A Generalized Version of the Earle-Hamilton Fixed Point Theorem for the Hilbert Ball

Let $D$ be a bounded domain in a complex Banach space. According to the Earle-Hamilton fixed point theorem, if a holomorphic mapping $F: D mapsto D$ maps $D$ strictly into itself, then it has a unique fixed point and its iterates converge to this fixed point locally uniformly. Now let $mathcal{B}$ be the open unit ball in a complex Hilbert space and let $F : mathcal{B} mapsto mathcal{B}$ be holomorphic. We show that a similar conclusion holds even if the image $F(mathcal{B})$ is not strictly inside $mathcal{B}$, but is contained in a horosphere internally tangent to the boundary of $mathcal{B}$. This geometric condition is equivalent to the fact that $F$ is asymptotically strongly nonexpansive with respect to the hyperbolic metric in $mathcal{B}$.