  We define classes of pseudodifferential operators on $G$-bundles with compact
base and give a generalized $L^2$ Fredholm theory for invariant operators in
these classes in terms of von Neumann's $G$-dimension. We combine this
formalism with a generalized Paley-Wiener theorem, valid for bundles with
unimodular structure groups, to provide solvability criteria for invariant
operators. This formalism also gives a basis for a $G$-index for these
operators. We also define and describe a transversal dimension and its
corresponding Fredholm theory in terms of anisotropic Sobolev estimates, valid
also for similar bundles with nonunimodular structure group.
