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 $\{F_{t}\}_{t\geq0}$, 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 $\{F_{t}\}_{t\geq0}$ and $\{G_{t}\}_{t\geq0}$, 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 $\Delta$.