Conditions such that $T(t)x \in D(A)$ for $C_0$-Semigroup

Question

Consider a $C_0$-Semigroup $(T(t))_{t \ge 0}$ on the Banachspace $X$. Are there any conditions on $x \in X$ or the semigroup such that for any $t > 0$ it holds $T(t)x \in D(A)$? Here $(A,D(A))$ denotes the Generator with its domain $D(A)$.

Answer 1

The semigroups which satisfy the property you mention are called immediately differentiable (see Engel-Nagel's book, Section 4.b, "Differentiable semigroups"). Corollary 4.15 provides a characterization in terms of a spectral property of the generator.

Analytic semigroups, which are holomorphic in the time variable on a sector in the complex plane, are immediately differentiable semigroups (see also this comment ). See Section 4.a of the aforementioned book.