  Starting from a general operator representation in the time-frequency domain,
this paper addresses the problem of approximating linear operators by operators
that are diagonal or band-diagonal with respect to Gabor frames. A
characterization of operators that can be realized as Gabor multipliers is
given and necessary conditions for the existence of (Hilbert-Schmidt) optimal
Gabor multiplier approximations are discussed and an efficient method for the
calculation of an operator's best approximation by a Gabor multiplier is
derived. The spreading function of Gabor multipliers yields new error estimates
for these approximations. Generalizations (multiple Gabor multipliers) are
introduced for better approximation of overspread operators. The Riesz property
of the projection operators involved in generalized Gabor multipliers is
characterized, and a method for obtaining an operator's best approximation by a
multiple Gabor multiplier is suggested. Finally, it is shown that in certain
situations, generalized Gabor multipliers reduce to a finite sum of regular
Gabor multipliers with adapted windows.
