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

David Shoikhet

doi:10.5539/jmr.v4n2p45

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 $\mathcal{B}$.

Highlights

Introduction and NotionsLet ∆ be the open unit disk in the complex plane C, and let F be a holomorphic self-mapping of ∆
According to the Earle-Hamilton fixed point theorem, if a holomorphic mapping F : D → D maps D strictly into itself, it has a unique fixed point and its iterates converge to this fixed point locally uniformly
We show that a similar conclusion holds even if the image F(B) is not strictly inside B, but is contained in a horosphere internally tangent to the boundary of B

Summary

Introduction and Notions

Let ∆ be the open unit disk in the complex plane C, and let F be a holomorphic self-mapping of ∆. 1011-1021) (see Goebel, K. et al, 1984, Theorem 25.2) asserts that: If a ρ-nonexpansive (holomorphic) self-mapping F : B → B is fixed point free, there is a unique sink point τ ∈ ∂B for F. 97-106), that if dim H < ∞, iterates of a fixed point free holomorphic self-mapping B converge uniformly on compact subsets on B to a sink point τ ∈ ∂B. If dim H > 1, even for holomorphic mappings, a converse assertion is no longer true: if F has a boundary sink point, F is not necessarily fixed point free (see Theorem 25.1 in Goebel, K. et al, 1984 and examples therein). 88-90) shows that for the infinite dimensional case iterates of a holomorphic self-mapping of B do not necessarily converge to a sink point even if it is a unique boundary regular fixed point of F. D(F(z), τ) ≤ d 1 (z + F(z)), τ provides the pointwise convergence (in the norm of H) of iterates Fn(z) to the point τ for all z ∈ B

Local Uniform Convergence of Iterates

Asymptotically Strongly Nonexpansive Mappings

Proof of Main Result