Factorizations of Characteristic Functions

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 $Θ_T$ be the characteristic function of $T = \begin{bmatrix} A& D_{A^*}L D_B\\ 0 & B \end{bmatrix}$. Then $Θ_T$ coincides with the product of the characteristic function $Θ_A$ of $A$, the Julia-Halmos matrix corresponding to $L$ and the characteristic function $Θ_B$ of $B$. More precisely, $Θ_T$ coincides with $\begin{bmatrix} Θ_B & 0 \\ 0 & I \end{bmatrix} (I_Γ\otimes \begin{bmatrix} L^* & (I - L^* L)^{\frac{1}{2}} \\ (I - L L^*)^{\frac{1}{2}} & - L \end{bmatrix}) \begin{bmatrix} Θ_A & 0\\ 0& I\end{bmatrix},$ where $Γ$ is the full Fock space. Similar results hold for constrained row contractions.

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