This question comes from a paper 2015(Kolokoltsov) Theorem 4.1. In the end of the proof i), “Applying to $T_t$ the procedure applied above to $T_t^h$ shows that $T_t$ defines also a strongly continuous semigroup in $C^1(\infty,b]$, and hence $C^1(\infty,b]$ is an invariant core for $T_t$ ”.
My question is Why $C^1(\infty,b]$ is an invariant core for $T_t$ ?
