## On Toeplitz operators and localization operators

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- by Luís Daniel Abreu and Nelson Faustino PDF
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**143**(2015), 4317-4323 Request permission

## 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
- 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
- © Copyright 2015
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3373930