Journal of Operator Theory

Volume 77, Issue 2, Spring 2017 pp. 377-390.

Factorizations of characteristic functions

Authors: Kalpesh J. Haria (1), Amit Maji (2), and Jaydeb Sarkar (3)
Author institution: (1) Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India
(2) Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India
(3) Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India

Summary: Let $A = (A_1, \ldots, A_n)$ and $B = (B_1, \ldots, B_n)$ be row contractions on Hilbert spaces $\mathcal {H}_1$ and $\mathcal{H}_2$, respectively, and $L$ be a contraction from $ \mathcal D_B = \overline{\mbox{ran}} D_B$ to $\mathcal D_{A^*}= \overline{\mbox{ran}} D_{A^*} $ where $D_{B} = (I - B^* B)^{{1}/{2}}$ and $D_{A^*} = (I - A A^*)^{{1}/{2}}$. Let $\Theta_T$ 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 \[ \begin{bmatrix} \Theta_B & 0 \\ 0 & I \end{bmatrix} \left(I_\Gamma \otimes \begin{bmatrix} L^* & (I - L^* - L)^{{1}/{2}} \\ (I - L - L^*)^{{1}/{2}} - & - - L \end{bmatrix}\right) \begin{bmatrix} \Theta_A & 0\\ 0 & I\end{bmatrix}, \] where $\Gamma$ is the full Fock space. Similar results hold for constrained row contractions.

DOI: http://dx.doi.org/10.7900/jot.2016apr20.2132
Keywords: row contractions, Fock space, invariant subspaces, characteristic functions, factorizations of analytic functions, upper triangular block operator matrices

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