## Complex geodesics on convex domains.(English)Zbl 0785.46044

Progress in functional analysis, Proc. Int. Meet. Occas. 60th Birthd. M. Valdivia, Peñíscola/Spain, North-Holland Math. Stud. 170, 333-365 (1992).
Let $$\rho$$ denote the Poincaré metric on the open unit disk $$\mathbb{D}$$ of $$\mathbb{C}$$. Let $${\mathcal D}$$ be a convex bounded domain in a complex Banach space $$X$$. For $$p,q\in {\mathcal D}$$ let $$d(p,q)=\sup \rho(f(p),f(q))$$, where the sup is taken over all holomorphic mappings $$f: {\mathcal D}\to \mathbb{D}$$; this is called the Carathéodory metric on $${\mathcal D}$$. A complex geodesic is a holomorphic mapping $$\phi: \mathbb{D}\to {\mathcal D}$$ such that $$\rho(u,v)=d(\varphi (u),\varphi(v))$$ for all $$u,v\in\mathbb{D}$$. In this paper existence and uniqueness of complex geodesics joining two points of a convex bounded domain in a Banach space $$X$$ are considered. Existence is proved for the unit ball of $$X$$ under the assumption that $$X$$ is 1- complemented in its bidual. Uniqueness (up to reparametrisation) is proved for strictly convex bounded domains in spaces with the analytic RNP.
{Reviewer’s remark: In Theorem 4.4 and elsewhere a bar above $$\mathbb{D}$$ is occasionally missing}.
For the entire collection see [Zbl 0745.00031].
Reviewer: D.Werner (Berlin)

### MSC:

 46G20 Infinite-dimensional holomorphy 46B20 Geometry and structure of normed linear spaces 46B22 Radon-Nikodým, Kreĭn-Milman and related properties 46B25 Classical Banach spaces in the general theory

Valdivia, M.
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