  We study the range of time-frequency localization operators acting on
modulation spaces and prove a lifting theorem. As an application we also
characterize the range of Gabor multipliers, and, in the realm of complex
analysis, we characterize the range of certain Toeplitz operators on weighted
Bargmann-Fock spaces. The main tools are the construction of canonical
isomorphisms between modulation spaces of Hilbert-type and a refined version of
the spectral invariance of pseudodifferential operators. On the technical level
we prove a new class of inequalities for weighted gamma functions.
