An $\mathcal{H}_\infty$ calculus of admissible operators

Given a Hilbert space and the generator A of a strongly continuous, exponentially stable, semigroup on this Hilbert space. For any $g(-s) \in H_{\infty}$ we show that there exists an infinite-time admissible output operator $g(A)$. If $g$ is rational, then this operator is bounded, and equals the “normal‿ definition of $g(A)$. In particular, when $g(s) = 1/(s + a), a \in {\mathbb C}_0^+$ , then this admissible output operator equals $(I - A)^{-1}$. Although in general $g(A)$ may be unbounded, we always have that $g(A)$ multiplied by the semigroup is a bounded operator for every (strictly) positive time instant. Furthermore, when there exists an admissible output operator $C$ such that $(C, A)$ is exactly observable, then $g(A)$ is bounded for all $g$’s with $g(-s) \in H_{\infty}$.