Interpolation and duality in spaces of pseudocontinuable functions

Abstract

Given an inner function $\theta$ on the unit disk, let $K_\theta^p:=H^p \cap \theta \bar{z} \overline{H^p}$ be the associated starinvariant subspace of the Hardy space $H^p$. Also, we put $K_{* \theta}:=K_\theta^2 \cap \mathrm{BMO}$. Assuming that $B=B_{\mathcal{Z}}$ is an interpolating Blaschke product with zeros $\mathcal{Z}=\left\{z_j\right\}$, we characterize, for a number of smoothness classes $X$, the sequences of values $\mathcal{W}=\left\{w_j\right\}$ such that the interpolation problem $\left.f\right|_{\mathcal{Z}}=\mathcal{W}$ has a solution $f$ in $K_B^2 \cap X$. Turning to the case of a general inner function $\theta$, we further establish a non-duality relation between $K_\theta^1$ and $K_{* \theta}$. Namely, we prove that the latter space is properly contained in the dual of the former, unless $\theta$ is a finite Blaschke product. From this we derive an amusing non-interpolation result for functions in $K_{* B}$, with $B=B_{\mathcal{Z}}$ as above.