A rigidity theorem for holomorphic generators on the Hilbert ball

Elin, Mark; Levenshtein, Marina; Reich, Simeon; Shoikhet, David

doi:10.1090/S0002-9939-08-09417-3

A rigidity theorem for holomorphic generators on the Hilbert ball
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by Mark Elin, Marina Levenshtein, Simeon Reich and David Shoikhet PDF

Proc. Amer. Math. Soc. 136 (2008), 4313-4320 Request permission

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 $K$-$\lim \limits _{z\rightarrow \tau }\frac {f(x)}{\|x-\tau \|^{3}}=0$, then $f$ vanishes identically on $\mathbb {B}$.

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