特征函数的因子分解
Factorizations of Characteristic Functions
摘要：
令$A = (A_1, \ldots, A_n)$和$B = (B_1, \ldots, B_n)$分别是$\mathcal{H}_1$和$\mathcal{H}_2$上的行收缩，$X$是从$\oplus_{i=1}^n \mathcal{H}_2$到$\mathcal{H}_1$的行运算符。设$D_{A^*} = (I - A A^*)^{\frac{1}{2}}$、$D_{B} = (I - B^* B)^{\frac{1}{2}}$和$Θ_T$是$T = \begin{bmatrix} A& D_{A^*}L D_B\\ 0 & B \end{bmatrix}$的特征函数。则$Θ_T$与$A$的特征函数$Θ_A$、对应于$L$的Julia-Halmos矩阵和$B$的特征函数$Θ_B$的乘积一致。更准确地说，$Θ_T$与\[ \begin{bmatrix} Θ_B & 0 \\ 0 & I \end{bmatrix} (I_Γ\otimes \begin{bmatrix} L^* & (I - L^* L)^{\frac{1}{2}} \\ (I - L L^*)^{\frac{1}{2}} & - L \end{bmatrix}) \begin{bmatrix} Θ_A & 0\\ 0& I\end{bmatrix}, \]一致，其中$Γ$是完整的Fock空间。类似的结果也适用于约束行收缩。
英文摘要：
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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