Krein-Langer factorization and related topics in the slice hyperholomorphic setting

Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Research output: Contribution to journalArticlepeer-review

36 Scopus citations


We study various aspects of Schur analysis in the slice hyperholomorphic setting. We present two sets of results: first, we give new results on the functional calculus for slice hyperholomorphic functions. In particular, we introduce and study some properties of the Riesz projectors. Then we prove a Beurling-Lax type theorem, the so-called structure theorem. A crucial fact which allows to prove our results is the fact that the right spectrum of a quaternionic linear operator and the point S-spectrum coincide. Finally, we study the Krein-Langer factorization for slice hyperholomorphic generalized Schur functions. Both the Beurling-Lax type theorem and the Krein-Langer factorization are far-reaching results which have not been proved in the quaternionic setting using notions of hyperholomorphy other than slice hyperholomorphy.

Original languageEnglish
Pages (from-to)843-872
Number of pages30
JournalJournal of Geometric Analysis
Issue number2
StatePublished - 1 Jan 2014


  • Realization
  • Reproducing kernels
  • S-resolvent operators
  • Schur functions
  • Slice hyperholomorphic functions

ASJC Scopus subject areas

  • Geometry and Topology


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