Connection between the inverse Schur transformation for generalized Nevanlinna functions with the rational matrix functions of special type

In this paper we consider classical Schur transformation and inverse Schur transformation for generalized Nevanlinna functions. The function $N(z)$ is called a generalized Nevanlinna functions with $\kappa$ nagative squares, if it is meromorphic in $\mathbb{C}^+$ and the kernel
$${{L}_{N}}(z,w)=\frac{N(z)-N{{(w)}^{*}}}{z-{{w}^{*}}}\left( =\frac{(1-N(z)){{J}_{l}}\left( \begin{matrix} 1 \\ -N{{(w)}^{*}} \\ \end{matrix} \right)}{z-{{w}^{*}}} \right)$$
has $\kappa$ negative squares in $hol_+(N)$ — the domain of holomorphy of $N(z)$ in $\mathbb{C}^+$. We denote this class of functions by N$_\kappa$. We often extend the domain of definition of $N(z)$ to the open lower half plane $\mathbb{C}^-$ by setting $N(z^*)=N(z)^*$ with $z\in hol_+(N)$ and by holomorphy to those points of the real axis where this is possible. We study rational $2\times 2$-matrix functions $\Theta(z)$ which have a pole only in the point $z_1^*$, that is their entries are polynomials in $1/(z-z_1^*)$, and which are $J_l$-unitary, that is, satisfy on the real line:
$$ \Theta(z)J_l\Theta(z)^*=J_l,\qquad z\in\mathbb{R},\qquad J_l:=\begin{pmatrix} 0&1\\-1&0 \end{pmatrix}. $$
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 the inverse Schur transformation which plays a main role. As fractional linear transformation, this inverse Schur transformation is according to (4) determined by a $2\times 2$-matrix function $\Theta(z)$. The connection between the Schur transformation and factorization of $2\times 2$-matrix functions is based on the fact that for generalized Nevanlinna functions the matrix functions $\Theta(z)$, corresponding to the inverse Schur transformation, are the elementary $J_l$-unitary factors. The minimal factorization of a given rational $J_l$-unitary $2\times 2$-matrix function $\Theta(z)$ can be obtained by a repeated application of the Schur transformation which we call the Schur algorithm. The reproducing kernel Pontryagin space associated with the kernel $L_N(z,w)$ with $z,w\in hol(N)$ will be denoted by $\mathcal{L}(N)$ and the reproducing kernel Pontryagin space associated with the same kernel but now with $z,w\in hol_+(N)$ will be denoted by $\mathcal{L}_+(N)$. The spaces coincide if there is a real interval where $N$ is holomorphic: the elements of the one are the analytic continuations of the elements of the other. In this paper with a given function $N(z)\in$ N the reproducing kernel Pontryagin space for the kernel $L_N(z,w)$ from (1) is introduced and studied. Theorems 1 and 2 are obtained from more general results from [10], [4] and [15].