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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A rigidity theorem for holomorphic generators on the Hilbert ball


Authors: Mark Elin, Marina Levenshtein, Simeon Reich and David Shoikhet
Journal: Proc. Amer. Math. Soc. 136 (2008), 4313-4320
MSC (2000): Primary 30C45, 30D05, 46T25, 47H20
Published electronically: June 25, 2008
MathSciNet review: 2431045
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Abstract: We present a rigidity property of holomorphic generators on the open unit ball $ \mathbb{B}$ of a Hilbert space $ H$. Namely, if $ f\in\operatorname{Hol}(\mathbb{B},H)$ is the generator of a one-parameter continuous semigroup $ \left\{F_t\right\}_{t\geq 0}$ on $ \mathbb{B}$ such that for some boundary point $ \tau\in\partial\mathbb{B}$, the admissible limit $ K$- $ \lim\limits_{z\rightarrow\tau}\frac{f(x)}{\Vert x-\tau\Vert^{3}}=0$, then $ f$ vanishes identically on $ \mathbb{B}$.


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Additional Information

Mark Elin
Affiliation: Department of Mathematics, ORT Braude College, P.O. Box 78, 21982 Karmiel, Israel
Email: mark.elin@gmail.com

Marina Levenshtein
Affiliation: Department of Mathematics, The Technion — Israel Institute of Technology, 32000 Haifa, Israel
Email: marlev@list.ru

Simeon Reich
Affiliation: Department of Mathematics, The Technion — Israel Institute of Technology, 32000 Haifa, Israel
Email: sreich@tx.technion.ac.il

David Shoikhet
Affiliation: Department of Mathematics, ORT Braude College, P.O. Box 78, 21982 Karmiel, Israel
Email: davs27@netvision.net.il

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09417-3
PII: S 0002-9939(08)09417-3
Keywords: Angular limit, Hilbert ball, holomorphic generator, $K$-limit, one-parameter continuous semigroup, rigidity
Received by editor(s): July 30, 2007
Received by editor(s) in revised form: October 20, 2007
Published electronically: June 25, 2008
Additional Notes: The third author was partially supported by the Fund for the Promotion of Research at the Technion and by the Technion President’s Research Fund.
All the authors thank the referee for several helpful comments and suggestions.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society