Factorizations of Characteristic Functions

Let $A=(A_1,\dots ,A_n)$ and $B=(B_1,\dots ,B_n)$ 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 $T = \begin{bmatrix} A& D_{A^*}L D_B\ 0 & B \end{bmatrix}$\mathrm{.}Then$\Theta_T$ coincides with the product of the characteristic function ${\mathrm{\Theta }}_A$ of $A$, the Julia-Halmos matrix corresponding to $L$ and the characteristic function ${\mathrm{\Theta }}_B$ of $B$. More precisely, ${\mathrm{\Theta }}_T$ coincides with $$\left[\begin{array}{cc}{\mathrm{\Theta }}_B& 0\\ 0& I\end{array}\right](I_{\mathrm{\Gamma }}\otimes \left[\begin{array}{cc}{L}^{\ast }& (I-L^{\ast }L{)}^{\frac{1}{2}}\\ (I-L{L}^{\ast }{)}^{\frac{1}{2}}& -L\end{array}\right])\left[\begin{array}{cc}{\mathrm{\Theta }}_A& 0\\ 0& I\end{array}\right],$$ where $\mathrm{\Gamma }$ is the full Fock space. Similar results hold for constrained row contractions.