Inverses of generators of integrated fractional resolvent operator functions

This paper is devoted to the inverse generator problem in the setting of generators of integrated resolvent operator functions. It is shown that if the operator $A$ is the generator of a tempered $\beta$-times integrated $\alpha$-resolvent operator function ($(\alpha,\beta)$-ROF) and it is injective, then the inverse operator $A^{-1}$ is the generator of a tempered $(\alpha,\gamma)$-ROF for all $\gamma > \beta+1/2,$ by means of an explicit representation of the integrated resolvent operator function based in Bessel functions of first kind. Analytic resolvent operator functions are also considered, showing that $A^{-1}$ is in addition the generator of a tempered $(\delta,0)$-ROF for all $\delta<\alpha.$ Moreover, the optimal decay rate of $(\alpha,\beta)$-ROFs as $t\to \infty$ is given. These result are applied to fractional Cauchy problem unsolved in the fractional derivative.