  We consider Gabor frames generated by a general lattice and a window function
that belongs to one of the following spaces: the Sobolev space $V_1 =
H^1(\mathbb R^d)$, the weighted $L^2$-space $V_2 = L_{1 + |x|}^2(\mathbb R^d)$,
and the space $V_3 = \mathbb H^1(\mathbb R^d) = V_1 \cap V_2$ consisting of all
functions with finite uncertainty product; all these spaces can be described as
modulation spaces with respect to suitable weighted $L^2$ spaces. In all cases,
we prove that the space of Bessel vectors in $V_j$ is mapped bijectively onto
itself by the Gabor frame operator. As a consequence, if the window function
belongs to one of the three spaces, then the canonical dual window also belongs
to the same space. In fact, the result not only applies to frames, but also to
frame sequences.
