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| A Schwarz Lemma at the Boundary of Hilbert Balls |
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Citation： |
Zhihua CHEN,Yang LIU,Yifei PAN.A Schwarz Lemma at the Boundary of Hilbert Balls[J].Chinese Annals of Mathematics B,2018,39(4):695~704 |
| Page view： 884
Net amount： 645 |
Authors： |
Zhihua CHEN; Yang LIU;Yifei PAN |
Foundation： |
This work was supported by the National Natural Science
Foundation of China (Nos.11671361, 11571256) and the Zhejiang
Provincial Natural Science Foundation of China (No.LY14A010008). |
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| 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
$f\in C^{1+\alpha}$ at $z_0\in \partial \mathbb B$ with
$f(z_0)=w_0\in
\partial \mathbb B'$, then the Fr\'echet derivative operator
${\rm D}f(z_0)$ maps the tangent space $T_{z_0}(\partial \mathbb
B^n)$ to $T_{w_0}(\partial \mathbb B')$, the holomorphic tangent
space $T^{(1,0)}_{z_0}(\partial \mathbb B^n)$ to
$T^{(1,0)}_{w_0}(\partial \mathbb B')$, respectively. |
Keywords： |
Boundary Schwarz lemma, Separable Hilbert space, Holomorphic mapping, Unit ball |
Classification： |
32H02, 46E20, 30C80 |
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