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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
DOI: https://doi.org/10.1090/proc/12211
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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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
Email: daniel@mat.uc.pt

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
Email: faustino@ime.unicamp.br

DOI: https://doi.org/10.1090/proc/12211
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.