The object of this study is the class of closable Gabor operators. That is the set of operators which map a Gabor function (or note) into a multiple of a Gabor function. By using the Bargman space (sometimes called Bargman representation) some general properties of these operators are derived. It is shown that the set of Gabor operators whose adjoint is also a Gabor operator establishes a six-dimensional complex manifold with a partial Lie-group structure and with an involution. The corresponding Lie-algebra and the infinitesimal generators are calculated. Further it turns out that, at least locally, a Gabor operator with a Gabor adjoint results from an evolution process. The proofs of the theorems are a hybridization of Hilbert space techniques and classical complex analysis (theorems of Osgood, Montel, etc.). The proofs will be published elsewhere in a wider context.