  We show that Hilbert-Schmidt operators can be used to define frame-like
structures for $L^2(\mathbb{R}^d)$ over lattices in $\mathbb{R}^{2d}$ that
include multi-window Gabor frames as a special case. These frame-like
structures are called Gabor g-frames, as they are examples of g-frames as
introduced by Sun. We show that Gabor g-frames share many properties of Gabor
frames, including a Janssen representation and Wexler-Raz biorthogonality
conditions. A central part of our analysis is a notion of Fourier series of
periodic operators based on earlier work by Feichtinger and Kozek, where we
show in particular a Poisson summation formula for trace class operators. By
choosing operators from certain Banach subspaces of the Hilbert Schmidt
operators, Gabor g-frames give equivalent norms for modulation spaces in terms
of weighted $\ell^p$-norms of an associated sequence, as previously shown for
localization operators by D\"orfler, Feichtinger and Gr\"ochenig.
