Semigroups of composition operators and integral operators in spaces of analytic functions

We study the maximal spaces of strong continuity on BMOA and the Bloch space B for semigroups of composition operators. Characterizations are given for the cases when these maximal spaces are V MOA or the little Bloch B0. These characterizations are in terms of the weak compactness of the resolvent function or in terms of a specially chosen symbol g of an integral operator Tg. For the second characterization we prove and use an independent result, namely that the operators Tg are weakly compact on the above mentioned spaces if and only if they are compact.