In this paper, we investigate a class of abstract degenerate fractional differential equations with Caputo derivatives. We consider subordinated fractional resolvent families generated by multivalued linear operators, which do have removable singularities at the origin. Semi-linear degenerate fractional Cauchy problems are also considered in this context.

This paper is concerned with α-times integrated C-semigroups for α > 0 and the associated abstract Cauchy problem: $u^{\text{'}}\left(t\right)=Au\left(t\right)+\frac{t^{\alpha -1}}{\Gamma \left(\alpha \right)}x$, t >0; u(0) = 0. We first investigate basic properties of an α-times integrated C-semigroup which may not be exponentially bounded. We then characterize the generator A of an exponentially bounded α-times integrated C-semigroup, either in terms of its Laplace transforms or in terms of existence of a unique solution of the above abstract Cauchy problem for every x in ${(\lambda -A)}^{-1}C\left(X\right)$.

Let $\left(T_t\right)$ be a C₀ semigroup with generator A on a Banach space X and let be an operator ideal, e.g. the class of compact, Hilbert-Schmidt or trace class operators. We show that the resolvent R(λ,A) of A belongs to if and only if the integrated semigroup $S_t:={\int }_0^t{T}_{s}ds$ belongs to . For analytic semigroups, $S_t\in$ implies $T_t\in$, and in this case we give precise estimates for the growth of the -norm of $T_t$ (e.g. the trace of $T_t$) in terms of the resolvent growth and the imbedding D(A) ↪ X.