Invariant subspaces of analytic functions. Analytic continuation

Let $G$ be a region in the complex plane; let $H$ be the space of functions analytic in $G$ with the topology of uniform convergence on compacta of $G$; let $W$ be a nontrivial invariant (with respect to differentiation) subspace in $H$ which admits a spectral synthesis. We investigate conditions for which all functions of $W$ can be analytically continued to a larger region $G'\supset G$.