Growth of semigroups in discrete and continuous time
Volume 206 / 2011
Abstract
We show that the growth rates of solutions of the abstract differential equations $\dot{x}(t) = A x(t)$, $\dot{x}(t)= A^{-1} x(t)$, and the difference equation $x_d(n+1) = (A+I)(A-I)^{-1} x_d(n)$ 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 $(e^{A^{-1}t})_{t \geq 0}$ is $O(\sqrt[4]{t})$, and for $( (A+I)(A-I)^{-1})^n$ 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 $( (A+I)(A-I)^{-1})^n$ is $O(1)$, i.e., the operator is power bounded.
