Fractional power of operators in $C_0$-semigroup

Question

Let $X$ be Banach space, and $\{Z(t)\}_{t\geq 0}\subseteq B(X)$ be the $C_0$-semigroup of operators defined on $X$. Moreover, let $A$ be the infinitesimal generator of $\{Z(t)\}_{t\geq 0}$. A fractional power of any closed linear operator $F$ is defined, when $(-\infty,0)\subset \rho(F)$(the resolvent set.) and the set $\{\lambda(\lambda-F)^{-1}:0<\lambda<\infty\}$ is bounded. It is noted that any such conditions do not imply that $F$ generates a semigroup.

My problem is, that I have to consider the fractional powers of operators in $\{Z(t)\}_{t\geq 0}$. i.e. $[Z(t)]^r$ , for $r\in \mathbb{R}$ and $t>0$. Then when is it possible? If I consider, only those semigroups whose generator satisfies above conditions and its fractional powers are defined. As we know $Z(t)=e^{tA}$, then is it possible to take its fractional powers? because integer powers are very well defined through Banach algebra. The problem is just with its fraction powers.

Answer 1

Observe that for $r>0$ the semigroup law implies $Z(t)^r = Z(r t)$. If $Z(t)$ is a $C_0$-group, then this is true for all $0\not=r\in\mathbb{R}$. The generator of $Z(t)^r$ is $(r A, D(A))$. You can find this construction sometimes called 'rescaled semigroup' e.g. in the book of Engel, Nagel, One-parameter semigroups for linear evolution equations, Springer 2000, p. 43 and p. 60. The book also contains a comprehensive treatment of fractional powers of generators pp. 137.