  Let T be a C_{\cdot 0}-contraction on a Hilbert space H and S be a
non-trivial closed subspace of H. We prove that S is a T-invariant subspace of
H if and only if there exists a Hilbert space D and a partially isometric
operator \Pi : H^2_D(\mathbb{D}) \raro H such that \Pi M_z = T \Pi and that S =
ran \Pi, or equivalently, P_S = \Pi \Pi^*. As an application we completely
classify the shift-invariant subspaces of C_{\cdot 0}-contractive and analytic
reproducing kernel Hilbert spaces over the unit disc. Our results also includes
the case of weighted Bergman spaces over the unit disk.
