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arXiv · 1401.0452

Duality theorems for coinvariant subspaces of $H^1$

R. V. Bessonov
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Abstract

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

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BibTeXRIS

R. V. Bessonov. 2014-01-02. Duality theorems for coinvariant subspaces of $H^1$. https://doi.org/10.1016/j.aim.2014.11.012

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