  This paper provides a complete characterization of quasicontractive
$C_0$-semigroups on Hardy and Dirichlet space with a prescribed generator of
the form $Af=Gf'$. We show that such semigroups are semigroups of composition
operators and we give simple sufficient and necessary condition on $G$. Our
techniques are based on ideas from semigroup theory, such as the use of
numerical ranges.
