Points fixes d'applications holomorphes dans un produit fini de boules- unités d'espaces de Hilbert
type: Item
instance of: scholarly article
title: Points fixes d'applications holomorphes dans un produit fini de boules- unités d'espaces de Hilbert
zbMATH Open document ID: 0567.46022
DOI: 10.1007/BF01789397
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1984-01-01T00:00:00Z
review text: Vesentini (1981, 1982) has studied complex geodesics of a bounded domain D in a complex Banach space E, that is, holomorphic mappings of the unit disc \(\Delta\) of \({\mathbb{C}}\) to D that are isometries for the Carathéodory distance. He showed in a number of cases that there is one and only one complex geodesic passing through two given points of D. He deduced that, if there is a unique complex geodesic passing through two fixed points a and b of a holomorphic mapping f of D into itself, the image of this complex geodesic in D is formed by fixed points of f. This allows to generalize results of Hervé (1963) and Renaud (1973). The author (1984) has shown that, if D is a convex bounded domain in \({\mathbb{C}}^ n\), and f is a holomorphic mapping of D into itself having two distinct fixed points a and b through which passes at least a complex geodesics, then there is a complex geodesics passing through them formed by fixed points of f. In this article, the author shows that, if \(B=B_ 1\times...\times B_ p\) is a finite product of open unit balls in Hilbert spaces, and f is a holomorphic mapping of B into itself having two distinct fixed points, then there is a complex geodesic passing by them formed by fixed points of f (this extends previous results of Vesentini (1982) and Hervé (1963) in finite dimensions). The proof rests on results of Earle and Hamilton, Goebel (1980, 1982), Sekowski (1980) and Stachura (1980). The author points out that Kuczumov and Stachura (preprint) have proved that, if f is a holomorphic mapping of the product \(B_ 1\times B_ 2\) of unit balls of Hilbert spaces into itself having two distinct fixed points, the set of fixed points of f contains some complex geodesic.
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 3905234
zbMATH Keywords: complex geodesics of a bounded domain in a complex Banach space
zbMATH Keywords: Carathéodory distance
zbMATH Keywords: fixed points
author: Jean-Pierre Vigué
MaRDI profile type: Publication
cites work: Q5604772
cites work: Q3892778
cites work: Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball
cites work: Q3949575
cites work: Q4196902
cites work: Quelques propriétés des applications analytiques d'une boule à \(m\) dimensions dans elle-même
cites work: Q3244654
cites work: Q3744762
cites work: Q4402615
cites work: Q3949259
cites work: Q4743072
cites work: Géodésiques complexes et points fixes d'applications holomorphes
Recommended article: Q3746097
Recommended article: Points Fixes D'Applications Holomorphes Dans Un Domaine Borne Convexe De C n
Recommended article: Q3744479
Recommended article: Q3712968
Recommended article: Géodésiques complexes et points fixes d'applications holomorphes
Recommended article: Fixed Points of Holomorphic Mappings in the Cartesian Product of n Unit Hilbert Balls
Recommended article: Fixed points of holomorphic mappings in the Hilbert ball
Recommended article: Q3728373
Recommended article: Fixed points of holomorphic mappings for domains in Banach spaces
Recommended article: Approximating fixed points of holomorphic mappings in the Hilbert ball
description: scientific article; zbMATH DE number 3905234