\rb
:\{y,f\}\in\tma\rb,\end{gathered}$$ where $\cH_0$ and $\cH_1$ are Hilbert spaces defined by means of $\bH, \cH_b$ and $\hat \cH_b$ and $\G_j': \dom\tma\to \cH_j, \; j\in\{0,1\}$ are linear maps constructed with the aid of $y(a)$ and the operators from . If $\Tmi$ has equal deficiency indices $n_+(\Tmi)=n_-(\Tmi)$, then $\cH_0=\cH_1=H\oplus\hat H\oplus \cH_b$ and the decomposing boundary relation can be written as $$\begin{gathered}
\label{1.19}
\G=\lb \lb {\pi y\choose \pi f}, {\{ y_0(a),\, \tfrac i {\sqrt 2}\ (\hat
y(a)-\hat\G_b y),\,\G_{0b}y \}\choose \{ y_1(a),\, \tfrac 1 {\sqrt 2} (\hat
y(a)+\hat\G_b y),\,-\G_{1b}y \} } \rb :\{y,f\}\in\tma\rb.\end{gathered}$$ In the case of the regular system one can put in $\G_{0b}y=y_0(b),
\; \G_{1b}y=y_1(b)$ and $\hat\G_b y=\hat y(b)$. If $b$ is not regular, then $\G_b y $ can be represented by means of certain limits at the point $b$ associated with the function $y\in\dom\tma$ (for more details see Remark \[rem5.2\] ). Therefore the decomposing boundary relation $\G$ is given by in terms of boundary values of the function $y\in\tma$ at the endpoint $a$ (regular value) and $b$ (singular value), which is of importance in our considerations of canonical systems.

Recall [@KogRof75] that the formal deficiency indices of the system are defined via $$\begin{gathered}
N_\pm=\dim \{y\in\lI:\, J y'(t)-B(t)y(t)=\l\D(t)y(t) \;\;\text{a.e. on}\;\;
\cI\}, \quad 