On characterization and perturbation of local $C$-semigroups

Let $S(\cdot)$ be a $(C_0)$-group with generator $-B$, and let $\{T(t);0\le t<\tau\}$ be a local $C$-semigroup commuting with $S(\cdot)$. Then the operators $V(t):=S(-t)T(t)$, $0\le t<\tau$, form a local $C$-semigroup. It is proved that if $C$ is injective and $A$ is the generator of $T(\cdot)$, then $A+B$ is closable and $\overline{A+B}$ is the generator of $V(\cdot)$. Also proved are a characterization theorem for local $C$-semigroups with $C$ not necessarily injective and a theorem about solvability of the abstract inhomogeneous Cauchy problem: $u'(t)=Au(t)+Cf(t), 0<t<\tau; u(0)=Cx.$