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 $n=2,3$.
Categories: math.CV math.DS