Integral resolvent for Volterra equations and Favard spaces

The objective of this work is to give a characterization of the domain $D\left(A\right)$ of $A$ in terms of the integral resolvent family of the equation $x\left(t\right)=x_{0}+\int\limits_{0}^{t}a\left(t-s\right)Ax(s)ds$, $t\geq0$, where $A$ is a linear closed densely defined operator, $a\in L_{loc}^{1}\left(\mathbb{R}^{+}\right)$ in a general Banach space $X$ and $x_{0}\in X$. Furthermore, we give a relationship between the Favard classes (temporal and frequency) for integral resolvents.