. Sb. (N.S.), 86(128):578-610, 1971. MR 0291839 (45:929)
  • 4. C. A. Berger and L. A. Coburn.
    Toeplitz operators on the Segal-Bargmann space.
    Trans. Amer. Math. Soc., 301(2):813-829, 1987. MR 882716 (88c:47044)
  • 5. C. A. Berger and L. A. Coburn.
    Heat flow and Berezin-Toeplitz estimates.
    Amer. J. Math., 116(3):563-590, 1994. MR 1277446 (95g:47038)
  • 6. P. Boggiatto, E. Cordero, and K. Gröchenig.
    Generalized anti-Wick operators with symbols in distributional Sobolev spaces.
    Integral Equations Operator Theory, 48(4):427-442, 2004. MR 2047590 (2005a:47088)
  • 7. J.-M. Bony and J.-Y. Chemin.
    Espaces fonctionnels associés au calcul de Weyl-Hörmander.
    Bull. Soc. Math. France, 122(1):77-118, 1994. MR 1259109 (95a:35152)
  • 8. L. A. Coburn.
    The Bargmann isometry and Gabor-Daubechies wavelet localization operators.
    In Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000), volume 129 of Oper. Theory Adv. Appl., pages 169-178. Birkhäuser, Basel, 2001. MR 1882695 (2003a:47054)
  • 9. J. B. Conway.
    A Course in Functional Analysis.
    Springer-Verlag, New York, second edition, 1990. MR 1070713 (91e:46001)
  • 10. E. Cordero and K. Gröchenig.
    Time-frequency analysis of localization operators.
    J. Funct. Anal., 205(1):107-131, 2003. MR 2020210 (2004j:47100)
  • 11. E. Cordero and K. Gröchenig.
    Symbolic calculus and Fredholm property for localization operators.
    J. Fourier Anal. Appl., 12(4):345-370, 2006. MR 2256930 (2007e:47030)
  • 12. I. Daubechies.
    Time-frequency localization operators: A geometric phase space approach.
    IEEE Trans. Inform. Theory, 34(4):605-612, 1988. MR 966733
  • 13. M. Engliš.
    Toeplitz operators and localization operators.
    Trans. Amer. Math. Soc., 361(2):1039-1052, 2009. MR 2452833 (2010a:47056)
  • 14. H. G. Feichtinger and K. Gröchenig.
    Banach spaces related to integrable group representations and their atomic decompositions. I.
    J. Funct. Anal., 86(2):307-340, 1989. MR 1021139 (91g:43011)
  • 15. H. G. Feichtinger and K. Nowak.
    A first survey of Gabor multipliers.
    In Advances in Gabor analysis, Appl. Numer. Harmon. Anal., pages 99-128. Birkhäuser Boston, Boston, MA, 2003. MR 1955933
  • 16. G. B. Folland.
    Harmonic Analysis in Phase Space.
    Princeton Univ. Press, Princeton, NJ, 1989. MR 983366 (92k:22017)
  • 17. K. Gröchenig.
    Foundations of Time-Frequency Analysis.
    Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1843717 (2002h:42001)
  • 18. K. Gröchenig.
    Time-frequency analysis of Sjöstrand's class.
    Revista Mat. Iberoam., 22(2):703-724, 2006. MR 2294795 (2008b:35308)
  • 19. K. Gröchenig.
    Weight functions in time-frequency analysis.
    Pseudodifferential Operators: Partial Differential Equations and Time-Frequency Analysis, (English Summary), volume 52, pages 343-366. Fields Institute Comm., 2007. MR 2385335 (2009c:42070)
  • 20. K. Gröchenig and Z. Rzeszotnik.
    Banach algebras of pseudodifferential operators and their almost diagonalization.
    Ann. Inst. Fourier (Grenoble), 58(7):2279-2314, 2008. MR 2498351 (2010h:47071)
  • 21. K. Gröchenig and J. Toft.
    Isomorphism properties of Toeplitz operators and pseudo-differential operators between modulation spaces.
    J. Anal. Math., 114 (1):255-283, 2011. MR 2837086
  • 22. A. Holst, J. Toft and P. Wahlberg.
