\left\{ \{f_{1},f_{2}\}, f_{1}' + f_{2}' \right\}
 :\, \{f_{j},f_{j}' \} \in A_{j}^{\half} , \,
 1 \le j \le 2\,\right\}.$$ Clearly, $\Phi$ is a relation whose domain and multivalued part are given by $$\dom \Phi = \dom A_{1}^{\half} \times \dom A_{2}^{\half},
 \quad \mul \Phi=\mul H_{1} +\mul H_{2}.$$ The relation $\Phi$ is not necessarily densely defined in $\sH
\times \sH$, so that in general $\Phi^*$ is a relation as $\mul
\Phi^*=(\dom \Phi)^\perp$. Furthermore, the adjoint $\Phi^*$ of $\Phi$ is the relation from $\sH$ to $\sH \times \sH$, given by $$\label{s-s-Zwei}
\Phi^* = \left\{ \left\{ h, \{ h_{1}', h_{2}'\} \right\} \, :
\{h, h_{j}'\} \in A_{j}^{\half}, \, 1 \le j \le 2 \right\}.$$ The identity shows that the (orthogonal) operator part $(\Phi^*)_\s$ of $\Phi^*$ is given by: $$\begin{aligned}
\label{s-s-qus}
 (\Phi^*)_\s & = &\left\{ \left\{ h, \{ h_{1}', h_{2}'\} \right\} \,
 : \{h, h_{j}'\} \in A_{j\s}^{\half}, \, 1 \le j \le 2  \right\}
 \\
& = & \left\{ \left\{ h, \{ A_{1\s}^{\half} h, A_{2\s}^{\half} h\} \right\}
\, : h \in \dom A_{1}^{\half} \cap \dom A_{2}^{\half} \right\}.
\nonumber\end{aligned}$$ The identities and show that $$\dom  \Phi^{*}  = \dom A_{1}^{\half} \cap \dom A_{2}^{\half},
\, \, \mul \Phi^*=\mul H_{1} \times \mul H_{2},
\, \, \ran (\Phi^{*})_\s  = \sF_{0},$$ where 