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 xi with \\xi\\ = 1. This extends results of Denjoy (1926), Wolff (1926), Herve (1963) and MacCluer (1983). The result is false without the compactness assumption, nor is it true for all open balls of J*-algebras.