RIGIDITY OF HOLOMORPHIC GENERATORS AND ONE-PARAMETER SEMIGROUPS

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 ∆. Namely,  if f ∈ Hol(∆, C) is the generator of a  one- parameter continuous semigroup  {F_t }_t ≥ 0, we show 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 ∆. Note, that if F is a self-mapping of ∆ 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 ≥ 0 and { G_t }_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 ∆.