On the Relation between stability of continuous- and discrete-time evolution equations via the Cayley transform

n this paper we investigate and compare the properties of the semigroup generated by $A$, and the sequence $A_d^n$, $n \in {\mathbb N}$, where $A_d= (I+A)(I-A)^{-1}$. We show that if $A$ and $A^{-1}$ generate a uniformly bounded, strongly continuous semigroup on a Hilbert space, then $A_d$ is power bounded. For analytic semigroups we can prove stronger results. If $A$ is the infinitesimal generator of an analytic semigroup, then power boundedness of $A_d$ is equivalent to the uniform boundedness of the semigroup generated by $A$.