A generalized version of the Earle-Hamilton fixed point theorem for the Hilbert ball

TL;DR

This paper extends the Earle-Hamilton fixed point theorem to holomorphic self-maps of the Hilbert ball, showing fixed point existence under a geometric condition related to horospheres and hyperbolic metrics.

Contribution

It generalizes the fixed point theorem by replacing the strict contraction condition with a horosphere containment condition in the Hilbert ball.

Findings

Fixed points exist under horosphere containment.

The geometric condition is equivalent to asymptotic strong nonexpansiveness.

The result applies to holomorphic maps not strictly mapping into the interior.

Abstract

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 $\mathcal{B}$.