  Exact functional renormalization group (FRG) flow equations for quantum
systems can be derived directly within an operator formalism without using
functional integrals. This simple insight opens new possibilities for applying
FRG methods to models for strongly correlated electrons with projected Hilbert
spaces, such as quantum spin models, the $t$-$J$ model, or the Hubbard model at
infinite on-site repulsion. By representing these models in terms of Hubbard
X-operators, we derive exact flow equations for the time-ordered correlation
functions of the X-operators (X-FRG), which allow us to calculate the
electronic correlation functions in the projected Hilbert space of these
models. The Hubbard-I approximation for the single-particle Green function of
the Hubbard model is recovered from a trivial truncation of the flow equations
where the two-point vertex is approximated by its atomic limit. We use our
approach to calculate the quasi-particle residue and damping in the ``hidden
Fermi liquid'' state of the Hubbard model at infinite on-site repulsion where
the Hamiltonian consists only of the projected kinetic energy.
