Admissible operators and ${\mathcal{H}}_{\mathrm{\infty }}$ calculus

Given a Hilbert space and the generator $A$ of a strongly continuous, exponentially stable, semigroup on this Hilbert space. For any $g(-s) \in {\mathcal H}_{\infty}$ 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 $g(s)=1/(s + \alpha)$,$ \alpha \in {\mathbb C}_0^+$, then this admissible output operator equals $(\alpha I-A{)}^{-1}$. 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 $g(-s)\in {\mathcal{H}}_{\mathrm{\infty }}$, i.e., there exists a bounded ${\mathcal{H}}_{\mathrm{\infty }}$-calculus. Moreover, we rediscover some well-known classes of generators also having a bounded ${\mathcal{H}}_{\mathrm{\infty }}$-calculus.