Rigidity of holomorphic generations and one-parameter semigroups

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 is an element of Hol(Delta, C) is the generator of a one-parameter continuous semigroup {F-t}(t >= 0), we show that the equality f (z) = o (vertical bar z - tau vertical bar(3)) when z -> tau in each non-tangential approach region at tau implies that f vanishes identically on Delta. Note, hat 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 >= 0) and {G(t)}(t >= 0), with generators f and g respectively, commute if and only if the equality f = alpha g holds for some complex constant a. 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.