A free interpolation problem for a subspace of $H^\infty$

Abstract

Given an inner function $\theta$, the associated star-invariant subspace $K^\infty_\theta$ is formed by the functions $f\in H^\infty$ that annihilate (with respect to the usual pairing) the shift-invariant subspace $\theta H^1$ of the Hardy space $H^1$. Assuming that $B$ is an interpolating Blaschke product with zeros $\{a_j\}$, we characterize the traces of functions from $K^\infty_B$ on the sequence $\{a_j\}$. The trace space that arises is, in general, non-ideal (i.e., the sequences $\{w_j\}$ belonging to it admit no nice description in terms of the size of $|w_j|$), but we do point out explicit -- and sharp -- size conditions on $|w_j|$ which make it possible to solve the interpolation problem $f(a_j)=w_j$ ($j=1,2,\dots$) with a function $f\in K^\infty_B$.