Commuting semigroups of holomorphic mappings - Elin, Mark; Levenshtein, Marina; Reich, Simeon; Shoikhet, David (MATHEMATICA SCANDINAVICA, 2008)
Let $S_{1}=\left\{F_t\right\}_{t≥ 0}$ and $S_{2}=\left\{G_t\right\}_{t≥ 0}$ be two continuous semigroups of holomorphic self-mappings of the unit disk $Δ=\{z:|z|<1\}$ 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., $F_{t}° G_{s}=G_{s}° F_{t}$ 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 $\left\{F_t\right\}_{t≥ 0}$ 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 $n=2,3$.