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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

Biographic References:

Valdivia, M.
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