On Toeplitz operators and localization operators

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 2 F2(C) (the Fock space of analytic functions square-integrable on the complex plane), there exists a di¤erential operator of in nite order D, with constant coe¢ cients 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 Engli s, 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. 1. Introduction Let Lp(C) (1 p <1) be the weighted L space of functions with norm kukLp(C) = Z C ju(z)je 2 pjzj2dz 1 p ; dz being the area measure on C and L1(C) the space of measurable functions on C such that ju(z)je 2 jzj2 is bounded, endowed with the norm kukL1(C) = sup z2C ju(z)je 2 jzj2 : Let H be a Hilbert space contained in L2(C), with reproducing kernel K( ; z). For f 2 L1(C), the Toeplitz operator Toepf with symbol f(z) is the operator acting on H de ned by (Toepfg)( ) = Z C f(z)g(z)K( ; z)e jzj 2 dz; 8 g 2 H: Date: March 27, 2013. 1991 Mathematics Subject Classi cation. Primary 47B32,30H20; Secondary 81R30,81S30.