A Carathéodory theorem for the bidisk via Hilbert space methods

By Jim Agler, John E. McCarthy, N. Young. Mathematische Annalen, 2010

72 citations recorded by the source corpus.

If φ is an analytic function bounded by 1 on the bidisk \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb D}^2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau\in\partial({\mathbb D}^2)}$$\end{document} is a point at which φ has an angular gradient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\nabla\varphi(\tau)}$$\end{document} then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\nabla\varphi(\lambda) \to \nabla\varphi(\tau)}$$\end{document} as λ → τ nontangentially in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb D}^2}$$\end{document}. This is an analog for the bidisk of a classical theorem of Carathéodory for the disk. For φ as above, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau\in\partial({\mathbb D}^2)}$$\end{document} is such that the lim inf of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(1-|\varphi(\lambda)|)/(1-\|\lambda\|)}$$\end{document} as λ → τ is finite then the directional derivative D−δφ(τ) exists for all appropriate directions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\delta\in{\mathbb C}^2}$$\end{document}. Moreover, one can associate with φ and τ an analytic function h in the Pick class such that the value of the directional derivative can be expressed in terms of h.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/fe76d1443ae98c4f8f77782c2049c31acd9ff45e

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The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

  1. Jim Agler
  2. N. Young
  3. John E. McCarthy
  4. Joseph A. Ball
  5. Victor Vinnikov
  6. Orr Shalit
  7. Gregory Marx
  8. Greg Knese

In Geometry and Topology

  1. Jim Agler
  2. N. Young
  3. John E. McCarthy
  4. Gautam Bharali
  5. Andrew M. Zimmer

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