COMMUTING SEMIGROUPS OF HOLOMORPHIC MAPPINGS

Let S-1 = {F-t)(t >= 0) and S-2 = {G(t)}(t >= 0) be two continuous semigroups of holomorphic self-mappings of the unit disk Delta = {z : vertical bar z vertical bar < 1) generated by f and g, respectively. We present conditions on the behavior of f (or g) in a neighborhood of a fixed point of S-1 (or S-2), under which the commutativity of two elements, say, F-1 and G(1) of the semigroups implies that the semigroups commute, i.e., F-t o G(s) = G(s) o F-t for all s, t >= 0. As an auxiliary result, we show that the existence of the (angular or unrestricted) n-th derivative of the generator f of a semigroup {F-t}(t >= 0) at a boundary null point of f implies that the corresponding derivatives of F-t, t >= 0, also exist, and we obtain formulae connecting them for n = 2, 3.