In my thesis I'm studying an article with which I'm having the following problem: let $E=\mathbb{R}^{d} \times L^{2}([0,+\infty))$ be the Hilbert space with norm $ \|\alpha\|^{2}:=|x|^{2}+\|z\|_{L^{2}}^{2} $ $\forall \alpha=(x, z) \in E$ and the corresponding inner product $\langle., .\rangle .$ Let $T$ be the linear unbounded operator on $E$ with domain $D(T)=\mathbb{R}^{d} \times H^{1}$ and defined by $$ T(y, w)=(y-w(0),-\dot{w}), \forall(y, w) \in D(T) $$
The unbounded operator $I+T^{*} T$ has domain $$ D\left(I+T^{*} T\right)=\left\{(y, w) \in E: w \in H^{2}, y=w(0)-\dot{w}(0)\right\} $$ and is given by $$ \left(I+T^{*} T\right)(y, w):=(2 y-w(0),-\ddot{w}+w), \forall(y, w) \in D\left(T^{*} T\right) $$ Now define $B=\left(I+T^{*} T\right)^{-1}$. For $\alpha=(x, z) \in E,(y, w)=B(\alpha)$ is defined as follows: firstly, $w \in H^{2}$ is the solution of $$ \left\{\begin{aligned} -\ddot{w}+w &=& z & \text { in }(0,+\infty) \\ -2 \dot{w}(0)+w(0) &=& x \end{aligned}\right. $$ secondly, $y$ is defined by $$ y=\frac{x+w(0)}{2}=w(0)-\dot{w}(0) $$ Set $B=\left(B_{1}, B_{2}\right)$ where $B_{2}(x, z)=w$ and $B_{1}(x, z)=y$. Setting: $$ \|\alpha\|_{B}^{2}:=\langle B(\alpha), \alpha\rangle, \forall \alpha=(x, z) \in E $$ Using the definition of $w,y$ we arrive at: $$ \|\alpha\|_{B}^{2}=\frac{|x|^{2}}{2}+\frac{|w(0)|^{2}}{2}+\|w\|_{H^{1}}^{2} $$ This is the setting in [Carlier, Guillaume, and Rabah Tahraoui. "Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory."], p. 758.
Now he claims that \begin{equation} \|z\|_{\left(H^{1}\right)^{\prime}}^{2}=\|w+\varphi\|_{H^{1}}^{2} \end{equation} where $$ \left\{\begin{aligned} -\ddot{\varphi}+\varphi &=0 \\ \dot{\varphi}(0) &=\frac{x-w(0)}{2}, \end{aligned} \quad \text { in }(0,+\infty)\right. $$ so that he can have the following bounds for $m,C \geq 0$ : $$ \|z\|_{\left(H^{1}\right)^{\prime}}^{2} \leq 2\left(\|w\|_{H^{1}}^{2}+\|\varphi\|_{H^{1}}^{2}\right) \leq 2\|w\|_{H^{1}}^{2}+m\left(|x|^{2}+|w(0)|^{2}\right) $$ $$ \|(x, z)\|_{B}^{2} \geq C\left(|x|^{2}+\|z\|_{\left(H^{1}\right)^{\prime}}^{2}\right), \forall(x, z) \in E $$ My question is: why is \begin{equation} \|z\|_{\left(H^{1}\right)^{\prime}}^{2}=\|w+\varphi\|_{H^{1}}^{2} \end{equation} true?
The quetion is related to Why is this the adjoint operator? where you can find the contruction of the adjoint.