Duality theorems for coinvariant subspaces of $H^1$

Abstract

Let $\theta$ be an inner function satisfying the connected level set condition of B. Cohn, and let $K^{1}_{\theta}$ be the shift-coinvariant subspace of the Hardy space $H^1$ generated by $\theta$. We describe the dual space to $K^{1}_{\theta}$ in terms of a bounded mean oscillation with respect to the Clark measure $\sigma_\alpha$ of $\theta$. Namely, we prove that $(K^{1}_{\theta} \cap zH^1)^* = {\rm BMO}(\sigma_\alpha)$. The result implies a two-sided estimate for the operator norm of a finite Hankel matrix of size $n\times n$ via ${\rm BMO}(\mu_{2n})$-norm of its standard symbol, where $\mu_{2n}$ is the Haar measure on the group $\{\xi \in \mathbb{C}: \xi^{2n} = 1\}$.