  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 $g(-s) \in {\mathcal H}_{\infty}$, i.e.,
there exists a bounded ${\mathcal H}_{\infty}$-calculus. Moreover, we
rediscover some well-known classes of generators also having a bounded
${\mathcal H}_{\infty}$-calculus.
