We use a projective groups representation $\rho$ of the
unimodular group $\mathcal{G} \times \hat{\mathcal{G}}$ on
$L^2(\mathcal{G}$) to define Gabor wavelet transform of a
function $f$ with respect to a window function $g$, where
$\mathcal{G}$ is a locally compact abelian group and
$\hat{\mathcal{G}}$ its dual group. Using these transforms, we
define a weighted Banach $\mathcal{H}^{1,
\rho}_w(\mathcal{G})$ and its antidual space
$\mathcal{H}^{{1}^{\sim}, \rho}_w(\mathcal{G})$, $w$ being a
moderate weight function on $\mathcal{G} \times
\hat{\mathcal{G}}$. These spaces reduce to the well known
Feichtinger algebra $S_0(\mathcal{G})$ and Banach space of
Feichtinger distribution $S'_0(\mathcal{G})$ respectively for
$w\equiv 1$. We obtain an atomic decomposition of
$\mathcal{H}^{1, \rho}_w(\mathcal{G})$ and study some
properties of Gabor multipliers on the spaces
$L^2(\mathcal{G}), \mathcal{H}^{1, \rho}_w(\mathcal{G})$ and
$\mathcal{H}^{{1}^{\sim}, \rho}_w(\mathcal{G})$. Finally, we
prove a theorem on the compactness of Gabor multiplier
operators on $L^2(\mathcal{G})$ and $\mathcal{H}^{1,
\rho}_w(\mathcal{G})$, which reduces to an earlier result of
Feichtinger [Fei 02, Theorem 5.15 (iv)] for $w=1$ and
$\mathcal{G}=R^d$.