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Admissible Majorants for Model Subspaces of H2, Part I: Slow Winding of the Generating Inner Function

Published online by Cambridge University Press:  20 November 2018

Victor Havin  and
Javad Mashreghi
Victor Havin
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2K6 e-mail: [email protected] Department of Mathematics and Mechanics, St. Petersburg State University, Russia 198904
Javad Mashreghi
Affiliation:
Département de mathématiques et de statistique, Université Laval, Laval, Québec, G1K 7P4 e-mail: [email protected]
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Abstract

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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.

Keywords

30D5547A15Hardy spaceinner functionshift operatormodel subspaceHilbert transformadmissible majorant
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 6 , 01 December 2003 , pp. 1231 - 1263
Copyright
Copyright © Canadian Mathematical Society 2003

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