Rigidity of holomorphic generators and one-parameter semigroups

Abstract

In this paper we establish a rigidity property of holomorphic generators by using their local behavior at a boundary point $\tau$ of the open unit disk $\Delta$. Namely, if $f\in \mathrm{Hol}(\Delta ,\mathbb{C})$ is the generator of a one-parameter continuous semigroup $\{F_t{\}}_{t\ge 0}$, we state that the equality $f(z)=o(|z-\tau {|}^3)$ when $z\to \tau$ in each non-tangential approach region at $\tau$ implies that $f$ vanishes identically on $\Delta$. Note, that if $F$ is a self-mapping of $\Delta$ then $f=I-F$ is a generator, so our result extends the boundary version of the Schwarz Lemma obtained by D. Burns and S. Krantz. We also prove that two semigroups $\{F_t{\}}_{t\ge 0}$ and $\{G_t{\}}_{t\ge 0}$, with generators $f$ and $g$ respectively, commute if and only if the equality $f=\alpha g$ holds for some complex constant $\alpha$. This fact gives simple conditions on the generators of two commuting semigroups at their common null point $\tau$ under which the semigroups coincide identically on $\Delta$.

20 pages

Cited by in corpus (2)

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