Points fixes d'applications holomorphes dans un produit fini de boules- unités d'espaces de Hilbert
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.
complex geodesics of a bounded domain in a complex Banach space
scientific article; zbMATH DE number 3905234
scientific article