Existence of a holomorphic retraction onto a common fixed point set of a family of commuting holomorphic self-mappings of \(B _{H} ^{n}\)
Let \(B_{H}\) be the open unit ball of a complex Hilbert space and let \(k_{B_{H}^{n}}\) be the Kobayashi distance on \(B_{H}^{n}\). In this paper, the following theorem is proved. Theorem. For any infinite family \(F\) of commuting holomorphic (\(k_{B_{H}^{n}}\)-nonexpansive) self-mappings of \(B_{H}^{n}\) with nonempty set \(\text{Fix} (F) = \bigcap_{f \in F} \text{Fix} (f)\) of common fixed points of \(F\), the set \(\text{Fix} (F)\) is a holomorphic (\(k_{B_{H}^{n}}\)-nonexpansive) retract of \(B_{H}^{n}\). This is a generalization of a theorem due to \textit{T. Kuczumow} [Nonlinear Anal. 9, 601-604 (1985; Zbl 0607.47053)], where \(F\) is here a finite family. In this case, \(\text{Fix} (F)\not= \emptyset\) is automatically fulfilled. The proof of the above theorem is divided into two cases. In case 1 (\(n=1\)), Kuczumow's theorem can be applied using ultranets and weak compactness of \(\overline{B_{H}}\). The case 2 (\(n > 1\)) is treated by iteration using the asymptotic center method in addition. Some remarks finish this paper. Here it is mentioned inter alia that the assumption \(\text{Fix} (F) \not= \emptyset\) is essential if \(F\) is infinite.
scientific article; zbMATH DE number 1901884
scientific article