Toeplitz operators and localization operators

Abstract

We show that for any localization operator on the Fock space with polynomial window, there exists a constant coefficient linear partial differential operator$D$such that the localization operator with symbol$f$coincides with the Toeplitz operator with symbol$Df$. An analogous result also holds in the context of Bergman spaces on bounded symmetric domains. This verifies a recent conjecture of Coburn and simplifies and generalizes recent results of Lo.