    Weyl product algebras and modulation spaces.
    J. Funct. Anal., 251:463-491, 2007. MR 2356420 (2008i:42047)
  • 23. A. J. E. M. Janssen.
    Bargmann transform, Zak transform, and coherent states.
    J. Math. Phys., 23(5):720-731, 1982. MR 655886 (84h:81041)
  • 24. N. Lerner.
    The Wick calculus of pseudo-differential operators and energy estimates.
    In New trends in microlocal analysis (Tokyo, 1995), pages 23-37. Springer, Tokyo, 1997. MR 1636234 (99i:35187)
  • 25. N. Lerner.
    The Wick calculus of pseudo-differential operators and some of its applications.
    Cubo Mat. Educ., 5(1):213-236, 2003. MR 1957713 (2004a:47058)
  • 26. N. Lerner and Y. Morimoto.
    A Wiener algebra for the Fefferman-Phong inequality.
    In Seminaire: Equations aux Dérivées Partielles. 2005-2006, Sémin. Équ. Dériv. Partielles, Exp. No. XVII, 12pp. École Polytech., Palaiseau, 2006. MR 2276082 (2008c:47077)
  • 27. M. A. Shubin.
    Pseudodifferential Operators and Spectral Theory.
    Springer-Verlag, Berlin, second edition, 2001.
    Translated from the 1978 Russian original by Stig I. Andersson. MR 1852334 (2002d:47073)
  • 28. M. Signahl and J. Toft.
    Remarks on mapping properties for the Bargmann transform on modulation spaces.
    Integral Transforms Spec. Funct., 22(4-5):359-366, 2011. MR 2801288 (2012e:44006)
  • 29. S. Thangavelu.
    Lectures on Hermite and Laguerre Expansions, volume 42 of Mathematical Notes.
    Princeton University Press, Princeton, NJ, 1993.
    With a preface by Robert S. Strichartz. MR 1215939 (94i:42001)
  • 30. J. Toft.
    Regularizations, decompositions and lower bound problems in the Weyl calculus.
    Comm. Partial Differential Equations, 25 (7&8): 1201-1234, 2000. MR 1765143 (2001i:47081)
  • 31. J. Toft.
    Subalgebras to a Wiener type algebra of pseudo-differential operators.
    Ann. Inst. Fourier (Grenoble), 51(5):1347-1383, 2001. MR 1860668 (2002h:47071)
  • 32. J. Toft.
    Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I.
    J. Funct. Anal., 207(2):399-429, 2004. MR 2032995 (2004j:35312)
  • 33. J. Toft.
    Continuity and Schatten-von Neumann properties for Toeplitz operators on modulation spaces. In: J. Toft, M. W. Wong, H. Zhu (Eds.), Modern Trends in Pseudo-Differential Operators, Operator Theory Advances and Applications, Vol. 172, Birkhäuser Verlag, Basel: pp. 313-328, 2007. MR 2308518 (2008d:47064)
  • 34. J. Toft.
    Multiplication properties in pseudo-differential calculus with small regularity on the symbols.
    J. Pseudo-Differ. Oper. Appl. 1(1): 101-138, 2010. MR 2679745 (2011h:47094)
  • 35. H. Triebel.
    Theory of Function Spaces.
    Birkhäuser Verlag, Basel, 1983. MR 781540 (86j:46026)
  • 36. M. W. Wong.
    Wavelet Transforms and Localization Operators, volume 136 of Operator Theory Advances and Applications.
    Birkhäuser, 2002. MR 1918652 (2003i:42003)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 46B03, 42B35, 47B35

Retrieve articles in all journals with MSC (2010): 46B03, 42B35, 47B35

Additional Information

Karlheinz Gröchenig
Affiliation: Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria

Joachim Toft
Affiliation: Department of Computer Science, Physics and Mathematics, Linnæus University, Växjö, Sweden

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.

American Mathematical Society