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

David Shoikhet


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 $\mathcal{B}$.

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DOI: https://doi.org/10.5539/jmr.v4n2p45

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Journal of Mathematics Research   ISSN 1916-9795 (Print)   ISSN 1916-9809 (Online)

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