The problem of factorization of rational matrix functions for the case of a generalized Nevanlinna class

In the present paper, we consider $z_1$ as a fixed point in the open upper half plane $\mathbb{C}^+$. We study rational $2\times2$-matrix functions $\Theta(z)$ which have a pole only in the point $z_1^*$. Their entries are polynomials in $1/(z-z_1^*)$, and which are $J_l$-unitary, that is, satisfy on the real line:
$$ \Theta(z) J_l\Theta(z)^*=J_l,\qquad z\in\mathbb{R},\qquad J_l:=\begin{pmatrix} 0&1\\-1&0 \end{pmatrix}. $$
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 $2\times2$ matrix polynomials, which was proved in [1], and which corresponds to the case that $z_1=\infty$; 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 $z_1=\infty$ 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
$$ J_c=\begin{pmatrix}1&0\\0&-1 \end{pmatrix}. $$
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 $2\times2$ matrix function $\Theta(z)$ with a single-pole this space consists of exactly one Jordan chain of the difference-quotient or backward-shift operator
$$ R_0{f}(z)=\frac{{f}(z)-{f}(0)}{z}, \quad {f}(z)\in P(\Theta). $$
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).