Krein-Langer factorization and related topics in the slice   hyperholomorphic setting
  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 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 
Daniel Alpay, Fabrizio Colombo, Irene Sabadini
math.FA math.CV