$A$. Denote by $\exa$ the set of all proper extensions of $A$, i.e., the set of all relations $\wt A\in \C (\gH)$ such that $A\subset\wt A\subset
A^*$.

Next assume that $\cH_0$ is a Hilbert space, $\cH_1$ is a subspace in $\cH_0$ and $\cH_2:=\cH_0\ominus\cH_1$, so that $\cH_0=\cH_1\oplus\cH_2$. Denote by $P_j$ the orthoprojector in $\cH_0$ onto $\cH_j,\; j\in\{1,2\} $.

\[def2.3\]$\,$[@Mog06.2] A collection $\Pi=\bta$, where $\G_j: A^*\to \cH_j, \; j\in\{0,1\}$ are linear mappings, is called a boundary triplet for $A^*$, if the mapping $\G :\hat f\to
\{\G_0 \hat f, \G_1 \hat f\}$ from $A^*$ into $\cH_0\oplus\cH_1$ is surjective and the following Green’s identity $$\label{2.3}
(f',g)-(f,g')=(\G_1  \hat f,\G_0 \hat g)_{\cH_0}- (\G_0 \hat f,\G_1 \hat
g)_{\cH_0}+i(P_2\G_0 \hat f,P_2\G_0 \hat g)_{\cH_2}$$ holds for all $\hat
f=\{f,f'\}, \; \hat g=\{g,g'\}\in A^*$.

In the following propositions some properties of boundary triplets are specified (see [@Mog06.2]).

\[pr2.4\] If $\Pi=\bta$ is a boundary triplet for $A^*$, then $$\label{2.4}
\dim \cH_1=n_-(A)\leq n_+(A)=\dim \cH_0.$$ Conversely for any symmetric linear relation $A\in \C (\gH)$ with $n_-(A)\leq
n_+(A)$ there exists a boundary triplet for $A^*$.

\[pr2.5\] Let $\Pi=\bta$ be a boundary triplet for $A^*$. Then:

1\) $\ker \G_0\cap\ker\G_1=A$ and $\G_j$ is a bounded operator from $A^*$ into $\cH_j, \;  j\in\{0,1\}$;

2\) the set of 