Factorizations of Characteristic Functions

Abstract

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 $D_{A^{\ast }}=(I-A{A}^{\ast }{)}^{\frac{1}{2}}$ and $D_B=(I-B^{\ast }B{)}^{\frac{1}{2}}$ and ${\Theta }_T$ be the characteristic function of $T=\left[\begin{array}{cc}A& D_{A^{\ast }}L{D}_B\\ 0& B\end{array}\right]$. 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 \[ \begin{bmatrix} Θ_B & 0 \\ 0 & I \end{bmatrix} (I_Γ\otimes \begin{bmatrix} L^* & (I - L^* L)^{\frac{1}{2}} \\ (I - L L^*)^{\frac{1}{2}} & - L \end{bmatrix}) \begin{bmatrix} Θ_A & 0\\ 0& I\end{bmatrix}, \] where $\Gamma$ is the full Fock space. Similar results hold for constrained row contractions.

13 pages