The Schur algorithm for generalized Schur functions III: J-unitary matrix polynomials on the circle

Abstract

The main result is that for J = 1 0 0 −1 every J-unitary 2 × 2-matrix polynomial on the unit circle is an essentially unique product of elementary J-unitary 2 × 2-matrix polynomials which are either of degree 1 or 2k. This is shown by means of the generalized Schur transformation introduced in [

Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions

“…In [4] it was shown that for J = 1 0 0 −1 aĴ -unitary 2 × 2 matrix polynomial on the unit circle admits an essentially unique factorization into elementaryĴ -unitary matrix polynomials. The essential tool was the theory of reproducing kernel Pontryagin spaces and the Schur algorithm for generalized Schur functions as developed in [2][3][4][5][6][7]12,14,17,19].…”

Generalized Schur Functions and Augmented Schur Parameters

“…In [8] it is applied to solve the problem: When is a formal power series around z = 0 the Taylor expansion of a generalized Schur function. In [1,2,4,6,17] it is studied for its effect on the coisometric and unitary operator realizations of a generalized Schur function, including those whose state spaces are the reproducing kernel Pontryagin spaces with kernels K s (z, w) and D s (z, w); in [3] it is shown to provide an algorithm for the unique factorization of a 2 × 2 matrix polynomial which is J-unitary on T (for the definition, see below) in normalized elementary factors; and, finally, in [5] (see also [12]) it is used in solving a basic interpolation problem for generalized Schur functions.…”

Indefinite Inner Product Spaces, Schur Analysis, and Differential Equations

“…Heinz proposed that Aad, Henk, their student Piet Bruinsma and this editor consider the interpolation problem using Krein's formula and the theory of resolvent matrices for the description of the self-adjoint extensions of a given Hermitian operator (see [22,33]); this led in particular to the publications [4,5]. A bit later, collaboration between Aad, Heinz and DA began (mainly on the Schur algorithm for generalized Schur functions) and lead to seventeen publications, some of them written in collaboration with Thomas Azizov, R. Buursema, Simeon Reich, David Shoikhet, Yuri Shondin, Dan Volok, and Gerald Wanjala; see for instance [2,3,8,9,10]. The encounter, and the subsequent collaboration with Heinz was fascinating on numerous grounds.…”