Krein–Langer Factorization and Related Topics in the Slice Hyperholomorphic Setting

Abstract

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.