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.



Fractional power of operators in $C_0$-semigroup
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.


