A Schwarz Lemma at the Boundary of Hilbert Balls

Abstract

In this paper, the authors prove a general Schwarz lemma at the boundary for the holomorphic mapping f between unit balls $\mathbb{B}$ and $\mathbb{B'}$ in separable complex Hilbert spaces $\mathcal{H}$ and $\mathcal{H'}$, respectively. It is found that if the mapping f ∈ C ^1+α at ${z_0} \in \partial \mathbb{B}$ with $f\left( {{z_0}} \right) = {w_0} \in \partial \mathbb{B}'$, then the Fréchet derivative operator Df(z ₀) maps the tangent space ${T_{{z_0}}}(\partial {\mathbb{B}^n})$ to ${T_{{w_0}}}(\partial {\mathbb{B}'})$, the holomorphic tangent space $T_{{z_0}}^{(1,0)}(\partial {\mathbb{B}^n})$ to $T_{{w_0}}^{(1,0)}(\partial {\mathbb{B}'})$, respectively.

Keywords

Boundary Schwarz lemma / Separable Hilbert space / Holomorphic mapping / Unit ball

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