We study relations of an infinitesimal generator $A$ and its restrictions $\tilde A^\omega$, $\omega>0$ which correspond to a family of unbounded operators $(S(t))_{t\geq0}$ and a family of Banach spaces $(\mathcal D_\omega,\|\cdot\|_\omega)$. Results are applied to a Cauchy problem $\frac d{dt}u(t)=Au(t)+f(t)$, $u(0)=u_0$ with appropriate $f$ and $u_0$.