The abstract Cauchy problem for the fractional evolution equation

We give necessary and suflicient conditions for an unbounded closed operator A in a Banach space X such that the abstract Cauchy problem for the fractional evolution equation Dau = Au, 0 <a <1, can be solved. We also present conditions on A such that the fractional time evolution is holomorphic. Arelation between the solutions of the problem for different a, 0 < a = 1, is obtained. In particular, this relation shows that the problem has a holomorphic solution whenever A generates just a Co-semigroup. Some examples are given.