Relation between transfer function and resolvent set

Question

I am studying an infinite dimensional system in the form $$ \begin{array}{l} \dot{x}= Ax+Bu, \\ y=Cx, \end{array} $$ where $A$ is the generator of a analytic semigroup and $B$ and $C$ are bounded operators. My question is, what is the relation between the domain transfer function $ G(s)=C(Is-A)^{-1}B $ and the resolvent set $\rho(A)$ ?

Thanks for the time.

Answer 1

You have $$G(s)=CR(s,A)B,$$ where $R(s,A)$ is the resolvent of the operator $A$. Since $B$ and $C$ are bounded, the resolvent set $\rho(A)$ is the domain of definition of $G(s)$.