I am reading Lemma 2.8 of "Semigroups of Linear Operators and Applications to Partial Differential Equations" by Pazy:
Let $A$ be the infinitesimal generator of a strongly continuous semigroup $T(t)$ satisfying $\|T(t)\| \leq M$ for $t \geq 0$. If $x \in D(A^2)$, then $\|Ax\|^2 \leq 4M^2 \|A^2x\| \|x\|$.
In part of the proof, it says: Using the fact that for any $x \in D(A)$, $T(b)x - T(a)x = \int_a^b T(s) Ax d s = \int_a^b AT(s)x ds$, it is easy to check that $$T(t)x - x = tAx + \int_0^t(t-s) T(s)Ax^2 ds.$$
But I don't know how to derive it from the fact it mentions. Could anyone help me with that?
Apply the fact given to you onto the term $AT(s)x$ in the integrand. That is \begin{aligned} \int^t_0A(T(s)x)\,ds &=\int^t_0A\Big(x+\int^s_0AT(v)x\,dv\Big)ds\\ &=tAx +\int^t_0\Big(\int^s_0A^2T(v)xdv\Big)ds \end{aligned} The rest should follow by changing order of integration. The validity of this relies on the strong continuity of the semigroup $T$.