\cH_0\oplus\cH_1\; (h_0\in\cH_0,\; h_1\in\cH_1)$. In this case it will be convenient to write $$\label{3.1a}
\hat\f=\{\hat f,\hat h\}=\lb\hat f, {h_0 \choose h_1} \rb=\lb
\begin{pmatrix} f\cr f' \end{pmatrix},
\begin{pmatrix} h_0\cr h_1 \end{pmatrix}\rb.$$ If in addition $\cH_j$ is decomposed as $\cH_j=\cH_{j1}\oplus\cH_{j2}\oplus
\dots  \cH_{j,n_j}, \; j\in\{0,1\}$, then the equality will be also written as $$\hat\f=\lb\hat f, {\{h_{01}, h_{02}, \dots, h_{0,n_0}\}\choose \{h_{11},
h_{12}, \dots, h_{1,n_1}\}} \rb,$$ where $h_{0k}=P_{\cH_{0k}}h_0$ and $h_{1k}=P_{\cH_{1k}}h_1$ are components of $h_0$ and $h_1$ respectively.

For a linear relation $\BR$ its multivalued part is a linear relation from $\cH_0$ into $\cH_1$ given by $$\mul\G =\lb \{h_0,h_1\}\in\cH_0\oplus\cH_1: \lb 0,{h_0\choose h_1}\rb \in\G \rb.$$ Using $\mul\G$ we define linear manifolds $\cK_\G'$ and $\cK_G''$ in $\cH_1$ via $$\begin{gathered}
\cK_\G'=P_1\,\dom (\mul\G)=\lb k'\in\cH_1: \lb 0,{k'+h_2\choose h_1} \rb \in\G
\;\;\text{for some}\;\; h_2\in\cH_2 \;\; \text{and}\; \; h_1\in\cH_1 \rb,
\label{3.3}\\
\cK_\G''=\mul (\mul\G)=\lb k''\in\cH_1:\lb 0,{0\choose
k''}\rb\in\G\rb.\label{3.4}\end{gathered}$$ Next introduce the signature operators $$\label{3.5}
J_\gH=\begin{pmatrix} 0 & -i I_\gH \cr iI_\gH & 0\end{pmatrix}:\gH\oplus\gH\to
\gH\oplus\gH, \;\;\;\; J_{01}=\begin{pmatrix} P_2 & -i I_{\cH_1} \cr iP_1 &
0\end{pmatrix}:\cH_0\oplus\cH_1\to \cH_0\oplus\cH_1$$ and denote by $(\gH^2,J_\gH)$ and $(\HH,J_{01})$ the corresponding Krein spaces. Recall [@Shm76] that a linear relation $\BR$ is called an isometric relation from $(\gH^2,J_\gH)$ into $(\HH,J_{01})$ if $$\label{3.6}
(J_\gH \hat f,\hat g)_{\gH^2}=(J_{01} \hat h, \hat x)_{\cH_0\oplus \cH_1},
\quad \{\hat f, \hat 