Analyticity and compactness of semigroups of composition operators

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This paper provides a complete characterization of quasicontractive groups and analytic C₀-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.