I study semigroups and their application in PDEs, and I'm stuck with an idea that I can't understand. Let $T(t)$ and $S(t)$ be $C_0$ semigroup of bounded linear operator with the same infinitesimal generator $A$. Then $S = T$.
Proof:
Let $x\in D(A)$ and
$U[0;t] \to X $
$\, \,\,\,$ $s\to u(s) = T(t-s)S(s)x $
$\frac{d}{ds}U(s) = 0$
So $U$ is a constant
If $U$ was a real or a complex function, it's obvious but $U$ is an operator from $X$ to $X$.
My question is: Who allows us to use the property of derivatives from the real world to the operator world. And if the derivitive of $U$ is $0$, how do we justify, that $U$ is a constant?