Commuting semigroups of holomorphic mappings

Authors

Let

$S_{1}=\left\{F_t\right\}_{t\geq 0}$ and

$S_{2}=\left\{G_t\right\}_{t\geq 0}$ be two continuous semigroups of holomorphic self-mappings of the unit disk

$\Delta=\{z:|z|<1\}$ 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.,

$F_{t}\circ G_{s}=G_{s}\circ F_{t}$ for all

$s,t\geq 0$ . As an auxiliary result, we show that the existence of the (angular or unrestricted)

$n$ -th derivative of the generator

$f$ of a semigroup

$\left\{F_t\right\}_{t\geq 0}$ at a boundary null point of

$f$ implies that the corresponding derivatives of

$F_{t}$ ,

$t\geq 0$ , also exist, and we obtain formulae connecting them for

$n=2,3$ .