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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 admissibleoutput operator $g(A)$. If $g$ is rational, then this operator is bounded, andequals the "normal" definition of $g(A)$. In particular, when $g(s)=1/(s +\alpha)$, $ \alpha \in {\mathbb C}_0^+$, then this admissible output operatorequals $(\alpha I - A)^{-1}$. Although in general $g(A)$ may be unbounded, wealways have that $g(A)$ multiplied by the semigroup is a bounded operator forevery (strictly) positive time instant. Furthermore, when there exists anadmissible output operator $C$ such that $(C,A)$ is exactly observable, then$g(A)$ is bounded for all $g$'s with $g(-s) \in {\mathcal H}_{\infty}$, i.e.,there exists a bounded ${\mathcal H}_{\infty}$-calculus. Moreover, werediscover some well-known classes of generators also having a bounded${\mathcal H}_{\infty}$-calculus.

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