Factorizations of Characteristic Functions

TL;DR

This paper explores the factorization of characteristic functions for certain operator matrices, revealing a structured product involving individual characteristic functions and the Julia-Halmos matrix, with extensions to constrained cases.

Contribution

It provides a new factorization formula for characteristic functions of block operator matrices, connecting them to individual characteristic functions and the Julia-Halmos matrix, including constrained scenarios.

Findings

Characteristic function of the block operator equals a product of simpler functions.

Explicit factorization involving Julia-Halmos matrices is established.

Results extend to constrained row contractions.

Abstract

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.