We formulate fractional difference equations of Riemann-Liouville and Caputo type in a functional analytical framework. Main results are existence of solutions on Hilbert space-valued weighted sequence spaces and a condition for stability of linear fractional difference equations. Using a functional calculus, we relate the fractional sum to fractional powers of the operator $1 - τ^{-1}$ with the right shift $τ^{-1}$ on weighted sequence spaces. Causality of the solution operator plays a crucial role for the description of initial value problems.