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 $K_{\theta }^1$ be the shift-coinvariant subspace of the Hardy space $H^1$ generated by $\theta$. We describe the dual space to $K_{\theta }^1$ in terms of a bounded mean oscillation with respect to the Clark measure ${\sigma }_{\alpha }$ of $\theta$. Namely, we prove that $(K_{\theta }^1\cap z{H}^1{)}^{\ast }=\mathrm{BMO}({\sigma }_{\alpha })$. The result implies a two-sided estimate for the operator norm of a finite Hankel matrix of size $n\times n$ via $\mathrm{BMO}({\mu }_{2n})$-norm of its standard symbol, where ${\mu }_{2n}$ is the Haar measure on the group $\{\xi \in \mathbb{C}:{\xi }^{2n}=1\}$.

24 pages

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