\l\in\bC_+,
\label{2.8}\\
(\G_1+iP_2\G_0)\up \hat\gN_z (A)=M_{\Pi -}(z)P_1\G_0 \up \hat\gN_z (A),\quad
z\in \bC_-.\label{2.9}\end{gathered}$$ (here $\pi_1$ is the orthoprojection in $\gH\oplus\gH$ onto $\gH\oplus \{0\}$). According to [@Mog06.2] all functions $\g_{\Pi\pm}$ and $M_{\Pi\pm} $ are holomorphic on their domains and $(M_{\Pi +}(\l))^*=M_{\Pi-}(\ov \l),
\;\l\in\bC_+$.

\[rem2.6\] In the case $\cH_0=\cH_1:=\cH$ Definition \[def2.3\] coincides with that of the boundary triplet (boundary value space) $\Pi=\{\cH,\G_0,\G_1\}$ for $A^*$ given in [@GorGor]. For such a triplet $n_+(A)=n_-(A)=\dim \cH$, $\,A_0(=\ker \G_0 )$ is a self-adjoint extension of $A$ and the relations $$\label{2.10}
\g_\Pi(\l)=\pi_1(\G_0\up\hat\gN_\l(A))^{-1}, \qquad \G_1\up\hat
\gN_\l(A)=M_\Pi(\l)\G_0 \up\hat\gN_\l(A), \qquad \l\in\rho (A_0)$$ define the $\g$-field $\g_\Pi(\cd):\rho (A_0)\to [\cH,\gH]$ and the Weyl function $M_\Pi(\cd):\rho (A_0)\to [\cH]$ [@DM91] associated with operator functions – via $\g_\Pi(\l)=\g_{\Pi\pm}
(\l)$ and $M_\Pi(\l)=M_{\Pi\pm}(\l), \;\l\in\bC_\pm $.

Boundary relations and their Weyl families
==========================================

Let $\gH $ and $\cH_0$ be Hilbert spaces, let $\cH_1$ be a subspace in $\cH_0$, let $\cH_0=\cH_1\oplus\cH_2$ be the corresponding orthogonal decomposition of $\cH_0$ with $\cH_2:=\cH_0\ominus\cH_1$ and let $P_j$ be the orthoprojector in $\cH_0$ onto $\cH_j, \; j\in \{0,1\}$. In the sequel we deal with linear relations from $\gH^2$ into $\cH_0\oplus\cH_1$. If $\G$ is such a relation, then an element $\hat\f\in\G$ will be denoted by $\hat\f =\{\hat f,\hat h\}$, where $\hat f=\{f,f'\}\in\gH^2\; (f,\,f'\in \gH)$ and $\hat h=\{h_0,h_1\}\in
