Growth of semigroups in discrete and continuous time

We show that the growth rates of solutions of the abstract differential equations $\dot{x}(t)=Ax(t), \dot{x}(t)=A^{-1}x(t)$, and the difference equation $x_d(n+1)=(A+I)(A-1)^{-1}x_d(n)$ are closely related. Assuming that $A$ generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup $\left(e^{A^{-1}t}\right)_{t\geq 0}$ is $O(\sqrt[4]{t})$, and for $\left((A+I)(A-I)^{-1}\right)^n$ it is $O(\sqrt[4]{n})$. The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are $O(\log(t))$ and $O(\log(n))$, respectively. Furthermore, we give conditions on $A$ such that the growth rate of $\left((A+I)(A-I)^{-1}\right)^n$ is $O(1)$, i.e., the operator is power bounded.