Admissible Majorants for Model Subspaces of H 2, Part I: Slow Winding of the Generating Inner Function

Havin, Victor ; Mashreghi, Javad

Canadian journal of mathematics, Tome 55 (2003) no. 6, pp. 1231-1263

Résumé

A model subspace ${{K}_{\Theta }}$ of the Hardy space ${{H}^{2}}={{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)$ for the upper half plane ${{\mathbb{C}}_{+}}$ is ${{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)\ominus \Theta {{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)$ where $\Theta $ is an inner function in ${{\mathbb{C}}_{+}}$ . A function $\omega :\,\mathbb{R}\mapsto [0,\,\infty )$ is called an admissible majorant for ${{K}_{\Theta }}$ if there exists an $f\,\in \,{{K}_{\Theta }},\,f\,\not{\equiv }\,0,\,|f\left( x \right)|\,\le \,\omega \left( x \right)$ almost everywhere on $\mathbb{R}$ . For some (mainly meromorphic) $\Theta $ 's some parts of Adm $\Theta $ (the set of all admissible majorants for ${{K}_{\Theta }}$ ) are explicitly described. These descriptions depend on the rate of growth of arg $\Theta $ along $\mathbb{R}$ . This paper is about slowly growing arguments (slower than $x$ ). Our results exhibit the dependence of Adm $B$ on the geometry of the zeros of the Blaschke product $B$ . A complete description of Adm $B$ is obtained for $B$ 's with purely imaginary (“vertical”) zeros. We show that in this case a unique minimal admissible majorant exists.

Détail

DOI : 10.4153/CJM-2003-048-8

Mots-clés : 30D55, 47A15, Hardy space, inner function, shift operator, model subspace, Hilbert transform, admissible majorant

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