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

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 → 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 B be the open unit ball in a complex Hilbert space and let F : B → B be holomorphic. 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. This geometric condition is equivalent to the fact that F is asymptotically strongly nonexpansive with respect to the hyperbolic metric in B.

Some Inequalities for the Horosphere Function and Hyperbolically Nonexpansive Mappings on the Hilbert Ball

“…The following two propositions provide necessary and sufficient conditions for some estimates of the horosphere function (we will use them in the proof of Theorem 5.2). The first of them has already been proven in [28]. Proposition 5.1.…”

“…For the case where F has no interior fixed point, the theorem has already been proved in [28]. Assume now that F has an interior fixed point ζ ∈ B.…”

“…In the setting of the Hilbert ball it has recently been shown in [28] that a similar conclusion holds even if the image F (B) is not strictly inside B but instead is contained in a horosphere in B. F is a holomorphic self-mapping of the open unit ball B with F (B) ⊂ E(τ, 1/a) for some a > 0, then its iterates F n converge to a point ζ ∈ B locally uniformly.…”

“…In the case where F is fixed point free, Theorem 5.2 was proved in [28]. Thus, it remains to prove the result when F has an interior fixed point ζ ∈ B.…”