On Toeplitz operators and localization operators

Abreu, Luís; Faustino, Nelson

doi:10.1090/proc/12211

On Toeplitz operators and localization operators
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by Luís Daniel Abreu and Nelson Faustino

Proc. Amer. Math. Soc. 143 (2015), 4317-4323

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

Published electronically: June 16, 2015

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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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