The Schur–Potapov Algorithm for Sequences of Complex p × q Matrices. II

Abstract

This is the second and final part of a paper which appeared in a preceding issue of this journal. Herein the methods developed in the earlier sections of this paper are used first to develop a number of applications. A central theme of this paper is to study the interplay between functions from Sp×q,∞(D) and their sequence of SP-parameters. In particular, we describe how certain summability properties of the SP-parameters are expressed in terms of the associated functions from Sp×q,∞(D). As a byproduct of our investigations on the interplay between the SP-algorithm for p × q Schur functions and the SP-algorithm for sequences of complex p×q matrices we present a new approach to the nondegenerate matricial Schur problem. Our method complies with the basic strategy in Schur's classical paper [138] because it does not make use of any tools outside of the theory of Schur functions and Schur sequences. A closer look at the behaviour of distinguished subclasses of Sp×q(D) with respect to the SP-algorithm enables us to handle the corresponding versions of the matricial Schur problem restricted to these subclasses. Mathematics Subject Classification (2000). Primary 30E05, 47A57.Keywords. Schur function, matricial Schur problem, Schur-Potapov matrix polynomials, Arov-Kreȋn matrix polynomials. IntroductionIn the first part [57] of this paper we have established an appropriate adaptation of the Schur-Potapov algorithm for functions belonging to the matricial Schur class S p×q (D) to infinite sequences of complex p × q matrices. Using the fact that the Taylor coefficients sequences of functions from S p×q (D) are infinite p × q Schur sequences we mainly focused on the subclass S p×q,∞ (D) of S p×q (D) which consists of all p × q Schur functions for which the corresponding Taylor coefficient sequences are nondegenerate p × q Schur sequences. Thus, an one-to-one correspondence between infinite nondegenerate p × q Schur sequences and the set of all

The Schur–Potapov Algorithm for Sequences of Complex p × q Matrices. I

“…In the second part [3] of this paper we will give a number of applications of the above mentioned Schur-Potapov parametrization of a nondegenerate p × q Schur sequence and of a S p×q,∞ (D)-function, respectively. First, we will provide a new approach to the nondegenerate matricial Schur problem.…”

“…Note that all references to propositions within this paper carrying numbers larger than 8 refer to the second part [3] of this paper. We study distinguished subclasses of matrix-valued Schur functions from the viewpoint of their Taylor coefficients.…”