Points fixes d'une application holomorphe d'un domaine borné dans lui- même. (Fixed points of a holomorphic mapping of a bounded domain into itself)
For a convex bounded domain X in \({\mathbb{C}}^ n\) and a holomorphic map \(f:X\to X\), the set Fix f of fixed points of f is an analytic variety in X, and if Fix \(f\neq 0\) then there is a holomorphic retraction \(X\to Fix f\). This result of the second author [Trans. Am. Math. Soc. 289, 345-353 (1985; Zbl 0589.32043)] has been generalized in certain situations for infinite dimensions by M. Abd-Alla and E. Vesentini. Now the authors prove: for X a bounded domain in a complex Banach space E and a holomorphic map \(f:X\to X\) with \(f(a)=a\) for some point \(a\in X\), if one of the conditions \((H_ 1)\) or \((H_ 2)\) is satisfied \(where\) (H\({}_ 1)\) \(E=Ker(f'_ a-id)+Im(f'_ a-id)\) \([f'_ a=derivative\) of f at the point \(a],\) (H\({}_ 2)\) \(E=dual\) \((E_*)\) for some \(E_*\), and \(f'_ a\) is continuous in the weak topology \(\sigma (E,E_*),\) then Fix f is a direct complex variety in a neighborhood of a, tangent to \(Ker(f'_ a-id).\) As a consequence, Fix f is shown to be a direct complex variety in X for X convex bounded in E under suitable conditions. This is akin to H. Cartan's uniqueness theorem: a holomorphic map \(f:X\to X\) with \(f(a)=a\), \(f'(a)=id\) is the identity.
scientific article; zbMATH DE number 4212446
scientific article