Rigidity of holomorphic generators and one-parameter semigroups

TL;DR

This paper proves a boundary rigidity property for holomorphic generators of semigroups, showing that certain boundary behavior implies the generator vanishes, and characterizes when two semigroups commute based on their generators.

Contribution

It establishes a boundary rigidity criterion for holomorphic generators and characterizes commuting semigroups via their generators, extending classical results.

Findings

Boundary behavior $f(z)=o(|z- au|^{3})$ implies $f$ vanishes identically.

Two semigroups commute iff their generators are scalar multiples.

Conditions for semigroups to coincide based on generator relations.

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 $\Delta$.