Commutation of $c_0$-semigroups

Question

Task is: given two $c_0$-semigroups $T(t)$ and $S(t)$ in Banach space X with property $\forall \;t \geq 0 : T(t)S(t) = S(t)T(t)$, show that $\forall \; t, s \geq 0 : T(t)S(s) = S(s)T(t)$.

I've tried using some addition-subtraction tricks and differentiating $\varphi(t,s) = T(t)S(s)x - S(s)T(t)x$ (defined for arbitrary $x \in X$), but none of these methods succeeded.

Answer 1

Hint: Fix $t \in (0,\infty)$ and set $$ A := \{s \in [0,\infty): T(t)S(s) = S(s)T(t)\}. $$ Show that if $s \in A$ and $k \in \mathbb{N} := \{1,2,\dots\}$, then $ks \in A$ and $s/k \in A$, too.