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A Julia–Carathéodory theorem for hyperbolically monotone mappings in the Hilbert ball (2006)
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BibTeX
@MISC{Elin06ajulia–carathéodory,
author = {Mark Elin and Simeon Reich and David Shoikhet},
title = {A Julia–Carathéodory theorem for hyperbolically monotone mappings in the Hilbert ball},
year = {2006}
}
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Abstract
We establish a Julia–Carathéodory theorem and a boundary Schwarz– Wolff lemma for hyperbolically monotone mappings in the open unit ball of a complex Hilbert space. Let B be the open unit ball of a complex Hilbert space H with inner product 〈·, · 〉 and norm ‖ · ‖, and let ρ: B × B ↦ → R + be the hyperbolic metric on B ([8], p. 98), i.e., where ρ(x, y) = tanh −1 √ 1 − σ(x, y), (1) σ(x, y) = (1 − ‖x‖2)(1 − ‖y ‖ 2) |1 − 〈x, y〉 | 2, x, y ∈ B. (2) We denote by Nρ the class of all those self-mappings F: B ↦ → B which are nonexpansive with respect to ρ (ρ-nonexpansive), i.e., ρ F(x), F(y) ≤ ρ (x, y). (3) Note that the class Nρ properly contains the class Hol(B) of all holomorphic self-mappings of B ([7, 8]). Definition 1 A family S = {F(t)}t≥0 of self-mappings of B is said to be a one-parameter continuous semigroup (flow) on B if F(t + s) = F(t) ◦ F(s), t, s ≥ 0, (4) and lim t→0 + F(t) = I, (5) where I is the restriction of the identity mapping of H to B and the limit is taken pointwise with respect to the strong topology of H. 1 We denote F(t)x, the value of F(t) at x ∈ B, by Ft(x), t ≥ 0. Definition 2 A flow S = {F(t)}t≥0 on B is said to be generated if for each x ∈ B, there exists the strong limit f(x): = lim t→0 + 1
