The Schur–Potapov Algorithm in the General Matrix Case and Its Application to the Matricial Schur Problem

• Published in: Complex analysis and operator theory Vol. 18; no. 5; p. 109
• Main Authors: Dubovoy, Vladimir K., Fritzsche, Bernd, Kirstein, Bernd, Mädler, Conrad, Müller, Karsten
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.07.2024; Springer Nature B.V
• Subjects: Algorithms; Analysis; Mathematics; Mathematics and Statistics; Operator Theory; Sequences; 30E05; Schur–Potapov transform; Schur–Potapov algorithm; 47A57; Matricial Schur problem; Schur parameters
• ISSN: 1661-8254; 1661-8262; 1661-8262
• DOI: 10.1007/s11785-024-01545-x

This paper is a generalization of the topic handled in Bogner et al. (Oper Theory 1(1):55–95, 2007a, Oper Theory 1(2):235–278, 2007b) where the Schur–Potapov algorithm (SP-algorithm) was handled in the context of non-degenerate p × q  Schur sequences and non-degenerate p × q  Schur functions. In particular, the interplay between both types of algorithms was intensively studied there. This was itself a generalization of the classical Schur algorithm (Schur in J Reine Angew Math 148:122–145, 1918) to the non-degenerate matrix case. In treating the matrix case a result due to Potapov (Potapov in Trudy Moskov Mat Obšč 4:125–236, 1955) concerning particular linear fractional transformations of contractive p × q  matrices was used. For this reason, the notation SP-algorithm was already chosen in Dubovoj et al. (Matricial version of the classical Schur problem, volume 129 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1992). We are going to introduce both types of SP-algorithms as well for arbitrary p × q  Schur sequences as for arbitrary p × q  Schur functions. Again we will intensively discuss the interplay between both types of algorithms. Applying the SP-algorithm, a complete treatment of the matricial Schur problem in the most general case is established. A one-step extension problem for finite p × q  Schur sequences is considered. Central p × q  Schur sequences are studied under the view of SP-parameters.