A Julia–Carathéodory theorem for hyperbolically monotone mappings in the Hilbert ball

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 7→ R be the hyperbolic metric on B ([8], p. 98), i.e., ρ(x, y) = tanh √ 1− σ(x, y), (1) where σ(x, y) = (1 − ‖x‖)(1 − ‖y‖) |1− 〈x, y〉|2 , x, y ∈ B. (2) We denote by Nρ the class of all those self-mappings F : B 7→ B which are nonexpansive with respect to ρ (ρ-nonexpansive), i.e.,