Commuting semigroups of holomorphic mappings

Abstract

Let $S_1={\left\{F_t\right\}}_{t\ge 0}$ and $S_2={\left\{G_t\right\}}_{t\ge 0}$ 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 $s,t\ge 0$. As an auxiliary result, we show that the existence of the (angular or unrestricted) $n$-th derivative of the generator $f$ of a semigroup ${\left\{F_t\right\}}_{t\ge 0}$ at a boundary null point of $f$ implies that the corresponding derivatives of $F_t$, $t\ge 0$, also exist, and we obtain formulae connecting them for $n=2,3$.