Angular derivatives of holomorphic maps in infinite dimensions (Q1353470)

The authors generalize to infinite dimensions the classical Pick-Julia-Carathéodory theory of angular limits and angular derivative of holomorphic functions on balls and half-planes. Previous results in this direction have been obtained e.g. by \textit{Ky Fan} [Pac. J. Math. 121, 67-72 (1986; Zbl 0588.47018)] and by \textit{K. Włodarczyk} [Indagationes Math. 50, No. 4, 455-468 (1988; Zbl 0665.46034)]. In the present paper, the authors look at holomorphic self maps of the infinite dimensional balls \(H_0= \{x\in H: |x |<1\}\) and \({\mathcal B}_0 =\{T\in {\mathcal L} (H,H): |T|<1\}\) of a Hilbert space \(H\) and its associated space \({\mathcal L} (H,H)\) of linear bounded endomorphisms and the analoguous problems for the infinite dimensional half-planes \(\{x\in H: \text{Re} \langle x,v \rangle- |x-\langle x,v \rangle v|^2 >0\}\) and \(\{T\in {\mathcal L} (H,H): \text{Re} T>0\}\). The results are too technical to be cited here.