Toeplitz operators and $H_{\infty}$ calculus

Let $A$ be the generator of a strongly continuous, exponentially stable, semigroup on a Hilbert space. Furthermore, let the scalar function $g$ be bounded and analytic on the left-half plane, i.e., $g(-s) \in {\mathcal H}_{\infty}$. By using the Toeplitz operator associated to $g$, we construct an infinite-time admissible output operator $g(A)$. If $g$ is rational, then this operator is bounded, and equals the ``normal'' definition of $g(A)$. Although in general $g(A)$ may be unbounded, we always have that $g(A)$ multiplied by the semigroup is a bounded operator for every 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 $g$ 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.