is a subspace in $\cH_0$ and $\G_j: A^*\to \cH_j, \;
j\in\{0,1\}$ are linear maps. Then a collection $\Pi=\bta$ is a boundary triplet (a $D$-triplet in terminology of [@Mog06.2]) for $A^*$ if the map $\G:=(\G_0\;\;\G_1)^\top:A^*\to \cH_0\oplus\cH_1$ is surjective and the identity $$\label{1.3}
(f',g)-(f,g')=(\G_1  \hat f, \G_0 \hat g)-(\G_0\hat f,\G_1\hat g)+i(P_2\G_0\hat
f,P_2\G_0\hat g), \quad \hat f=\{f,f'\},\;\hat g=\{g,g'\}\in  A^*$$ holds in place of (here $P_2$ is the orthoprojector in $\cH_0$ onto $\cH_2:=\cH_0\ominus\cH_1$). Associated with such a triplet $\Pi$ is the Weyl function $M_{\Pi +} (\l)$ defined for all $\l\in\bC_+$ by $$\begin{gathered}
\label{1.4}
\G_1\{f_\l,\l f_\l\}=M_{\Pi +}(\l)\G_0\{f_\l,\l f_\l\}, \quad f_\l\in\ker
(A^*-\l)\end{gathered}$$ The function $M_{\Pi +}(\l)$ is holomorphic on $\bC_+$, takes on values in $[\cH_0,\cH_1]$ and possesses a number of properties similar to those of the Weyl function . In particular, the function $M_\Pi(\l)=M_{\Pi
+}(\l)\up \cH_1$ is a uniformly strict Nevanlinna function with values in $[\cH_1]$.

A boundary triplet $\bta$ for $A^*$ enables to parametrize efficiently all proper extensions of $A$. Namely, if $\cK$ is a Hilbert space and $\{(C_0,C_1);\cK\}$ is a pair of operators $C_j\in [\cH_j,\cK]$, then the equality (the abstract boundary condition) $$\label{1.5}
 \wt A= \{ \hat f\in A^*: C_0\G_0  \hat f + C_1 \G_1 \hat f =0\}$$ defines the proper extension $A\subset \wt 