Complemented invariant subspaces of $H ^p,\;0 < p < 1$, and the Hahn-Banach extension property

Abstract:

Let $0{\text { < }}p{\text { < }}1$ and let ${H^p}$ denote the usual Hardy class of functions analytic on the disc. In this note we show that an invariant subspace of ${H^p}$ is complemented in ${H^p}$ if and only if it has the form $B{H^p}$ where $B$ is a Blaschke product whose zero sequence is a Carleson sequence. We also prove that this occurs if and only if the invariant subspace has the Hahn-Banach extension property.