Toeplitz operators and localization operators

We show that for any localization operator on the Fock space with polynomial window, there exists a constant coefficient linear partial differential operator D D such that the localization operator with symbol f f coincides with the Toeplitz operator with symbol D f 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.