  We study fractional differential equations of Riemann-Liouville and Caputo
type in Hilbert spaces. Using exponentially weighted spaces of functions
defined on $\mathbb{R}$, we define fractional operators by means of a
functional calculus using the Fourier transform. Main tools are extrapolation-
and interpolation spaces. Main results are the existence and uniqueness of
solutions and the causality of solution operators for non-linear fractional
differential equations.
