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 (Formula Presented) with the right shift (Formula Presented) on weighted sequence spaces. Causality of the solution operator plays a crucial role for the description of initial value problems.