  We construct and study a time--semidiscretization scheme for the Cauchy
problem associated with a linear homogeneous differential equation with the
Caputo fractional time derivative of order $\alpha\in(0,1)$ and a spatial
sectorial operator in a Banach space. For this scheme, we obtain
rate--of--convergence and error estimates in terms of the discretization step.
We use properties of Mittag--Leffler functions, hypergeometric functions, and
the calculus of sectorial operators in a Banach space. Results of numerical
experiments are also reported.
