  We consider general difference equations $u_{n+1} = F(u)_n$ for $n \in
\mathbb{Z}$ on exponentially weighted $\ell_2$ spaces of two-sided Hilbert
space valued sequences $u$ and discuss initial value problems. As an
application of the Hilbert space approach, we characterize exponential
stability of linear equations and prove a stable manifold theorem for causal
nonlinear difference equations.
