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On Toeplitz operators and localization operators

Authors: Luís Daniel Abreu and Nelson Faustino
Journal: Proc. Amer. Math. Soc. 143 (2015), 4317-4323
MSC (2010): Primary 47B32, 30H20; Secondary 81R30, 81S30
Published electronically: June 16, 2015
MathSciNet review: 3373930
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Abstract: This note is a contribution to a problem of Lewis Coburn concerning the relation between Toeplitz operators and Gabor-Daubechies localization operators. We will show that, for any localization operator with a general window $ w\in \mathcal {F}_{2}({\mathbb{C}})$ (the Fock space of analytic functions square-integrable on the complex plane), there exists a differential operator of infinite order $ D$, with constant coefficients explicitly determined by $ w,$ such that the localization operator with symbol $ f$ coincides with the Toeplitz operator with symbol $ Df$. This extends results of Coburn, Lo and Engliš, who obtained similar results in the case where $ w$ is a polynomial window. Our technique of proof combines their methods with a direct sum decomposition in true polyanalytic Fock spaces. Thus, polyanalytic functions are used as a tool to prove a theorem about analytic functions.

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  • [1] Luís Daniel Abreu, Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions, Appl. Comput. Harmon. Anal. 29 (2010), no. 3, 287-302. MR 2672228 (2011d:42076),
  • [2] Luis Daniel Abreu and Hans G. Feichtinger, Function spaces of polyanalytic functions, Harmonic and complex analysis and its applications, Trends Math., Birkhäuser/Springer, Cham, 2014, pp. 1-38. MR 3203099,
  • [3] Luís Daniel Abreu and Karlheinz Gröchenig, Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group, Appl. Anal. 91 (2012), no. 11, 1981-1997. MR 2983996,
  • [4] L. A. Coburn, The Bargmann isometry and Gabor-Daubechies wavelet localization operators, (Bordeaux, 2000) Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 169-178. MR 1882695 (2003a:47054)
  • [5] Elena Cordero and Karlheinz Gröchenig, Time-frequency analysis of localization operators, J. Funct. Anal. 205 (2003), no. 1, 107-131. MR 2020210 (2004j:47100),
  • [6] Ingrid Daubechies, Time-frequency localization operators: a geometric phase space approach, IEEE Trans. Inform. Theory 34 (1988), no. 4, 605-612. MR 966733,
  • [7] Miroslav Engliš, Toeplitz operators and localization operators, Trans. Amer. Math. Soc. 361 (2009), no. 2, 1039-1052. MR 2452833 (2010a:47056),
  • [8] Hans G. Feichtinger, On a new Segal algebra, Monatsh. Math. 92 (1981), no. 4, 269-289. MR 643206 (83a:43002),
  • [9] H. G. Feichtinger, G. A. Zimmermann, A Banach space of test functions for Gabor analysis. Gabor analysis and algorithms, 123-170, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1998.
  • [10] H. G. Feichtinger and K. Nowak, A Szegő-type theorem for Gabor-Toeplitz localization operators, Michigan Math. J. 49 (2001), no. 1, 13-21. MR 1827072 (2003b:47049),
  • [11] Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366 (92k:22017)
  • [12] A. Haimi, H. Hendenmalm, The polyanalytic Ginibre ensembles. preprint arXiv:1106.2975.
  • [13] Min-Lin Lo, The Bargmann transform and windowed Fourier localization, Integral Equations Operator Theory 57 (2007), no. 3, 397-412. MR 2307818 (2008b:47049),
  • [14] Li Zhong Peng, Richard Rochberg, and Zhi Jian Wu, Orthogonal polynomials and middle Hankel operators on Bergman spaces, Studia Math. 102 (1992), no. 1, 57-75. MR 1164633 (93b:47054)
  • [15] N. L. Vasilevski, Poly-Fock spaces, Differential operators and related topics, Vol. I (Odessa, 1997) Oper. Theory Adv. Appl., vol. 117, Birkhäuser, Basel, 2000, pp. 371-386. MR 1764974 (2002a:46026)

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Additional Information

Luís Daniel Abreu
Affiliation: Acoustic Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, 1040, Vienna, Austria – and – CMUC, Department of Mathematics, University of Coimbra, Coimbra, Portugal

Nelson Faustino
Affiliation: CMUC, Department of Mathematics, University of Coimbra, Coimbra, Portugal
Address at time of publication: Departamento de Matemática Aplicada, IMECC-Unicamp, CEP 13083-859, Campinas, SP, Brasil

Keywords: Localization operators, polyanalytic Fock spaces, Toeplitz operators
Received by editor(s): December 26, 2012
Received by editor(s) in revised form: March 20, 2013, and August 7, 2013
Published electronically: June 16, 2015
Additional Notes: Both authors were supported by CMUC and FCT (Portugal), through European program COMPETE/FEDER and by FCT project PTDC/MAT/114394/2009. The first author was also supported by Austrian Science Foundation (FWF) project “Frames and Harmonic Analysis” and START-project FLAME (“Frames and Linear Operators for Acoustical Modeling and Parameter Estimation”, Y 551-N13). The second author was also supported by São Paulo Research Foundation (FAPESP) through the grant 13/07590-8.
Communicated by: Pamela B. Gorkin
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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