2005/12/21 by Elin, M., Levenshtein, M., Shoikhet, D. +1
#30D05 #47H20 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.math/0512482
In this paper we establish a rigidity property of holomorphic generators by using their local behavior at a boundary point τ of the open unit disk Δ. Namely, if f\inHol(Δ,ℂ) is the generator of a one-parameter continuous semigroup \Ft\t≥0, we state that the equality f(z)=o(|z-τ|3) when z→τ in each non-tangential approach region at τ implies that f vanishes identically on Δ. Note, that if F is a self-mapping of Δ 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 \Ft\t≥0 and \Gt\t≥0, with generators f and g respectively, commute if and only if the equality f=αg holds for some complex constant α. This fact gives simple conditions on the generators of two commuting semigroups at their common null point τ under which the semigroups coincide identically on Δ.