  This paper provides a complete characterization of quasicontractive groups
and analytic $C_0$-semigroups on Hardy and Dirichlet space on the unit disc
with a prescribed generator of the form $Af=Gf'$. In the analytic case we also
give a complete characterization of immediately compact semigroups. When the
analyticity fails, we obtain sufficient conditions for compactness and
membership in the trace class. Finally, we analyse the case where the unit disc
is replaced by the right-half plane, where the results are drastically
different.
