Let $X$ be a complex Hilbert space; and let $A:D(A)\subset X \to X$ be a $\mathbb C-$linear operator. Assume that $A$ is self-adjoint(so that $D(A)$ is a dense subset of $X$) and that $A\leq 0$ (i.e., $(Ax,x)\leq 0$ for all $x\in D(A)$).\ Since $A$ is self-adjoint, $iA:D(A)\subset X \to X$ defined by $(iA)x:=iAx$ for $x\in D(A)$ is also $\mathbb C-$linear and is skew-adjoint. In particular, $iA$ generates the group of isometries $\{\mathcal{T}(t)\}_{t\in \mathbb R}$ on $X.$ We deduce easily from the skew-adjointness of $iA$ that, $$\mathcal{T}(t)^{\ast}=\mathcal{T}(-t), \forall t \in \mathbb R.$$
For every $x\in X$ and every $f\in C([0,T], X)$ (where $T\in \mathbb R$), there exists a unique solution of the problem $$ (NH) \begin{cases} u\in C([0, T], X) \cap C^{1}([0, T], (D(A))'),\\ i\frac{du}{dt}+\bar{A}u+f=0,\\ u(0)=x. \end{cases} $$
My Question is:
How to show, $u\in C([0, T], X) $ is a solution of the above problem $(NH)$ if and only if $u$ satisfies $$u(t)=\mathcal{T}(t)x+i\int_{0}^{t}\mathcal{T}(t-s)f(s) ds;$$ for all $t\in [0, T].$
Note. For more detail of the above problem see "Semilinear Schr\"odinger Equations" by Thierray Cazenave, section 1.6
Thanks,
Edit: We put, $(D(A))'=$ The dual of $D(A);$ and $A$ can be extended to a self adjoint operator $\bar{A}$ on $(\mathcal{D}(A))'$ with domain $X.$ We note, $\bar{A}|_{D(A)}=A.$