Proposition 2

Let B be a continuous endomorphism of E, independent of t, and commuting with A(t) for all $t \in J$; then B commutes with $C(t, t_0)$ for all t and $t_0$ in J.

Indeed, by (5)

so $\frac{d}{dt}(BC - CB) = A(BC - CB)$; but $BC(t_0, t_0) - C(t_0, t_0)B = 0$, so (IV, p. 179, th. 1) $BC(t, t_0) - C(t, t_0)B = 0$ for all $t \in J$.

An important instance of prop. 2 is that where E is endowed with the structure of a normed vector space with respect to the field of complex numbers $\mathbf{C}$, and where, for every $t \in J$, $A(t)$ is an endomorphism of E for this vector space structure; this means that $A(t)$ commutes with the continuous endomorphism $x \mapsto tx$ of E (for the vector space structure over $\mathbf{R}$); then $C(t, t_0)$ commutes with this endomorphism, which means that for any $t$ and $t_0$ in J, the map $C(t, t_0)$ is a continuous endomorphism for the normed vector space structure of E over $\mathbf{C}$.