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We study the range of time-frequency localization operators acting onmodulation spaces and prove a lifting theorem. As an application we alsocharacterize the range of Gabor multipliers, and, in the realm of complexanalysis, we characterize the range of certain Toeplitz operators on weightedBargmann-Fock spaces. The main tools are the construction of canonicalisomorphisms between modulation spaces of Hilbert-type and a refined version ofthe spectral invariance of pseudodifferential operators. On the technical levelwe prove a new class of inequalities for weighted gamma functions.

The Range of Localization Operators and Lifting Theorems for Modulation and Bargmann-Fock Spaces