The Carathéodory isometries between the products of balls
type: Item
instance of: scholarly article
title: The Carathéodory isometries between the products of balls
zbMATH Open document ID: 0859.32012
DOI: 10.1007/BF01198075
author: Włodzimierz Zwonek
published in: Archiv der Mathematik
publication date: 1997-04-07T00:00:00Z
review text: On sait que, si \(F=(F_1, F_2)\) est une isométrie (de Carathéodory) du disque unité ouvert \(B_1\) dans le bidisque \(B_1\times B_1\), l'une au moins des composantes \(F_1, F_2\) est une isométrie \(B_1\to B_1\), et l'Auteur [Arch. Math. 60, No. 2, 167-176 (1993; Zbl 0791.32010)] a déjà étendu ce fait aux isométries \(B_\alpha\to B_{\beta_1} B_{\beta_2}\), où les lettres telles que \(\alpha,\beta\) désignent des entiers \(>0\), et \(B_\alpha\) la boule unité euclidienne ouverte dans \(\mathbb{C}^\alpha\). Il considère maintenant une isométrie \(F= (F_{\beta_1}, \dots, F_{\beta_n})\) de \(B_{\alpha_1} \times \cdots \times B_{\alpha_m}\) dans \(B_{\beta_1} \times \cdots \times B_{\beta_n}\) et montre dans quels cas il existe un entier \(s\in [1,\inf (m,n)]\), \(s\) entiers distincts \(j_1, \dots, j_s\in \{1, \dots, m\}\) et \(s\) entiers distincts \(k_1, \dots, k_s \in\{1, \dots, n\}\) tels que, \(\forall t\in \{1, \dots, s\}\), \(F_{\beta_{k_t}}\) s'obtienne en composant la projection \(B_{\alpha_1} \times \cdots \times B_{\alpha_m} \to B_{\alpha_{j_t}}\) avec une isométrie \(B_{\alpha_{j_t}} \to B_{\beta_{k_t}}\). La construction de \(s\) donne en particulier: 1) \(n<3m\) assure l'existence de \(s\); 2) si \(m\leq n<3m\) et \(\sup (\alpha_1, \dots, \alpha_m) \leq\inf (\beta_1, \dots, \beta_n)\), alors \(s=m -[{n-m \over 2}]\), où \([ ]\) désigne la partie entière.
Mathematics Subject Classification ID: 32F45
zbMATH DE Number: 937033
zbMATH Keywords: products of balls
zbMATH Keywords: Carathéodory isometry
reviewed by: Michel Hervé
MaRDI profile type: Publication
cites work: Invariant distances and metrics in complex analysis
cites work: Isometers in the Cartesian product of n unit open Hilbert balls with hyperbolic metric
cites work: A note on Carathéodory isometries
full work available at URL: https://doi.org/10.1007/bf01198075
OpenAlex ID: W2078671550
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description: scientific article; zbMATH DE number 937033