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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