  Certain mathematical objects appear in a lot of scientific disciplines, like
physics, signal processing and, naturally, mathematics. In a general setting
they can be described as frame multipliers, consisting of analysis,
multiplication by a fixed sequence (called the symbol), and synthesis. They are
not only interesting mathematical objects, but also important for applications,
for example for the realization of time-varying filters. In this paper we show
a surprising result about the inverse of such operators, if existing, as well
as new results about a core concept of frame theory, dual frames. We show that
for semi-normalized symbols, the inverse of any invertible frame multiplier can
always be represented as a frame multiplier with the reciprocal symbol and dual
frames of the given ones. Furthermore, one of those dual frames is uniquely
determined and the other one can be arbitrarily chosen. We investigate
sufficient conditions for the special case, when both dual frames can be chosen
to be the canonical duals. In connection to the above, we show that the set of
dual frames determines a frame uniquely. Furthermore, for a given frame, the
union of all coefficients of its dual frames is dense in $\ell^2$. We also
investigate invertible Gabor multipliers. Finally, we give a numerical example
for the invertibility of multipliers in the Gabor case.
