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

TL;DR

This paper advances the understanding of slice hyperholomorphic functions by establishing new functional calculus results, a Beurling-Lax theorem, and Krein-Langer factorization in the quaternionic setting, which are novel contributions in hyperholomorphic analysis.

Contribution

It introduces new functional calculus properties, proves a Beurling-Lax type theorem, and develops Krein-Langer factorization specifically for slice hyperholomorphic functions, filling gaps in quaternionic analysis.

Findings

Right spectrum and S-spectrum coincide for quaternionic operators

Established a Beurling-Lax type structure theorem in the hyperholomorphic setting

Proved Krein-Langer factorization for slice hyperholomorphic generalized Schur functions

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 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