  We show that, given a Banach space and a generator of an exponentially stable
$C_{0}$-semigroup, a weakly admissible operator $g(A)$ can be defined for any
$g$ bounded, analytic function on the left half-plane. This yields an
(unbounded) functional calculus. The construction uses a Toeplitz operator and
is motivated by system theory. In separable Hilbert spaces, we even get
admissibility. Furthermore, it is investigated when a bounded calculus can be
guaranteed. For this we introduce the new notion of exact observability by
direction. Finally, it is shown that the calculus coincides with one for
half-plane-operators.
