Connection between the inverse Schur transformation for generalized Nevanlinna functions with the rational matrix functions of special type
E. N. Andreishcheva Black Sea Higher Naval School
Abstract:
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].
Keywords:
indefinite metrics, Nevanlinna function, Pontryagin space, Schur transformation, reproducing kernel, factorization of rational matrix function.
UDC:
517.58
MSC: 47A58
DOI:
10.37279/1729-3901-2021-20-1-32-47