  Let $A = (A_1, \ldots, A_n)$ and $B = (B_1, \ldots, B_n)$ be row contractions
on $\mathcal{H}_1$ and $\mathcal{H}_2$, respectively, and $X$ be a row operator
from $\oplus_{i=1}^n \mathcal{H}_2$ to $\mathcal{H}_1$. Let $D_{A^*} = (I - A
A^*)^{\frac{1}{2}}$ and $D_{B} = (I - B^* B)^{\frac{1}{2}}$ and $\Theta_T$ be
the characteristic function of $T = \begin{bmatrix} A& D_{A^*}L D_B\\ 0 & B
\end{bmatrix}$. Then $\Theta_T$ coincides with the product of the
characteristic function $\Theta_A$ of $A$, the Julia-Halmos matrix
corresponding to $L$ and the characteristic function $\Theta_B$ of $B$. More
precisely, $\Theta_T$ coincides with \[ \begin{bmatrix} \Theta_B & 0 \\ 0 & I
\end{bmatrix} (I_\Gamma \otimes \begin{bmatrix} L^* & (I - L^* L)^{\frac{1}{2}}
\\ (I - L L^*)^{\frac{1}{2}} & - L \end{bmatrix}) \begin{bmatrix} \Theta_A &
0\\ 0& I\end{bmatrix}, \] where $\Gamma$ is the full Fock space. Similar
results hold for constrained row contractions.
