ON TOEPLITZ OPERATORS AND LOCALIZATION OPERATORS

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 is an element of F-2(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 Englis, 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.