Pauline Mellon

Iteration of compact holomorphic maps on a Hilbert ball

Given a compact holomorphic fixed-point-free self-map, f , of the open unit ball of a Hilbert space, we show that the sequence of iterates, (fn), converges locally uniformly to a constant map ξ with ‖ξ‖ = 1. This extends results of Denjoy (1926), Wolff (1926), Hervé (1963) and MacCluer (1983). The result is false without the compactness assumption, nor is it true for all open balls of J∗-algebras.

Holomorphic functions on symmetric Banach manifolds of compact type are constant

1. Throughout, we use C to denote the one-point compactification C∪{∞} of C. Let (Z, ‖ · ‖) denote a Banach space over the complex numbers C. Let L(Z) and L(Z) denote, respectively, the space of continuous linear and anti-linear (or conjugate linear) mappings from Z to Z endowed with the operator norm. We consider a continuous triple product {·, ·, ·} : Z3 → Z, which is symmetric and linear in the outer variables and conjugate linear in the middle variable. In particular, this implies

Angular derivatives on bounded symmetric domains

In this paper we generalise the classical Julia-Wolff-Carathéodory theorem to holomorphic functions defined on bounded symmetric domains.