Invariant subspaces in quasi-Banach spaces of analytic functions

Let $X$ be a quasi-Banach space of analytic functions on a finitely connected bounded domain $\Omega$ on the complex plane. We prove a theorem that reduces the study of the hyperinvariant subspaces of $X$ to that of the hyperinvariant subspaces of $X_1$ where $X_1$ is a quasi-Banach space of analytic functions on a domain $\Omega_1$ obtained from $\Omega$ by adding some of the bounded connectivity components of $\mathbb C\setminus\Omega$. In particular, the lattice structure (incident to the hyperinvariant subspaces) of a quasi-Banach space $X$ of analytic functions on the annulus $\{z\in\mathbb C:\rho<|z|<1\}$, $0<\rho<1$, is understood in terms of the lattice structure of the space $X_1$, the counterpart of $X$ for the unit disk.