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Schur analysis in the quaternionic setting: the Fueter regular and the slice regular case. (English) Zbl 1337.47015

Alpay, Daniel (ed.), Operator theory. With 51 figures and 2 tables. In 2 volumes. Basel: Springer (ISBN 978-3-0348-0666-4/print; 978-3-0348-0667-1/ebook; 978-3-0348-0668-8/print+ebook; 978-3-0348-0692-3/online (updated continuously)). Springer Reference, 1745-1786 (2015).
Classically, Schur functions are functions which are holomorphic and contractive on the unit disk. This class is important in operator theory and its applications, such as the theory of linear discrete systems. The authors consider quaternionic analogs of Schur functions. After introductory sections devoted to rational functions, quaternionic Pontryagin spaces, and an analog of the Hardy space, they consider various classes of quaternionic functions of the Schur type and find counterparts of the Krein-Langer factorization and \(J\)-unitary rational functions. Quaternionic moment problems, Fueter series and de Branges-Rovnyak spaces are considered. Possible directions of future work are outlined.
For the entire collection see [Zbl 1325.47001].

MSC:

47A48 Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
30G35 Functions of hypercomplex variables and generalized variables
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47A57 Linear operator methods in interpolation, moment and extension problems
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