# The Problem of Factorization of Rational Matrix Functions for the Case of a Generalized Nevanlinna Class

**Андреищева Е. Н. The Problem of Factorization of Rational Matrix Functions for the Case of a Generalized Nevanlinna Class // Taurida Journal of Computer Science Theory and Mathematics, – 2019. – T.18. – №4. – P. 7-26**

https://doi.org/10.37279/1729-3901-2019-18-4-7-26

https://doi.org/10.37279/1729-3901-2019-18-4-7-26

In the present paper, we consider $z_1$ as a fixed point in the open upper half

plane $\mathbb{C} ^+$. We study rational 2 × 2- matrix functions $\Theta(z)%$ which have a pole only in the point

$z^1_*$. Their entries are polynomials in $\frac{1}{(z-z^1_*)}$, and which are $J_l$ -unitary, that is, satisfy on the

real line:

$\Theta(z)J_l\Theta(z)^* = J_l$, $z \in \mathbb{R}$, $\mathbf{J_l} := \left(

\begin{array}{cc}

0 & 1 \\

-1 & 0 \\

\end{array} \right)$.

The main results are the existence and essential uniqueness of a minimal factorization of

such a matrix function into elementary factors which have the same properties, and the analytic

description of the elementary factors, see theorems 4 and 5. The assumption (1) on $\Theta(z)$ does

not imply that $\Theta(z)$ is $J_l$-inner, which would mean that the kernel

$K_{\Theta(z,w)} = \frac{J_l - \Theta(z)J_l\Theta(w)^*}{2\pi(z - w^)*}$

is positive. However, due to our assumption that $\Theta(z)$ is a polynomial in $1/(z − z^∗_1)$ this kernel

has a finite number of negative (and positive) squares. This indefinite setting implies that

the elementary factors can become more complicated than in the positive definite case, see

formula (22).

The results of the present paper can be viewed as analogs of the results obtained in [1]. There

the extension of the classical Schur transformation to generalized Schur functions as defined and

studied for example, in the papers [3], [4], [5] and [6], played an important role.

In this paper, we use a corresponding for Nevanlinna functions and generalized Neanlinna

functions, which we also call Schur transformation and which to our knowledge, appears here

for the first time. The factorization result of this paper is also an analog of the factorization

for $J_l$ -unitary 2 × 2 matrix polynomials, which was proved in [1], and which corresponds to the

case that $z_1 = ∞$; there the role of the Schur transformation was played by a generalization of a lemma of N. I. Akhiezer for Nevanlinna functions to generalized Nevanlinna functions which

tend to zero if $z_1$ = ∞ along the imaginary axis. A corresponding result for a real point $z_1$ will

be considered elsewhere; it is the analog to the case of rational matrix function with a pole on

the unit circle which is $J_l$-unitary outside the pole, where

$\mathbf{J_c} := \left(

\begin{array}{cc}

1 & 0 \\

0 & -1 \\

\end{array} \right)$.

Similar to [1], [3], and [5] , the main tool for proving the factorization in the present paper

is a result on finite-dimensional reproducing kernel Pontryagin spaces $P(\Theta)$ with reproducing

kernel (2) see theorem 2. It states that for a rational $J_l$-unitary 2 × 2 matrix function $\Theta(z)$

with a single-pole this space consists of exactly one Jordan chain of the difference-quotient or

backward-shift operator

$R_0\boldsymbol{f}(z) = \frac {\boldsymbol{f}(z) - \boldsymbol{f} (0)} {z}$, $\boldsymbol{f}(z) \in P(\Theta)$.

Theorem 2 is obtained from more general factorization and realization results from [9], [5]

and [10], see Theorem 1. Note that $P(\Theta)$ is the state space for an underlying minimal realization

of $\Theta(z)$ , which is given in formula (12).

**Keywords**:* indefinite metrics, Nevanlinna function, Pontryagin space, Schur transformation,
reproducing kernel, factorization of rational matrix function.
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