On the Properties of the Semigroup Generated by the RL Fractional Integral

For operators $A$, it is sometimes possible to define $e^{At}$ as an operator in and of itself provided it meets certain regularity conditions. Like $e^{\lambda x}$ for ODEs, this operator is useful for solving PDEs involving the operator $A$. We call the set of $e^{At}$ a semigroup generated by $A$. In this paper, we discuss the properties of semigroups generated by the fractional integral, an operator appearing in PDEs in increasingly many fields, over Bochner-Lebesgue spaces.