Maximal subspaces of strong continuity for composition semigroups

Let $(\varphi_t)_{t\geq 0}$ a semigroup of holomorphic self-maps of the unit disk and $C_t f = f \circ \varphi_t$ the semigroup of composition operators which corresponds to $\varphi_t.$ Given a non-separable Banach space of analytic functions $X$ we study the properties of the maximal subspace of $X$ on which the semigroup $C_t$ is strongly continuous. In particular when $X$ contains the polynomials an interesting question is for which semigroups the maximal subspace of strong continuity coincides with the norm closure of the polynomials. This problem has been investigated in several function spaces including $BMOA$, $BMOA_p$, the Bloch space, $Q_s$ space and analytic Morrey spaces. However, in most cases only partial results are available. We offer a unified approach to this problem which encompasses all of the above spaces as particular examples. Moreover, we completely characterize the semigroups for which the maximal subspace of strong continuity coincides with the norm-closure of the polynomials in the space, giving therefore sharp versions of a number of results in the literature.