K & = &
 \big\{ \{\{
(I+iB_{1})A_{1\s}^{\half} f, (I+iB_{2})A_{2\s}^{\half}f \}, f_{1}' + f_{2}' \}\, :
 \\
&&
\quad \quad \quad \{ (I+iB_{1})A_{1\s}^{\half} f, f_{1}' \} \in A_{1}^{\half},
\{ (I+iB_{2})A_{2\s}^{\half} f, f_{2}' \} \in A_{2}^{\half} \big\}
\nonumber \\
&&
\hspace{-0.6cm} \subset \hspace{0.2cm} (\sH \times \sH) \times \sH. \nonumber\end{aligned}$$ Clearly, the domain and multivalued part of $K$ are given by $$\dom K= \sD_0, \quad \mul K=\mul (H_{1} + H_{2}),$$ where $$\label{domK}
   \sD_{0}= \left\{\,\{(I+iB_{1}) A_{1\s}^{1/2}f,
   (I+iB_{2}) A_{2\s}^{1/2}f\} :
   \,  f\in \dom H_{1} \cap \dom H_{2}
  \,\right\} .$$ The closure of $\sD_{0}$ in $\sH \times \sH$ will be denoted by $\sD$.

\[lem3.1\] The relations $K$, $\Phi$, and $\Psi$ satisfy the following inclusions: $$\label{s-s-trits}
  K \subset \Phi \subset \Psi^*, \quad \Psi \subset \Phi^* \subset
K^*.$$

To see this note that $K \subset \Phi$ follows from and , and that $\Psi \subset \Phi^*$ follows from and . Therefore, also $\Phi^* \subset K^*$ and $\Phi \subset \Phi^{**} \subset \Psi^*$.

The Friedrichs and the Kreĭn extensions of $H_1+H_2$ {#sec3.2}
----------------------------------------------------

The descriptions of the Friedrichs extension and the Kreĭn extension $(H_{1}+H_{2})_F$ and $(H_{1}+H_{2})_K$ of $H_1+H_2$ are now recalled from [@HSS19]. For this, define the orthogonal sum of the operators $B_{1}$ and $B_{2}$ in $\sH \times \sH$ by $$B_\oplus:=B_1 